The relativistic Euler equations: Remarkable null structures and

arXiv:1809.06204v1 [math.AP] 17 Sep 2018

THE RELATIVISTIC EULER EQUATIONS: REMARKABLE NULL STRUCTURES AND REGULARITY PROPERTIES MARCELO M. DISCONZI∗# AND JARED SPECK∗∗† Abstract. We derive a new formulation of the relativistic Euler equations that exhibits remarkable properties. This new formulation consists of a coupled system of geometric wave, transport, and elliptic equations, sourced by nonlinearities that are null forms relative to the acoustical metric. Our new formulation is well-suited for various applications, in particular for the study of stable shock formation, as it is surveyed in the paper. Moreover, using the new formulation presented here, we establish a local well-posedness result showing that the vorticity and the entropy of the fluid are one degree more differentiable compared to the regularity guaranteed by standard estimates (assuming that the initial data enjoy the extra differentiability). This gain in regularity is essential for the study of shock formation without symmetry assumptions. Our results hold for an arbitrary equation of state, not necessarily of barotropic type. Keywords: Relativistic Euler; Null structure; Acoustical metric; Shock formation. Mathematics Subject Classification (2010) Primary: 35Q75; Secondary: 35L10, 35Q35, 35L67.

September 18, 2018 Contents 1. Introduction 1.1. Rough statement of the new formulation 1.2. Connections to the study of shock waves 2. Setup and background 2.1. Notation and conventions 2.2. First-order formulation of the relativistic Euler equations 3. The new formulation of the relativistic Euler equations 4. Preliminary identities 5. Wave equations 5.1. Covariant wave operator 5.2. Covariant wave equation for h

3 5 7 16 16 19 20 23 27 27 28

# MMD gratefully acknowledges support from NSF grant # 1812826, from a Sloan Research Fellowship provided by the Alfred P. Sloan foundation, and from a Discovery grant administered by Vanderbilt University. † JS gratefully acknowledges support from NSF grant # 1162211, from NSF CAREER grant # 1454419, from a Sloan Research Fellowship provided by the Alfred P. Sloan foundation, and from a Solomon Buchsbaum grant administered by the Massachusetts Institute of Technology. ∗ Vanderbilt University, Nashville, TN, USA. [email protected]. ∗∗ Vanderbilt University, Nashville, TN, USA & Massachusetts Institute of Technology, Cambridge, MA, USA. [email protected].

1

2

Relativistic Euler 5.3. Covariant wave equation for the rectangular four-velocity components 6. Transport equations for the entropy gradient and the modified divergence of the entropy 7. Transport equation for the vorticity 8. The transport-div-curl system for the vorticity 8.1. Preliminary identities 8.2. The div-curl-transport system 9. Well-posedness 9.1. Notation, norms, and basic tools from analysis 9.2. The regime of hyperbolicity 9.3. Standard local well-posedness 9.4. A new inverse Riemannian metric and the classification of various combinations of solution variables 9.5. Elliptic estimates and the corresponding energies 9.6. Energies for the wave equations via the vectorfield multiplier method 9.7. Proof of Theorem 9.2 References

29 32 35 38 38 49 56 57 61 61 62 65 70 76 81

M. Disconzi, J. Speck

3

1. Introduction The relativistic Euler equations are the most well-studied PDE system in relativistic fluid mechanics. In particular, they play a prominent role in cosmology, where they are often used to model the evolution of the average matter-energy content of the universe; see, for example, Weinberg’s well-known monograph [36] for an account of the role that the relativistic Euler equations play in the standard model of cosmology. The equations are also widely used in astrophysics and high-energy nuclear physics, as is described, for example, in [26]. Our main result in this article is our derivation of a new formulation of the relativistic Euler equations that reveals remarkable new regularity and null structures that are not visible relative to the standard first-order formulation. The new formulation is available for an arbitrary equation of state, not necessarily of barotropic1 type. Below we will describe potential applications that we anticipate will be the subject of future works. We mention already that our new formulation of the equations provides a viable framework for the rigorous mathematical study of stable shock formation without symmetry assumptions in solutions to the relativistic Euler equations; for reasons to be explained, standard first-order formulations is not adequate for tracking the behavior of solutions (without symmetry assumptions) all the way to the formation of a shock or for extending the solution (uniquely, in a weak sense tied to suitable selection criteria) past the first singularity. We derive the new formulation by differentiating a standard first-order formulation with various geometric differential operators and observing remarkable cancellations.2 The calculations are rather involved and make up the bulk of the article. We have carefully divided them into manageable pieces; see Sects. 4-8. Readers can jump ahead to Theorem 1.1 for a rough statement of the equations and Theorem 3.1 for the precise version. As we alluded to above, the relativistic Euler equations are typically formulated as a firstorder quasilinear hyperbolic PDE system. In our new formulation, the equations take the form of a system of covariant wave equations coupled to transport equations and to two transport-div-curl systems. The new formulation is well-suited for various applications in ways that first-order formulations are not. In particular, the equations of Theorem 3.1 can be used to prove that the vorticity and entropy are one degree more differentiable than one might naively expect (assuming that the gain in differentiability is present in the initial data). This gain in differentiability is crucial for the rigorous mathematical study of some fundamental phenomena that occur in fluid dynamics. In particular, this gain, as well as other structural aspects of the new formulation, is essential for the study of shock waves (without symmetry assumptions) in relativistic fluid mechanics; see Subsect. 1.2 for further discussion. Although the gain in differentiability for the vorticity had previously been observed relative to Lagrangian coordinates [12, 14], Lagrangian coordinates are inadequate, for example, for the study of the formation of shock singularities because they are not adapted to the acoustic characteristics, whose intersection corresponds to a shock. Hence, it is of fundamental importance that our new formulation allows one to prove the gain in differentiability relative to 1Barotropic

equations of state are such that the pressure is a function of the density alone. observing many of the cancellations, the precise numerical coefficients in the equations are important; roughly, these cancellations lead to the presence of the null form structures described below. However, for most applications, the overall coefficient of the null forms is not important; what matters is that the cancellations lead to null forms. 2In

4

Relativistic Euler

arbitrary vectorfield differential operators (with suitably regular coefficients). In this vein, we also mention the works [8–10] on the non-relativistic compressible Euler equations, in which a gain in differentiability for the vorticity was shown relative to Lagrangian coordinates, and the first author’s joint work [11], in which elliptic estimates were used to show that for the non-relativistic barotropic compressible Euler equations, it is possible to gain one derivative on the density relative to the velocity (again, assuming that the gain is present in the initial data). We also highlight the following key advantage of our new formulation: It dramatically enlarges the set of energy estimate techniques that can be applied to the study of the relativistic Euler equations. More precisely, the new formulation partially decouples the “wave parts” and “transport parts” of the system and unlocks our ability to apply the full power of the commutator and multiplier vectorfield methods to the study of the wave part; see Subsect. 9.6 for further discussion. For applications to shock waves, it is fundamentally important that one is able to use the full scope of the vectorfield method on the wave part of the system; see the introduction of [22] for a discussion of this issue in the related context of the non-relativistic barotropic compressible Euler equations with vorticity. In particular, our new formulation of the equations allows one to derive a coercive energy estimate for the wave part of the system for any multiplier vectorfield that is causal relative to the acoustical metric g of Def. 2.1 and on any hypersurface that is null or spacelike relative to g; see Subsubsect. 9.6.1 for further discussion. In contrast, for first-order hyperbolic systems (a special case of which is the relativistic Euler equations) without additional structure, there is, up to scalar function multiple, only one3 available energy estimate on each causal or spacelike hypersurface. Our second result in this article is that we provide a proof of local well-posedness for the relativistic Euler equations that relies on the new formulation; see Theorem 9.2. The new feature of Theorem 9.2 compared to standard proofs of local well-posedness for the relativistic Euler equations is that it provides the aforementioned gain in differentiability for the vorticity and entropy. For convenience, throughout the article, we restrict our attention to the special relativistic Euler equations, that is, the relativistic Euler equations on the Minkowski spacetime background (R1+3 , η), where η is the Minkowski metric. However, using arguments similar to the ones given in the present article, our results could be extended to apply to the relativistic Euler equations on a general Lorentzian manifold; such an extension could be useful, 3Here

we further explain how standard first-order formulations of the relativistic Euler equations limit the available energy estimates. In deriving energy estimates for the relativistic Euler equations in their standard first-order form, one is effectively controlling the wave and transport parts of the system at the same time, and, up to a scalar function multiple, there is only one energy estimate available for transport equations. To see this limitation in a more concrete fashion, one can rewrite the relativistic Euler equations in first-order symmetric hyperbolic form as Aα (V)∂α V = 0, where V is the array of solution variables and the Aα are symmetric matrices with A0 positive definite; see, for example, [25] for a symmetric hyperbolic formulation of the general relativistic Euler equations in the barotropic case. The standard energy estimate for symmetric hyperbolic systems is obtained by taking the Euclidean dot product of both sides of the equation with V and then integrating by parts over an appropriate spacetime domain foliated by spacelike hypersurfaces. The key point is that for systems without additional structure, no other energy estimate is known, aside from rescaling the standard one by a scalar function.


5

for example, in applications to fluid mechanics in the setting of general relativity. For use throughout the article, we fix a standard rectangular coordinate system {xα }α=0,1,2,3 , relative to which ηαβ := diag(−1, 1, 1, 1). See Subsect. 2.1 for our index conventions. We clarify that in Sect. 9, we prove local well-posedness for the relativistic Euler equations (including the aforementioned gain in regularity for the vorticity and entropy) on the flat spacetime background (R × T3 , η), where the “spatial manifold” T3 is the three-dimensional torus and we recycle the notation in the sense that {xα }α=0,1,2,3 denotes standard coordinates on R × T3 (see Subsubsect. 9.1.1 for further discussion) and η again denotes the Minkowski metric; the compactness of T3 allows for a simplified approach to some technical aspects of the argument while allowing us to illustrate the ideas needed to exhibit the gain in regularity for the vorticity and entropy. Our work here can be viewed as extensions of the second author’s previous joint work [21], in which the authors derived a similar formulation of the non-relativistic compressible Euler equations under an arbitrary barotropic equation of state, as well as the second author’s work [31], which extended the results of [21] to a general equation of state. However, since the geo-analytic structures revealed by [21, 31] are rather delicate (that is, quite unstable under perturbations of the equations), it is far from obvious that similar results hold in the relativistic case. We also stress that compared to the non-relativistic case, our work here is substantially more intricate in that it extensively relies on decompositions of various spacetime tensors into tensors that are parallel to the four-velocity u and tensors that are η-orthogonal to u. In particular, we heavily exploit that many of the tensorfields appearing in our analysis exhibit improved regularity under u-directional differentiation or contraction against u. 1.1. Rough statement of the new formulation. In this subsection, we provide a schematic version of our new formulation of the equations; in Subsect. 1.2, we will refer to the schematic version when describing potential applications. In any formulation of the relativistic Euler equations, there is great freedom in choosing state-space variables (i.e., the fundamental unknowns in the system). In this article, as state space variables, we use the logarithmic enthalpy h, the entropy s, and the four-velocity u, which is a future-directed timelike vectorfield normalized by ηαβ uα uβ = −1. Other fluid quantities such as the density ρ, pressure p, etc. will also play a role in our discussion, but these quantities can be viewed as functions of the state space variables; see see Sect. 2 for detailed descriptions of all of these variables as well as the first-order formulation of the equations that forms the starting point for our ensuing analysis. As we mentioned earlier, our new formulation comprises a system of covariant wave equations coupled to transport equations and to two transport-div-curl systems. Roughly, the wave equations correspond to the propagation of sound waves, while the transport equations correspond to the transporting of vorticity and entropy along the integral curves of u. The transport-div-curl systems are needed to control the top-order derivatives of the vorticity and the entropy and to exhibit the aforementioned gain in differentiability. In addition to the state space variables h, s, and u, our formulation also involves a collection of auxiliary4 fluid variables, including the entropy gradient one-form Sα := ∂α s and the vorticity ̟ α , which is a vectorfield that is η-orthogonal to u (see Def. 2.3). Among these auxiliary variables, of 4By

“auxiliary”, we mean that they are determined by h, s, and u.

6

Relativistic Euler

crucial importance for our work is that we have identified new combinations of fluid variables that solve transport equations with unexpectedly good structure, which allow for a proof that the combinations exhibit a gain in regularity compared to what can be inferred from a standard first-order formulation of the equations. We refer to these special combinations as “modified variables”, and throughout, we denote them by C α and D; see Def. 2.8. The remaining discussion in this subsection relies on some schematic notation and refers to some geometric objects that are not precisely defined until later in the article: • The notation “∼” below means that we are only highlighting the maximum number of derivatives of the state-space variables that the auxiliary variables depend on. We note, however, that in practice, the precise structure of many of the terms that we encounter is important for observing the cancellations that lie behind our main results. • “∂” schematically denotes the spacetime gradient with respect to the rectangular coordinates, and “∂ 2 ” schematically denotes two differentiations with respect to the rectangular coordinates. • g = g(h, s, u) denotes the acoustical metric, which is Lorentzian (see Def. 2.1). • ̟ ∼ ∂u + ∂h is the vorticity vectorfield (see Def. 2.3). • Sα := ∂α s is the entropy gradient one-form. • C α ∼ ∂ 2 u + ∂ 2 h is a modified version of the vorticity of ̟, that is, the vorticity of the vorticity (see Def. 2.8). • D ∼ ∂ 2 s is a modified version of ∂α S α (see Def. 2.8). • Q(∂T1 , · · · , ∂Tm ) denotes special terms that are quadratic in the tensorfields ∂T1 , · · · , ∂Tm . More precisely, the Q(∂T1 , · · · , ∂Tm ) are linear combinations of the standard null forms relative to g; see Def. 1.1 for the definitions of the standard null forms relative to g and Subsubsect. 1.2.2 for a discussion of the significance that the special structure of these null forms plays in the context of the study of shock waves. • L(∂T1 , · · · , ∂Tm ) denotes linear combinations of terms that are at most linear in ∂T1 , · · · , ∂Tm ; see Subsubsect. 1.2.2 for a discussion of the significance of the linear dependence in the context of the study of shock waves. Before schematically stating our main theorem, we first provide the definitions of the standard null forms relative to g. Definition 1.1 (Standard null forms relative to g). We define the standard null forms relative to g (standard “g-null forms for short”) as follows (where 0 ≤ µ < ν ≤ 3): Q(g) (∂φ, ∂ψ) := (g −1 )αβ ∂α φ∂β ψ,

Qµν (∂φ, ∂ψ) := (∂µ φ)∂ν ψ − (∂ν φ)∂µ ψ.

(1)

We now present the schematic version of our main theorem; see Theorem 3.1 for the precise statements. Theorem 1.1 (New formulation of the relativistic Euler equation (schematic version)). Assume that (h, s, uα ) is a C 2 solution to the (first-order) relativistic Euler equations (24)(26) + (29). Then h and uα also verify the following covariant5 wave equations, where the 5Relative

p 1 ∂α |detg|

p to arbitrary coordinates, |detg|(g −1 )αβ ∂β f .

for

scalar

functions

f,

we

have

g f

=


7

schematic notation “≃” below means that we have ignored the coefficients of the inhomogeneous terms and also harmless (from the point of view of applications to shock waves) lower-order terms, which are allowed to depend on h, s, u, S, and ̟ (but not their derivatives): g h ≃ D + Q(∂h, ∂u) + L(∂h), α

α

g u ≃ C + Q(∂h, ∂u) + L(∂h, ∂u).

(2a) (2b)

In addition, s, S α , and ̟ α verify the following transport equations: uκ ∂κ s = 0,

(3a)

κ

α

(3b)

κ

α

(3c)

u ∂κ S ≃ L(∂u),

u ∂κ ̟ ≃ L(∂h, ∂u).

Moreover, S α verifies the following div-curl-transport system: uκ ∂κ D ≃ C + Q(∂S, ∂h, ∂u) + L(∂h, ∂u),

vortα (S) = 0,

(4a) (4b)

where the vorticity operator vort is defined in Def. 2.2. Finally, ̟ α verifies the following div-curl-transport system: ∂κ ̟ κ ≃ L(∂h),

uκ ∂κ C α ≃ C + D + Q(∂S, ∂̟, ∂h, ∂u) + L(∂S, ∂̟, ∂h, ∂u).

(5a) (5b)

1.2. Connections to the study of shock waves. As we have mentioned, the relativistic Euler equations are an example of a quasilinear hyperbolic PDE system. A central feature of the study of such systems is that initially smooth solutions can form shock singularities in finite time. By a “shock,” we roughly mean that one of the solution’s partial derivatives with respect to the standard coordinates blows up in finite time while the solution itself remains bounded. In the last decade, for interesting classes of quasilinear hyperbolic PDEs in multiple spatial dimensions, there has been dramatic progress [3, 7, 22–24, 30, 32–34] on our understanding of the formation of shocks as well as our understanding of the subsequent behavior of solutions past their singularities [4, 6] (where the equations are verified in a weak sense past singularities). The works cited above have roots in the work of John [15] on singularity formation for quasilinear wave equations in one spatial dimension as well as Alinhac’s foundational works [1, 2], which were the first to provide a constructive description of shock formation for quasilinear wave equations in more than one spatial dimension without symmetry assumptions. More precisely, Alinhac’s approach allowed him to follow the solution precisely to the time of first blowup, but not further. His work yielded sharp information about the first singularity, but only for a subset of “non-degenerate” initial data such that the solution’s first singularity is isolated in the constant-time hypersurface of first blowup; in particular, his proof did not apply to spherically symmetric initial data, where the “first” singularity typically corresponds to blowup on a sphere. Subsequently, Christodoulou [3] proved a breakthrough result on the formation of shocks for solutions to the relativistic Euler equations in irrotational (that is, vorticity-free) regions of spacetime. More precisely, for the family of quasilinear wave equations that arise in the

8

Relativistic Euler

study of the irrotational and isentropic relativistic Euler equations,6 Christodoulou gave a complete description of the maximal development of an open set (without symmetry assumptions) of initial data and showed in particular that an open subset of these data lead to shock-forming7 solutions. Moreover, he gave a precise geometric description of the set of spacetime points where blowup occurs by showing that the singularity formation is exactly characterized by the intersection of the characteristics. In practice, he accomplished this by constructing an acoustical eikonal function U, whose level sets are acoustic characteristics (see Subsubsect. 1.2.1 for further discussion), and then constructing an initially positive geometric quantity µ ∼ 1/∂U known as the inverse foliation density of the characteristics, such that µ → 0 corresponds to the intersection of the characteristics and the blowup of the solution’s derivatives. Analytically, µ plays the role of a weight that appears throughout the work [3], and the main theme of the proof is to control the solution all the way up to the region where µ = 0. We stress that [3] was the first work that provided sharp information about the boundary of the maximal development in more than one spatial dimension in the context of shock formation. Roughly, the maximal development is the largest possible classical solution that is uniquely determined by the initial data; see [27, 37] for further discussion. To prove his results, Christodoulou relied on a novel formulation of the relativistic Euler equations. However, since he studied the shock formation only in irrotational and isentropic regions, he was able to introduce a potential function Φ, and his new formulation of the equations was drastically simpler than the equations of Theorem 1.1. In fact, the equations are exactly the wave equation system eg ∂α Φ = 0 (with α = 0, 1, 2, 3), where e g is an appropriate scalar function multiple of the acoustical metric g and e g=e g (∂Φ). In particular, Christodoulou was able to avoid deriving/relying on the div-curl-transport equations from Theorem 1.1, and he therefore did not need to derive elliptic estimates for the fluid variables. In total, the potential formulation leads to dramatic simplifications compared to the equations of Theorem 1.1, especially in the context of the study of shock waves; it seems quite miraculous that the equations of Theorem 1.1 have structures that are compatible with extending Christodoulou’s results away from the irrotational and isentropic case (see below for further discussion). Although the sharp information that Christodoulou derived about the maximal development is of interest in itself, it is also an essential ingredient for setting up the shock development problem. The shock development problem, which was recently partially8 solved in the breakthrough work [4] (see also the precursor work [6] in spherical symmetry), is the problem of constructing the shock hypersurface of discontinuity (across which the solution 6For

solutions with vanishing vorticity and constant entropy, one can introduce a potential function Φ and reformulate the relativistic Euler equations as a quasilinear wave equation in Φ. 7 One of the key results of [3] is conditional: for small data, the only possible singularities in the solution are shocks driven by the intersection of the characteristics. Here “small” means a small perturbation of the data of a constant fluid state, where the size of the perturbation is measured relative to a high-order Sobolev norm. Another result of [3] is that there is an open subset of small data, perhaps smaller than the aforementioned set, such that the characteristics do in fact intersect in finite time. The results of [3] leave open the possibility that there might exist some non-trivial small global solutions. 8In [4], Christodoulou solved the “restricted” shock development problem, in which he ignored the jump in entropy and vorticity across the shock hypersurface.


9

jumps) as well as constructing a unique weak solution in a neighborhood of the shock hypersurface (uniqueness is enforced by selection criteria that are equivalent to the well-known Rankine–Hugoniot conditions). Christodoulou’s description of the maximal development provided substantial new information that was not available under Alinhac’s approach; as we mentioned above, due to some technical limitations tied to his reliance on Nash–Moser estimates, Alinhac was able to follow the solution only to the constant-time hypersurface of first blowup. In contrast, by exploiting some delicate tensorial regularity properties of eikonal functions for wave equations (see below for more details), Christodoulou was able to avoid Nash–Moser estimates; this was a key ingredient in his following the solution to the boundary of the maximal development. Readers can consult [13] for a survey of some of these works, with a focus on the geometric and analytic techniques that lie behind the proofs. We now aim to connect the works mentioned above to the new formulation of the relativistic Euler equations that we provide in this paper. To this end, for the equations in the works mentioned above, we first highlight the main structural features that allowed the proofs to go through. Specifically, the works [3, 7, 22–24, 30, 32–34] crucially relied on the following ingredients: (1) (Nonlinear geometric optics). The authors relied on geometric decompositions adapted to the characteristic hypersurfaces (also known as “characteristics” or “null hypersurfaces” in the context of wave equations) corresponding to the solution variable whose derivatives blow up. This was implemented with the help of an eikonal function U, whose level sets are characteristics. The eikonal function is a solution to the eikonal equation, which is a fully nonlinear transport equation that is coupled to the solution in the sense that the coefficients of the eikonal equation depend on the solution. Moreover, the authors showed that the intersection of the characteristics corresponds to the formation of a singularity in the derivatives of the eikonal function and in the derivatives of the solution. (2) (Nonlinear null structure). The authors found a formulation of the equations exhibiting remarkable null structures. This allows for sharp, fully nonlinear decompositions along characteristic hypersurfaces that reveal exactly which directional derivatives blow up and that precisely identify the terms driving the blowup (which are typically of Riccati-type, i.e., in analogy with the nonlinearities in the ODE y˙ = y 2 ). (3) (Regularity properties and singular high-order energy estimates). The authors’ formulation allows one to derive sufficient L2 -type Sobolev regularity for all unknowns in the problem, including the eikonal function, whose regularity properties are tied to the regularity of the solution through the dependence of the coefficients of the eikonal equation on the solution. In particular, to close these estimates, the authors had to show that various solution variables are one degree more differentiable compared to the degree of differentiability guaranteed by standard energy estimates. (4) (Structures amenable to commutations with geometric vectorfields). The authors’ formulation is such that one can commute all of the equations with geometric vectorfields constructed out of the eikonal function U, generating only controllable commutator error terms. By “controllable,” we mean both from the point of view of regularity and from the point of view of the strength of their singular nature. In the works [22, 30, 32, 33] that treat systems with multiple characteristic speeds, these are

10

Relativistic Euler

particularly delicate tasks that are quite sensitive to the structure of the equations; one key reason behind their delicate nature is that the eikonal function (and thus the geometric vectorfields constructed from it) can be fully adapted only to “one speed”, that is, to the characteristics whose intersection correspond to the singularity. In the remainder of this subsection, we explain why our new formulation of the relativistic Euler equations has all four of the features listed above and is therefore well-suited for studying shocks without symmetry assumptions. Readers can consult the works [21, 31] for related but extended discussion in the case of the non-relativistic compressible Euler equations. 1.2.1. Nonlinear geometric optics and geometric coordinates. First, to implement nonlinear geometric optics, one can construct an eikonal function. In the context of the relativistic Euler equations, one would construct an eikonal function U adapted to the acoustic characteristics, that is, a solution to the eikonal equation (g −1 )αβ ∂α U∂β U = 0,

(6)

supplemented by appropriate initial conditions, where g = g(h, s, u) is the acoustical metric (see Def. 2.1). Note that U is adapted to the “wave part” of the system and not the transport part. In the context of the relativistic Euler equations, this is reasonable in the sense that the transport part corresponds to the evolution of vorticity and entropy, and there are no known blowup results for these quantities, even in one spatial dimension.9 Therefore, in introducing such an eikonal function, one is essentially placing a bet that the singularity formation is tied to the wave part of the system and not the transport part. We also stress that introducing an eikonal function is essentially the same as relying on the method of characteristics. However, in more than one spatial dimension, the method of characteristics must be supplemented with an exceptionally technical ingredient that we further describe below: energy estimates that hold all the way up to the shock. The first instance of an eikonal function being used to study the global properties of solutions to a hyperbolic PDE is found in the celebrated work of Christodoulou–Klainerman [5] on the stability of the Minkowski spacetime as a solution to the Einstein-vacuum equations. Alinhac’s aforementioned works [1, 2] were the first instance in which an eikonal function was used to study a non-trivial set of solutions (without symmetry assumptions) to a quasilinear wave equation all the way up to the first singularity. Eikonal functions also played a fundamental role in all of the other shock formation results mentioned above. They have also played a role in other contexts, such as low-regularity local well-posedness for quasilinear wave equations [19, 20, 28, 35]. The eikonal equation is a fully nonlinear hyperbolic PDE that is coupled to the PDE system of interest (in our case the relativistic Euler equations) through its coefficients (in our case through the acoustical metric, since g = g(h, s, u)). As we mentioned earlier, in the case of the relativistic Euler equations, the level sets of U are characteristics for the “wave part” of the system. Following Alinhac [1, 2] and Christodoulou [3], in order to study the formation of shocks, one completes U to a geometric coordinate system (t, U, ϑ1 , ϑ2 ) 9In

one spatial dimension, the vorticity must vanish, but the entropy can be dynamic.

(7)


11

on spacetime, where t = x0 is the Minkowski time coordinate and the ϑA are solutions to the transport equation (g −1 )αβ ∂α U∂β ϑA = 0 supplemented by appropriate initial conditions on the initial constant-time hypersurface Σ0 . Note that (t, ϑ1 , ϑ2 ) can be viewed as a coordinate system along each characteristic hypersurface {U = const}. 1.2.2. Nonlinear null structure. We now aim to explain the role that the nonlinear null structure of the equations played in the works [3, 7, 22–24, 30, 32–34] and to explain why the equations of Theorem 1.1 enjoy the same good structures. In total, one could say that the equations of Theorem 1.1 have been geometrically decomposed into terms that are capable of generating shocks and “harmless” terms, whose nonlinear structure is such that they do not interfere with the shock formation mechanisms. To flesh out these notions, we first provide some background material. In the works cited above, the main idea behind proving shock formation is to study the solution relative to the geometric coordinates (7) and to show that in fact, the solution remains rather smooth in these coordinates, all the way up to the shock. This approach allows one to transform the problem of shock formation into a more traditional one in which one tries to derive long-time estimates for the solution relative to the geometric coordinates. One then recovers the blowup of the solution’s derivatives with respect to the original coordinates by showing that the geometric coordinates degenerate in a precise fashion relative to the standard rectangular coordinates as the shock forms; the degeneration is exactly tied to the vanishing of the inverse foliation density µ that we mentioned earlier. Although the above description might seem compellingly simple, as we explain in Subsubsect. 1.2.3, in implementing this approach, one encounters severe analytical difficulties. We now highlight another key aspect of the proofs in the works cited above: showing that Euclidean-unit-length derivatives of the solution in directions tangent to the characteristics remain bounded all the way up to the shock. It turns out that in terms of the geometric ∂ coordinates (7), this is equivalent to showing that the ∂t and ∂ϑ∂A derivatives of the solution remain bounded all the way up to the shock. Put differently, relative to the standard coordinates, the following holds: The singularity occurs only in derivatives that are transversal to the characteristics. In the works cited above, to prove all of these facts, the authors had to control various inhomogeneous error terms by showing that they enjoy a good nonlinear null structure relative to the wave characteristics. A key conclusion of the present article is that the derivative-quadratic inhomogeneous terms in the equations of Theorem 1.1 enjoy the same good structure (which we further describe just below). In fact, all terms on the RHSs of all equations of Theorem 1.1 are harmless in that they do not drive the Riccati-type blowup that lies behind shock formation. Consequently, the equations of Theorem 1.1 pinpoint the dangerous nonlinear terms in the relativistic Euler equations: The terms capable of driving shock formation are of Riccati-type and are hidden in the covariant wave operator terms on LHSs (2a)-(2b). These terms become visible only when the covariant wave operator terms are expanded relative to the standard coordinates.

12

Relativistic Euler

In view of the above remarks, one might wonder why it is important to “hide” the dangerous terms in the covariant wave operator. The answer is that there is an advanced framework for constructing geometric vectorfields adapted to wave equations, and the framework is tailored to covariant wave operators.10 As we explain later in this subsection, this geometric framework seems to be essential in more than one spatial dimension,11 when one is forced to commute the wave equations with suitable vectorfields and to derive energy estimates. We now further describe the good structure found in the terms on the RHS of the equations of Theorem 1.1. The good nonlinear “null structure” is found precisely in the (quadratic) null form terms Q appearing on the RHS of the equations of Theorem 1.1. More precisely, these Q are null forms relative to the acoustical metric g, which means that they are linear combinations (with coefficients that are allowed to depend on the solution variables – but not their derivatives) of the standard null forms relative to g (see Def. 1.1). The key property of null forms relative to g is that given any hypersurface H that is characteristic relative to g (e.g., any level set of any eikonal function U that solves equation (6)), we have the following well-known schematic decomposition: Q(∂φ, ∂ψ) = T φ · ∂ψ + T ψ · ∂φ,

(8)

where T denotes a differentiation in a direction tangent to H and ∂ denotes a generic directional derivative; see, for example, [21] for a standard proof of (8). Equation (8) implies that even though Q is quadratic, it never involves two differentiations in directions transversal to any characteristic. Since, in all known proofs, it is precisely the transversal derivatives that blow up when a shock forms (since the Riccati-type terms that drive the blowup are precisely quadratic in the transversal derivatives), we see that null forms are linear in the tensorial component of the solution that blows up. This can be viewed as the absence of the worst possible combinations of terms in Q. In terms of the geometric coordinates (7), ∂ ∂ null forms do not contain any “dangerous” terms proportional to ∂U φ · ∂U ψ. We also note that, obviously, the terms L from Theorem 1.1 cannot contain any dangerous quadratic terms since they are linear in the solution’s derivatives. In contrast, upon expanding the covariant wave operator terms on LHSs (2a)-(2b) relative to the standard coordinates, one typically encounters terms that are quadratic in derivatives of h and u that are transversal to the characteristics; as we highlighted above, it is precisely such “Riccati-type” terms that can drive the formation of a shock. We stress that near a shock, such transversalderivative-quadratic terms are much larger than the null form terms. We also stress that for the relativistic Euler equations, one encounters such transversal-derivative-quadratic terms on LHSs (2a)-(2b) under any equation of state aside from a single exceptional one. In the irrotational and isentropic case (in which case the relativistic Euler equations reduce to a quasilinear wave equation satisfied by a potential function), this exceptional equation of state was identified in [3]; it corresponds to the quasilinear wave equation satisfied by a 10Roughly,

these covariant wave operators are equivalent to divergence-form wave operators. In this way, one could say that a better theory is available for divergence-form wave operators than for non-divergenceform wave operators. This reminds one of the situation in elliptic theory, where better results are known for elliptic equations in divergence form compared to ones in non-divergence form. 11In one spatial dimension, one can rely exclusively on the method of characteristics and thus avoid energy estimates.


13

timelike minimal ( surface graph in an ambient ) Minkowski spacetime, which can be expressed −1 αβ (η ) ∂β Φ = 0. as follows: ∂α p 1 + (η−1 )κλ (∂κ Φ)(∂λ Φ) In view of the previous paragraph, we would like to highlight the following point: Proofs of shock formation are unstable under typical perturbations of the equations by nonlinear terms that are quadratic or higher-order in derivatives. However, proofs of shock formation for wave equations typically are stable under perturbations of the equations by null forms that are adapted to the metric of the shock-forming wave in the following sense: as the shock forms, null form terms become “asymptotically negligible” compared to the shockdriving terms (for the reasons described above). The reason that the precise structure of the nonlinearities is so important for the proofs is that the known framework is designed precisely to handle specific kinds of singularity-driving derivative-quadratic terms: the kind that are hidden in the covariant wave operator terms on LHSs (2a)-(2b). In the context of the relativistic Euler equations, this means that if any of the equations of Theorem 1.1 had contained, on the right-hand side, an inhomogeneous non-null-form quadratic term of type (∂h)2 , ∂u · ∂h, (∂u)2 , etc., or a term of type (∂h)3 , (∂h)4 , etc., then the only known framework for proving shock formation does not work. The difficulty is that adding such terms to the equation could in principle radically alter the expected blowup-rate or even altogether prevent the formation of a singularity; either way, this would invalidate12 the known approach for proving shock formation. One might draw an analogy with the Riccati ODE y˙ = y 2 , which we suggest as a caricature model for the shock formation (in the case of the relativistic Euler equations, y should be identified with ∂h and/or ∂u). Note that for all data y(0) = y0 with y0 > 0, the solution to the Riccati ODE blows up in finite time. Now if one perturbs the Riccati ODE to obtain the perturbed equation y˙ = y 2 ± ǫy 3 , with ǫ a small positive number, then depending on the sign of ±, the perturbed solutions with y0 > 0 will either exist for all time or will blow up at a quite different rate13 compared to the blowup-rate for the unperturbed equation. 1.2.3. Regularity properties and singular high-order energy estimates. In the rigorous mathematical study of quasilinear hyperbolic PDEs in more than one spatial dimension, one is forced to derive energy estimates for the solution’s higher derivatives by commuting the equations with appropriate differential operators. Indeed, all known approaches to studying even the basic local well-posedness theory for such equations rely on deriving estimates in L2 -based Sobolev spaces. In the works [3, 7, 22–24, 30, 32–34], the authors controlled the solutions’ higher geometric derivatives by differentiating the equations with geometric 12As

is explained in [21], in the known framework for proving shock formation, one crucially relies on the C fact that the derivatives of the solution blow up at a linear rate, that is like T(Lif espan)−t ; if one perturbs the equation by adding terms that are expected to alter this blowup-rate, then one should expect that the known approach for proving shock formation will not work (at least in its current form). 13It turns out that in the rigorous study of shock formation in more than one spatial dimension, it is of crucial importance to derive the sharp blowup-rate of the shock-forming variable. In all known results, the C shock-forming variable’s derivatives blow up like T(Lif espan) −t , where C is a constant and T(Lif espan) > 0 is the (future) classical lifespan of the solution; see, for example, [21] for further discussion of this point.

14

Relativistic Euler

commutator vectorfields Z that are adapted to the characteristics, more precisely to the characteristics corresponding to the variables that form a shock singularity. As we mentioned earlier, the Z are designed to avoid generating uncontrollable commutator error terms. It turns out that all Z that have been successfully used to study shock formation have the schematic structure Z α ∼ ∂U, where Z α denotes a rectangular component of Z and U is the eikonal function. Although the geometric vectorfields Z exhibit good commutation properties with the characteristics to which they are adapted, the regularity theory of the vectorfields themselves is very delicate and is intimately tied to that of the solution. We now further explain this fact in the context of wave equations, where the eikonal equation is the nonlinear transport equation (g −1)αβ ∂α U∂β U = 0. They key point is that the standard regularity theory of transport equations yields only that U is as regular as its coefficients. In the context of the relativistic Euler equations (where the formation of a shock corresponds to the intersection of the wave characteristics and g = g(h, s, u)), this suggests that one might expect U only to be as regular as h, s, and u. This leads to the following severe difficulty: in commuting equation the wave equation (2a) with Z, one obtains the wave equation g Zh = g Z α ·∂α h+· · · ∼ ∂ 3 U ·∂h+· · · (one would obtain a similar wave equation for Zu upon commuting equation (2b) with Z). The difficulty is that the above discussion suggests that the factor ∂ 3 U can be controlled only by three derivatives of h, s, and u, while standard energy estimates for the wave equations g Zh = · · · and g Zu = · · · yield control of only two derivatives of h and u (for convenience, here we will not discuss the regularity of the entropy s). This suggests that there is a loss of regularity and in fact, this is the reason that Alinhac used Nash–Moser estimates in his works [1, 2]. However, for wave equations, one can in fact overcome this loss of regularity by exploiting some delicate tensorial properties of the eikonal equation (g −1)αβ ∂α U∂β U = 0 and of the wave equation itself relative to geometric coordinates, which in conjunction can be used to show that in directions tangent to the characteristics, U is one degree more differentiable than one might expect. This crucial fact was first observed for Einstein’s equations in [5] and for more general wave equations in [19], in the context of low-regularity local well-posedness for quasilinear wave equations. In total, using this gain in regularity along the characteristics and carefully accounting for the precise tensorial structure of the product ∂ 3 U · ∂h highlighted above, one can completely avoid losing derivatives in the problem. Despite the fact that the procedure described above allows one to avoid losing derivatives, in the context of shock formation for wave equations,14 one pays a steep price: it turns out that upon implementing this procedure, one introduces a dangerous factor into the wave equation energy identities, one that in fact blows up as the shock forms. More precisely, the singular factor is 1/µ, where µ is the inverse foliation density mentioned earlier, with µ → 0 signifying the formation of a shock. This leads to singular top-order a priori energy estimates for the wave equation solutions relative to the geometric coordinates. At first glance, these singular geometric energy estimates might seem to obstruct the philosophy of obtaining regular estimates relative to the geometric coordinates. However, below the top derivative level, one can allow the loss of a derivative, and it turns out that this allows one to derive improved (i.e., less singular) energy estimates below the top derivative level. In fact, 14Actually, it is not known whether or not the derivative-loss-avoiding procedure can be implemented for general systems of wave equations featuring more than one distinct wave operator. From this perspective, we find it fortunate that the equations of Theorem 1.1 feature only one wave operator.


15

by an induction-from-the-top-down argument, one can show that the mid-derivative-level and below geometric energies remain bounded up to the shock. This allows one to show that indeed, the solution remains rather smooth relative to the geometric coordinates, which in practice is a crucial ingredient that is needed to close the proof. It also turns out that many steps are needed to descend to the level of a non-singular energy, which in practice means that one must assume that the data have a lot of Sobolev regularity to close the proof; see [13] for an in-depth overview of these issues in the context of quasilinear wave equations. In the above discussion, we have outlined why the eikonal function for the relativistic Euler equations has enough regularity to be consistent with the regularity of the fluid variables. However, we tacitly assumed that one would be able to prove that the fluid variables have a consistent amount of regularity among themselves. At first thought, the desired consistency of regularity might seem to follow from standard local well-posedness. However, all standard local-well posedness results for the relativistic Euler equations are based on firstorder formulations, which are not known to be sufficient for avoiding a loss of derivatives in the eikonal function U; the above outline for how to avoid derivative loss in U relied on the assumption that u and h solve wave equations whose source terms have an allowable amount of regularity, which, as we will explain, is a true – but deep – fact. Moreover, the first-order formulations do not seem to be sufficient for studying solutions all the way up to a shock; as we have mentioned, the known framework for studying shocks crucially relies on the special null structures exhibited by the equations of Theorem 1.1. For this reason, one must carefully check that (under suitable assumptions on the initial data), all terms in the equations of Theorem 1.1 have a consistent amount of regularity. We stress that this is not obvious, as we now illustrate by counting derivatives. For example, to control ∂uα in L2 using standard energy estimates for the wave equation (2b), one must control the source term C α on RHS (2b) where, from the point of view of regularity, we have (see (23a)) C α ∼ vortα (̟) ∼ ∂̟. Moreover, since ̟ solves the transport equation (3c), whose source term depends on ∂u, this suggests that ∂̟ should be no more regular than ∂ 2 u and thus C α should be no more regular than ∂ 2 u. In total, this discussion suggests that the wave equation for u has the following schematic structure from the point of view of regularity: g uα = ∂ 2 u + · · · . That is, this discussion suggests that in order to control ∂u in L2 using standard energy estimates for wave equations, we must control ∂ 2 u in L2 . This approach therefore seems to lead to a loss in derivatives, which is a serious obstacle to using the equations of Theorem 1.1 to prove any rigorous result. Similar difficulties arise in the study of h, due to the source term D in the wave equation (2a). A crucial feature of the equations of Theorem 1.1 is that one can in fact overcome the loss-of-derivative-problem noted in the previous paragraph. To this end, one must rely on the div-curl-transport equations for ̟ and S; see Subsect. 9.5, Prop. 9.6, and the proof of Theorem 9.2 for the details on how one can use these equations and elliptic estimates to avoid the loss of derivatives. Equally important for applications to shock waves is the fact that the elliptic estimates, which occur across space, are compatible with the proof of the formation of a spatially localized shock singularity and with the singular high-order geometric energy estimates described earlier in this subsubsection. These are delicate issues, especially since the elliptic estimates involve derivatives in directions transversal to the characteristics, i.e., in the singular directions; see [21] for an overview of how to derive the relevant elliptic

16

Relativistic Euler

estimates in the context of shock-forming solutions to the non-relativistic compressible Euler equations. 1.2.4. Structures amenable to commutations with geometric vectorfields. A key point is that the geometric vectorfields Z described in Subsubsect. 1.2.3 are adapted only to the principal part of the shock-forming solution variables, e.g., the operator g in the case that a wave equation solution is the shock-forming variable. However, to close the proof of shock formation for a system in which wave equations of the type g · = · · · are coupled to other equations, one must commute that Z through all of the equations in the system. One then has to handle the commutator terms generated by commuting the Z through the other equations. It turns out, perhaps not surprisingly, that commuting Z through a generic second-order differential operator ∂ 2 leads to uncontrollable error terms, from the point of view of regularity and from the point of view of the singular nature of the commutator error terms; see the work [21] on the non-relativistic compressible barotropic Euler equations for further discussion on this point. However, as was first shown in [21], it is possible to commute the Z through an arbitrary first-order differential operator ∂ by first weighting it by µ; this leads to commutator error terms that are controllable under the scope of the approach. It is for this reason that we have formulated Theorem 1.1 in such a way that all of the equations are of the type g · = · · · or are first-order; i.e., the equations of Theorem 1.1 are such that the approach described in [21] can be applied. Put differently, the geometric vectorfields Z that are of essential importance for commuting the wave equations of Theorem 1.1 can also be commuted through all of the remaining equations, generating only controllable error terms. 2. Setup and background 2.1. Notation and conventions. We somewhat follow the setup of [3], but there are some differences, including sign differences and notational differences. Greek “spacetime” indices α, β, · · · take on the values 0, 1, 2, 3, while Latin “spatial” indices a, b, · · · take on the values 1, 2, 3. Repeated indices are summed over (from 0 to 3 if they are Greek, and from 1 to 3 if they are Latin). Greek and Latin indices are lowered and raised with the Minkowski metric η and its inverse η−1 , and not with the acoustical metric g. ···αl is a type ml tensorfield, then If X α is a vectorfield and ξβα11···β m ···αl (LX ξ)βα11···β m

=X

κ

···αl ∂κ ξβα11···β m

−

l X a=1

(∂κ X

αa

α ···α κα ···α )ξβ11···βma−1 a+1 l

+

m X

···αl (∂βb X κ )ξβα11···β b−1 κβb+1 ···βm

b=1

(9)

denotes the Lie derivative of ξ with respect to X. Throughout we rely on a Minkowski-rectangular coordinate system {xα }α=0,1,2,3 in which the Minkowski metric η takes the form ηαβ := diag(−1, 1, 1, 1). {∂α }α=0,1,2,3 denotes the corresponding rectangular coordinate partial derivative vectorfields. We sometimes use the alternate notation x0 := t and ∂t := ∂0 . Indices are lowered and raised with the Minkowski metric and its inverse. Moreover, d denotes the exterior derivative operator. In particular, if f is a scalar function, then (df )α := ∂α f , and if V is a one-form, then (dV )αβ := ∂α Vβ −∂β Vα .

17


We use the notation V♭ to denote the one-form that is η-dual to the vectorfield V , i.e., (V♭ )α := ηακ V κ . The fluid four velocity uα is future-directed and normalized by uα uα = −1. p denotes the pressure, ρ denotes the proper energy density, n denotes the number density, s denotes the entropy per particle, θ denotes the temperature, and H = (ρ + p)/n

(10)

is the enthalpy per particle. Thermodynamics supplies the following two laws: θ=

1 ∂ρ | , n ∂s n

dH =

dp + θds, n

(11)

∂ | n denotes partial differentiation with respect to s at fixed n. Below we employ where n1 ∂s similar partial differentiation notation, and in Def. 2.7, we introduce alternate partial differentiation notation, which we use throughout the remainder of the article. ǫαβγδ is the fully antisymmetric symbol normalized by ǫ0123 = 1. Note that ǫ0123 = −1. We assume an equation of state of the form p = p(ρ, s). The speed of sound is defined by s ∂p c := | . (12) ∂ρ s

In this article, we will confine our study to equations of state and solutions that verify 0 < c ≤ 1.

(13)

The upper bound in (13) signifies that the speed of sound is no bigger than the speed of light. In this article, we exploit both inequalities in (13). We remark that we use the bound c ≤ 1 to ensure that we can always solve for time derivatives of the solution in terms of spatial derivatives; see the discussion surrounding equation (35). Παβ := (η−1 )αβ + uα uβ

(14)

denotes projection onto the η-orthogonal complement of u. The acoustical metric g is a Lorentzian15 metric that drives the propagation of sound waves. Definition 2.1 (Acoustical metric and its inverse). We define the acoustical metric gαβ and its inverse16 (g −1)αβ as follows: gαβ := c−2 ηαβ + (c−2 − 1)uα uβ ,

−1 αβ

(g )

2

αβ

:= c Π

α β

2

(15a)

−1 αβ

− u u = c (η )

2

α β

+ (c − 1)u u .

(15b)

It is straightforward to compute that relative to the rectangular coordinates, we have |detg| 15That

1/2

detg = −c−6 ,

−1 αβ

(g )

−1

−1 αβ

= c (η )

(16a) + (c

−1

−3

α β

− c )u u .

(16b)

is, the signature of the 4 × 4 matrix gαβ , viewed as a quadratic form, is (−, +, +, +). is straightforward to check that (g −1 )ακ gκβ = δβα , where δβα is the Kronecker delta. That is, g −1 is indeed the inverse of g. 16It

18

Relativistic Euler

Definition 2.2 (The vorticity of a one-form). Given a one-form V , we define the corresponding (u-orthogonal vorticity) vectorfield as follows: vortµ (V ) := −ǫµβγδ uβ ∂γ Vδ .

(17)

Definition 2.3 (Vorticity vectorfield). We define the vorticity vectorfield ̟ α as follows: ̟ α := vortα (Hu) = −ǫαβγδ uβ ∂γ (Huδ ).

(18)

We find it convenient to work with the log of the enthalpy. Definition 2.4 (Logarithmic enthalpy). Let H > 0 be a fixed constant value of the enthalpy. We define the (dimensionless) logarithmic enthalphy h as follows: h := ln H/H . (19)

Definition 2.5 (The quantity q). We define the quantity q as follows:

θ . (20) H Definition 2.6 (Entropy gradient one-form). We define the entropy gradient one-form Sα as follows: q :=

Sα := ∂α s.

(21)

The notation featured in the next definition will allow for a simplified presentation of various equations. Definition 2.7 (Partial derivatives with respect to h and s). If Q is a quantity that can be expressed as a function of (h, s), then ∂Q |s , ∂h ∂Q |h , Q;s = Q;s (h, s) := ∂s ∂ where ∂h |s denotes partial differentiation with respect to h at fixed s and partial differentiation with respect to s at fixed h. Q;h = Q;h (h, s) :=

(22a) (22b) ∂ | ∂s h

denotes

In our analysis, we will have to control the vorticity of the vorticity, that is, vortα (̟). The following modified version of vortα (̟), denoted by C α obeys a transport equation (see (45b)) with a better structure (from the point of view of the regularity of the RHS and also the null structure of the RHS) than the one satisfied by vortα (̟). Similar remarks apply to the modified version of the divergence of entropy gradient, which we denote by D (see equation (43a) for the transport equation verified by D). Definition 2.8 (Modified variables). C α := vortα (̟) + c−2 ǫαβγδ uβ (∂γ h)̟δ

+ (θ − θ;h )S α (∂κ uκ ) + (θ − θ;h )uα (S κ ∂κ h) + (θ;h − θ)S κ ((η−1 )αλ ∂λ uκ ), 1 1 1 D := (∂κ S κ ) + (S κ ∂κ h) − c−2 (S κ ∂κ h). n n n

(23a)

(23b)


19

2.2. First-order formulation of the relativistic Euler equations. In formulating the relativistic Euler equations as a first-order hyperbolic system, we will consider h, s, and {uα }α=0,1,2,3 to be the fundamental unknowns.17 In terms of these variables, the relativistic Euler equations are uκ ∂κ h + c2 ∂κ uκ = 0, κ

α

ακ

(24)

−1 ακ

(25)

κ

(26)

u ∂κ u + Π ∂κ h − q(η ) ∂κ s = 0, u ∂κ s = 0.

Note that (26) is equivalent to uκ Sκ = 0.

(27)

Equation (25) can be written more explicitly as uκ ∂κ uα + ∂α h + uα uκ ∂κ h − qSα = 0.

(28)

The following constraint is preserved by the flow of equation (25): uκ uκ = −1.

(29)

uκ ∂κ (Huα) + ∂α H − θSα = 0.

(30)

Also, from (28), we easily derive

Moreover, differentiating (26) with a rectangular coordinate partial derivative, we deduce uκ ∂κ Sα = −Sκ (∂α uκ ).

(31)

In our analysis, we will also use the following evolution equation for n: uκ ∂κ n + n∂κ uκ = 0.

(32)

To obtain (32), we first use equations (24) and (26), the thermodynamic relation dH = dp/n + θds, and the relation H = (ρ + p)/n to deduce uκ ∂κ p + c2 (ρ + p)∂κ uκ = 0. We then use this equation, (12), and equation (26) to deduce uκ ∂κ ρ + (ρ + p)∂κ uκ = 0. Next, using this equation and equation (26), we deduce ∂ρ(n,s) |s uκ ∂κ n + (ρ + p)∂κ uκ = 0. Finally, from ∂n this equation and the thermodynamic relation ρ + p = n ∂ρ(n,s) |s , we conclude (32). ∂n For future use, we also note that equations (24)-(26) can be written (using (29)) in the form   h u0   1 u  α (33) A ∂α  2  = 0, u  u3  s 17On

might argue that it is more accurate to think of u0 as being “redundant” in the sense that it is algebraically determined in terms of {ua }a=1,2,3 via the condition u0 > 0 and the normalization condition (29). In fact, in most of Sect. 9, we adopt this point of view. However, prior to Sect. 9, we do not adopt this point of view.

20

Relativistic Euler

where for α = 0, 1, 2, 3, Aα is a 6 × 6 matrix that (h, u0 , u1, u2 , u3 , s). In particular, we see that  0 u c2 0 a ua u u0 0  0 1  u u 0 u0 0 A = 0 2 u u 0 0  u0 u3 0 0 0 0 0

and we compute that

is a smooth function of the solution array  0 0 0 0 0 q  0 0 0 , u0 0 0  0 u0 0  0 0 u0

detA0 = (u0 )6 − c2 (u0 )4 ua ua = (1 + ua ua )4 1 + (1 − c2 )ubub .

(34)

(35)

In particular, in view of (13), we deduce from (35) that A0 is invertible. We view the speed of sound to be a function of h and s: c = c(h, s).

(36)

For the purpose of deriving the new formulation of the equations stated in Theorem 3.1, we assume that we have a C 2 solution (h, s, u0, u1 , u2 , u3) to (24)-(26) + (29). 3. The new formulation of the relativistic Euler equations In the next theorem, we provide the main result of the article: the new formulation of the relativistic Euler equations. Theorem 3.1 (New formulation of the relativistic Euler equations). For C 2 solutions (h, s, uα) to the relativistic Euler equations (24)-(26) + (29), the following equations hold, where the phrase “g-null form” refers to a linear combination of the standard g-null forms of Def. 1.1 such that the coefficients are allowed to depend on the quantities (h, s, uα, S α , ̟ α ) (but not their derivatives). Wave equations. The logarithmic enthalpy h verifies the following covariant wave equation (see Footnote 5 on pg. 6 for a formula for the covariant wave operator): g h = nc2 qD + Q(h) + L(h) ,

(37)

where Q(h) is the g-null form defined by Q(h) := −c−1 c;h (g −1 )κλ (∂κ h)(∂λ h) + c2 (∂κ uκ )(∂λ uλ) − (∂λ uκ )(∂κ uλ )

(38a)

and L(h) , which is at most linear in the derivatives of (h, s, uα, S α , ̟ α), is defined by L(h) := (1 − c2 )q + c2 q;h − cc;s (S κ ∂κ h) + c2 q;s Sκ S κ . (38b)

Moreover, the rectangular four-velocity components uα verify the following covariant wave equations: g uα = −

c2 α C + Q(uα ) + L(uα ) , H

(39)


21

where Q(uα ) is the g-null form defined by Q(uα ) := (η−1 )αλ {(∂κ uκ )(∂λ h) − (∂λ uκ )(∂κ h)} + c2 uα (∂κ uλ )(∂λ uκ ) − (∂λ uλ )(∂κ uκ ) (40a) − 1 + c−1 c;h (g −1)κλ (∂κ h)(∂λ uα )

and L(uα ) , which is at most linear in the derivatives of (h, s, uα, S α , ̟ α ), is defined by c2 αβγδ (1 − c2 ) αβγδ (1 − c2 )q αβγδ ǫ (∂β uγ )̟δ + ǫ uβ (∂γ h)̟δ + ǫ Sβ u γ ̟ δ H H H + {q − cc;s } (S κ ∂κ uα ) + q(c2 − 1)uα S κ (uλ∂λ uκ ) (θ − θ;h )c2 2 κ ((η−1 )αλ ∂λ uκ ) +S c q+ H + 2c−1 c;h qS α + 2c−1 c;s S α − q;h S α (uκ ∂κ h) (θ − θ;h )c2 α κ (θ − θ;h )c2 α − q (∂κ uκ ) + u (S ∂κ h). +S H H

L(uα ) := −

(40b)

Transport equations. The rectangular components of the entropy gradient vectorfield S α , whose η-dual is defined in (21), verify the following transport equation: uκ ∂κ S α = −Sκ ((η−1 )αλ ∂λ uκ ).

(41)

Moreover, the rectangular components of the vorticity vectorfield ̟ α , which is defined in (18), verify the following transport equation: uκ ∂κ ̟ α = −uα (̟ κ ∂κ h) + ̟ κ ∂κ uα − ̟ α (∂κ uκ ) + (θ − θ;h )ǫαβγδ uβ (∂γ h)Sδ + quα̟ κ Sκ . (42) Transport-div-curl systems. The modified divergence of the entropy gradient D (which is defined in (23b)) and the rectangular components vortα (S) of the u-perpendicular vorticity (see definition (17)) of the entropy gradient vectorfield verify the following transport-div-curl system: 1 2 (∂κ S κ )(∂λ uλ ) − (∂λ S κ )(∂κ uλ ) + c−2 uκ (∂κ h)(∂λ S λ ) − (∂λ h)(∂κ S λ ) n n (43a) κ Sκ C + Q(D) + L(D) , + nH vortα (S) = 0, (43b) uκ ∂κ D =

where Q(D) is the g-null form defined by Q(D) :=

1 −2 κ c S (∂κ uλ )(∂λ h) − (∂λ uλ)(∂κ h) n

(44a)

22

Relativistic Euler

and L(D) , which is linear in the derivatives of (h, s, uα, S α , ̟ α ), is defined by

1 αβγδ (1 − c−2 ) αβγδ ǫ Sα uβ (∂γ h)̟δ + ǫ Sα (∂β uγ )̟δ nH nH S κ S λ (θ − θ;h ) − 2q (∂κ uλ ) + n H Sκ S κ (θ;h − θ) −1 2 + + 2c c;s − c q;h + q (∂λ uλ ). n H

L(D) :=

(44b)

Finally, the divergence of the vorticity vectorfield ̟ α (which is defined in (18)) and the rectangular components C α of the modified vorticity of the vorticity (which is defined in (23a)) verify the following equations:

∂α ̟ α = −̟ κ ∂κ h + 2q̟ κ Sκ ,

κ

α

κ

α

α

κ

(45a) α κ

λ

u ∂κ C = C ∂κ u − 2C (∂κ u ) + u C (u ∂λ uκ ) αβγδ

(45b)

κ

− 2ǫ

uβ (∂γ ̟ )(∂δ uκ ) + (θ;h − θ) (η−1 )ακ + 2uα uκ (∂κ h)(∂λ S λ ) − (∂λ h)(∂κ S λ ) + (θ − θ;h )nuα (uκ ∂κ h)D

+ (θ − θ;h )qS α (∂κ S κ ) + (θ;h − θ)qSκ ((η−1 )αλ ∂λ S κ ) + Q(C α ) + L(C α ) ,

where Q(C α ) is the g-null form defined by

Q(C α ) := −c−2 ǫκβγδ (∂κ uα )uβ (∂γ h)̟δ + (c−2 + 2)ǫαβγδ uβ (∂γ h)̟ κ (∂δ uκ ) −2 αβγδ

κ

κ

+c ǫ uβ ̟δ {(∂κ u )(∂γ h) − (∂γ u )(∂κ h)} + (θ;h;h − θh ) + c−2 (θ − θ;h ) uκ (η−1 )αλ S β {(∂κ h)(∂λ uβ ) − (∂λ h)(∂κ uβ )} + (θ;h − θ)S κ uλ {(∂κ uα )(∂λ h) − (∂λ uα )(∂κ h)} + (θ;h − θ) (η−1 )ακ + uα uκ S β (∂κ uβ )(∂λ uλ ) − (∂λ uβ )(∂κ uλ ) + (θ;h − θ)S α (∂κ uλ )∂λ uκ − (∂λ uλ )(∂κ uκ ) + (θ;h − θ)S κ (∂κ uα )(∂λ uλ ) − (∂λ uα )(∂κ uλ ) + S α c−2 (θh − θ;h;h ) + c−4 (θ;h − θ) (g −1 )κλ (∂κ h)(∂λ h),

(46a)

23

M. Disconzi, J. Speck and L(C α ) , which is linear in the derivatives of (h, s, uα , S α , ̟ α), is defined by 2q κ 2 (̟ Sκ ̟ α ) − ̟ α (̟ κ ∂κ h) H H + 2c−3 c;s ǫαβγδ uβ Sγ ̟δ (uκ ∂κ h)

L(C α ) :=

(46b)

− 2qǫαβγδ uβ Sγ ̟ κ (∂δ uκ ) − qǫαβγδ Sβ uγ ̟δ (∂κ uκ ) 1 + (θ − θ;h )ǫκβγδ (∂κ uα )Sβ uγ ̟δ + c−2 qǫαβγδ Sβ (∂γ h)̟δ − c−2 quα ǫκβγδ Sκ uβ (∂γ h)̟δ H + (θ;h − θ)qSκ S κ (uλ ∂λ uα ) + uα Sκ S κ {(θ;h − θ)q + (θ;h;s − θ;s )} (uλ ∂λ h)

+ S α {(θ;s − θ;h;s ) + (θ − θ;h )q;h } (S κ ∂κ h) + Sκ S κ (θ;h;h − θh )q + (θ;h;s − θ;s ) + (θ − θ;h )qc−2 + (θ;h − θ)q;h ((η−1 )αλ ∂λ h).

Remark 3.1 (Special structure of the inhomogeneous terms). We emphasize the following two points, which are of crucial importance for applications to shock waves (see Subsubsect. 1.2.2 for further discussion): i) all inhomogeneous terms on the RHSs of the equations of the theorem are at most quadratic in the derivatives of (h, s, uα , S α , ̟ α) and ii) all derivative-quadratic terms on the RHSs of the equations of the theorem are linear combinations of standard g-null forms. In particular, the following are linear combinations of standard g-null forms, even though we did not explicitly state so in the theorem: the terms on the first line of RHS (43a) and the terms on the second and third lines of RHS (45b). We have separated these null forms, which involve the derivatives of ̟ and S, because they need to be handled with elliptic estimates, at least at the top derivative level (see the proof of Theorem 9.2). This is different compared to the terms Q(h) , Q(uα ) , Q(D) , and Q(C α ) , which can be handled with standard energy estimates at all derivative levels. Proof. Theorem 3.1 follows from a lengthy series of calculations, most of which we derive later in the paper, except that we have somewhat reorganized (using only simple algebra) the terms on the right-hand sides of the equations of the theorem. More precisely, we prove (37)-(38b) in Prop. 5.2. We prove (39)-(40b) in Prop. 5.3. (41) follows from raising the indices of (31) with the inverse Minkowski metric. We prove (42) in Prop. 7.1. Except for (43b), (43a)-(44b) follow from Prop. 6.2. (43b) is a simple consequence of definition (17) and the symmetry property ∂α Sβ = ∂β Sα (see (47)). Finally, we prove (45a)-(46b) in Prop. 8.2. 4. Preliminary identities In the next lemma, we derive some preliminary identities that we will later use when deriving the equations stated in Theorem 3.1. Lemma 4.1 (Some useful identities). Assume that (h, s, uα ) is a C 2 solution to (24)-(26) + (29), and let Vα be any one-form. Then the following identities hold: ∂α Sβ = ∂β Sα ,

(47)

24

Relativistic Euler

̟ κ uκ = 0,

(48)

uκ ∂α uκ = 0,

(49)

uκ ∂α Sκ = −S κ ∂α uκ ,

(50)

uκ ∂α ̟κ = −̟ κ ∂α uκ ,

(51)

∂α = −uα uκ ∂κ + Πακ ∂κ

(52)

∂κ V κ = −uκ uλ ∂λ V κ + Πκλ ∂κ Vλ ,

(53)

∂α Vβ − ∂β Vα = ǫαβγδ uγ vortδ (V ) + uα uκ ∂β Vκ − uβ uκ ∂α Vκ + uβ uκ ∂κ Vα − uα uκ ∂κ Vβ , Παβ Πγδ (∂α Vγ − ∂γ Vα )(∂β Vδ − ∂δ Vβ ) = 2Παβ vortα (V )vortβ (V ).

(54) (55)

κ

Moreover, if u Vκ = 0, then

∂α Vβ − ∂β Vα = ǫαβγδ uγ vortδ (V ) − uα Vκ ∂β uκ + uβ Vκ ∂α uκ + uβ uκ ∂κ Vα − uα uκ ∂κ Vβ .

(56)

In addition, the following identity holds, where the indices for ǫ on LHS (57) are raised before Lie differentiation: Lu (ǫαβγδ ) = (−∂κ uκ )ǫαβγδ .

(57)

Lu (u♭ )α = uκ ∂κ uα = −∂α h − uα uκ ∂κ h + qSα ,

(58)

Lu d(Hu♭) = dLu (Hu♭ ),

(59)

[Lu d(Hu♭)]αβ = θ;h (∂α h)∂β s − θ;h (∂α s)∂β h,

(60)

∂α (Huβ ) − ∂β (Huα ) = ǫαβγδ uγ ̟ δ + θ {Sα uβ − Sβ uα } ,

(61)

ǫαβγδ ∂γ (Huδ ) = ̟ αuβ − uα ̟ β + θǫαβγδ Sγ uδ ,

(62)

Furthermore, the following identities hold:

∂α uβ − ∂β uα =

1 ǫαβγδ uγ ̟ δ − (∂α h)uβ + (∂β h)uα + q {Sα uβ − Sβ uα } , H

(63)

25


(uκ ∂κ uλ )S λ = −S κ ∂κ h + qSκ S κ ,

(64)

(uκ ∂κ Sλ )uλ = S κ ∂κ h − qSκ S κ ,

(65)

S κ ∂α uκ = S κ ∂κ uα + (S κ ∂κ h)uα − qS κ Sκ uα + = S κ ∂κ uα − (S κ uλ ∂λ uκ )uα +

1 ǫακγδ S κ uγ ̟ δ H

1 ǫαβγδ S β uγ ̟ δ , H

̟ κ ∂κ uα = ̟ κ ∂α uκ − (̟ κ ∂κ h)uα + q̟ κ Sκ uα , ǫαβγδ ∂γ uδ =

1 1 α β ̟ u − uα ̟ β − ǫαβγδ (∂γ h)uδ + qǫαβγδ Sγ uδ , H H ǫαβγδ uβ ∂γ uδ = −

(66)

1 α ̟ , H

(67)

(68)

(69)

∂γ ̟δ − ∂δ ̟γ = ǫγδκλ uκ vortλ (̟) − (uκ ∂κ ̟δ )uγ + uκ (∂δ ̟κ )uγ + (uκ ∂κ ̟γ )uδ − uκ (∂γ ̟κ )uδ , (70) ǫαβγδ ∂γ ̟δ = vortα (̟)uβ − uα vortβ (̟) + ǫαβγδ (uκ ∂κ ̟γ )uδ

(71)

αβγδ κ

−ǫ

u ∂γ ̟κ uδ .

Proof. (47) follows from the symmetry property ∂α ∂β s = ∂β ∂α s. (48) is a simple consequence of definition (2.3). (49) follows from differentiating (29) with ∂α . (50) follows from differentiating (27) with ∂α . (51) follows from differentiating (48) with ∂α . (57) is a standard geometric identity, as is (59). To prove (60), we first use (30) and the Lie derivative formula (9) to deduce that Lu (Hu♭ )α = κ u ∂κ (Huα) + Huκ ∂α uκ = −∂α H + θ∂α s + Huκ ∂α uκ . From (49), we see that the last product on the RHS of this identity vanishes. Hence, taking the exterior derivative of the identity, we obtain [dLu (Hu♭)]αβ = θ;h (∂α h)∂β s − θ;h (∂α s)∂β h. The desired identity (60) now follows from this identity and (59). (52) follows directly from definition (14). (53) then follows from (52). To prove (58), we first note the Lie differentiation identity Lu (u♭ )α = uκ ∂κ uα + uκ ∂α uκ , which follows from (9). (58) follows from this identity, (28), and (49).

26

Relativistic Euler

To prove (61), we first use definition (18) to compute that ǫαβγδ uγ ̟ δ = −ǫαβγδ ǫδκθλ uγ uκ ∂θ (Huλ ). Using the identity ǫαβγδ ǫδκθλ = −ǫαβγδ ǫλκθδ = δαλ δβκ δγθ − δαλ δβθ δγκ + δακ δβθ δγλ − δακ δβλ δγθ + δαθ δβλ δγκ − δαθ δβκ δγλ , we deduce, in view of (29) and (48), that −ǫαβγδ ǫδκθλ uγ uκ ∂θ (Huλ) = ∂α (Huβ ) − ∂β (Huα) κ

(72) κ

κ

κ

− uβ u ∂κ (Huα ) − uα u ∂β (Huκ) + uα u ∂κ (Huβ ) + uβ u ∂α (Huκ ).

Using (29), (30), and (49), we compute that the last four products on RHS (72) sum to θ(uα Sβ − uβ Sα ), which yields the desired identity (61). To prove (54), we first use definition (17) to express the first product on RHS (54) as follows: ǫαβγδ uγ vortδ (V ) = −ǫαβγδ ǫδθκλ uγ uθ ∂κ Vλ .

(73)

Next, we observe the following identity for some of the factors on RHS (73): −ǫαβγδ ǫδθκλ = ǫαβγδ ǫθκλδ =

δαθ δγκ δβλ

−

(74) δαθ δγλ δβκ

+

δαλ δγθ δβκ

−

δαλ δγκ δβθ

+

δακ δγλ δβθ

−

δακ δγθ δβλ .

Using (74) to substitute on RHS (73), we deduce, in view of (29), the following identity: −ǫαβγδ ǫδθκλ uθ ∂κ Vλ uγ = uα uκ ∂κ Vβ − uαuκ ∂β Vκ − ∂β Vα − uβ uκ ∂κ Vα + uβ uκ ∂α Vκ + ∂α Vβ . (75) Combining (73) and (75) and rearranging the terms, we arrive at the desired identity (54). (56) then follows from (54) and the relation uκ ∂α Vκ = −Vκ ∂α uκ , which follows from differentiating the assumed identity uκ Vκ = 0 with ∂α . To prove (55), we first use (54) to deduce Παβ Πγδ (∂α Vγ − ∂γ Vα )(∂β Vδ − ∂δ Vβ ) = Παβ Πγδ ǫαγκλ ǫβδµν uκ vortλ (V )uµ vortν (V ).

(76)

Next, we note the following identity, which follows easily from definition (14): Παβ Πγδ ǫαγκλ ǫβδµν uκ vortλ (V )uµ vortν (V ) = (η−1 )αβ (η−1 )γδ ǫαγκλ ǫβδµν uκ vortλ (V )uµ vortν (V ) (77) = ǫαβκλ ǫαβµν uκ vortλ (V )uµ vortν (V ). From (77), the identity ǫαβκλ ǫαβµν = 2δµλ δνκ −2δµκ δνλ , (29), and the simple identity uα vortα (V ) = 0 (which follows easily from definition (17)), we find that RHS (77) = 2vortα (V )vortα (V ). Again using that uα vortα (V ) = 0, we conclude, in view of definition (14), the desired identity (55). To prove (62), we first contract 21 ǫαβγδ against (61) to obtain the identity 1 ǫαβγδ ∂γ (Huδ ) = ǫαβγδ ǫγδκλ uκ ̟ λ + θǫαβγδ Sγ uδ . 2

(78)

(62) now follows from using the identity 12 ǫαβγδ ǫγδκλ = δκβ δλα −δκα δλβ to substitute for the factor 1 αβγδ ǫ ǫγδκλ on RHS (78). 2 (63) follows from (61) and simple computations. To prove (64), we contract S α against equation (28) and use equation (26). (65) then follows from (50) and (64).

27


To prove the first equality in (66), we contract S β against (63) and use equation (26). To obtain the second equality in (66), we use the first equality and the identity (64). (67) follows from contracting (63) against ̟ α and using (48). To prove (68), we first use (63) to deduce that 1 1 αβγδ ǫαβγδ ∂γ uδ = ǫ ǫγδκλ uκ ̟ λ − ǫαβγδ (∂γ h)uδ + qǫαβγδ Sγ uδ . (79) 2H (68) now follows from using the identity 12 ǫαβγδ ǫγδκλ = δλα δκβ − δκα δλβ to substitute for the product 12 ǫαβγδ ǫγδκλ on RHS (79). To prove (69), we contract (68) against uβ and use (29) and (48). To prove (70), we first use definition (17) to express the first product on RHS (70) as follows: ǫγδκλ uκ vortλ (̟) = −ǫγδκλ ǫλθαβ uκ uθ ∂α ̟β .

(80)

Next, we use the identity −ǫγδκλ ǫλθαβ = ǫγδκλ ǫθαβλ = δγθ δδβ δκα − δγθ δδα δκβ + δγα δδθ δκβ − δγα δδβ δκθ + δγβ δδα δκθ − δγβ δδθ δκα to substitute on RHS (80), thereby obtaining, in view of (29), the following identity: ǫγδκλ uκ vortλ (̟) = uγ uκ ∂κ ̟δ − uγ uκ ∂δ ̟κ + uδ uκ ∂γ ̟κ + ∂γ ̟δ − ∂δ ̟γ − uδ uκ ∂κ ̟γ . (81) Finally, we note that it is straightforward to see that (81) is equivalent to the desired identity (70). To prove (71), we first contract (70) against 21 ǫαβγδ to deduce 1 ǫαβγδ ∂γ ̟δ = ǫαβγδ ǫγδκλ uκ vortλ (̟) + ǫαβγδ (uκ ∂κ ̟γ )uδ − ǫαβγδ uκ (∂γ ̟κ )uδ . 2

(82)

Using the identity 12 ǫαβγδ ǫγδκλ = 12 ǫγδαβ ǫγδκλ = δκβ δλα − δκα δλβ to substitute in the first product on RHS (82), we arrive at the desired identity (71). 5. Wave equations In this section, with the help of the preliminary identities of Lemma 4.1, we derive the covariant wave equations (37) and (39). 5.1. Covariant wave operator. We start by establishing a formula for the covariant wave operator of the acoustical metric acting on a scalar function. Lemma 5.1 (Covariant wave operator of g). Assume that (h, s, uα) is a C 2 solution to (24)(26) + (29). Then the covariant wave operator of the acoustical metric g = g(h, s, u) (see Def. 2.1) acts on scalar functions φ as follows, where RHS (83) is expressed relative to the rectangular coordinates: g φ = (c2 − 1)uκ ∂κ (uλ ∂λ φ) + c2 ((η−1 )κλ ∂κ ∂λ φ) 2

κ

λ

−1

κ

(83) λ

+ (c − 1)(∂κ u )(u ∂λ φ) + 2c c;h (u ∂κ h)(u ∂λ φ) − c−1 c;h (g −1 )κλ (∂κ h)(∂λ φ)

− cc;s (S κ ∂κ φ).

28

Relativistic Euler

Proof. It is a standard fact that relative to arbitrary coordinates relative p(and in particular 1 −1 κλ |detg|(g ) ∂λ φ . Using to the rectangular coordinates), we have g φ = p ∂κ |detg| this formula and (16a)-(16b), we compute that g φ = c3 ∂κ −(c−3 − c−1 )uκ (uλ ∂λ φ) + c−1 ((η−1 )κλ ∂λ φ) (84) = −(1 − c2 )uκ ∂κ (uλ ∂λ φ) − (1 − c2 )(∂κ uκ )(uλ ∂λ φ)

+ (3c−1 − c)(uκ ∂κ c)(uλ ∂λ φ) − c(η−1 )κλ (∂κ c)(∂λ φ) + c2 (η−1 )κλ ∂κ ∂λ φ.

The desired identity (83) now follows from (84), (15b), the evolution equation (26), and straightforward computations. 5.2. Covariant wave equation for h. We now derive the covariant wave equation (37). Proposition 5.2 (Covariant wave equation for the logarithmic enthalpy). Assume that (h, s, uα) is a C 2 solution to (24)-(26) + (29). Then the logarithmic enthalpy h verifies the following covariant wave equation: g h = nc2 qD − c−1 c;h (g −1 )κλ (∂κ h)(∂λ h) + c2 (∂κ uκ )(∂λ uλ ) − (∂κ uλ )(∂λ uκ ) (85) + (1 − c2 )q(S κ ∂κ h) − cc;s (S κ ∂κ h) + c2 q;h (S κ ∂κ h) + c2 q;s Sκ S κ .

Proof. From (83) with φ := h, we deduce g h = (c2 − 1)uκ ∂κ (uλ ∂λ h) + c2 ((η−1 )κλ ∂κ ∂λ h) 2

κ

λ

−1

κ

(86) λ

+ (c − 1)(∂κ u )(u ∂λ h) + 2c c;h (u ∂κ h)(u ∂λ h) − c−1 c;h (g −1)κλ (∂κ h)(∂λ h) − cc;s (S κ ∂κ h).

Next, we differentiate equation (28) with ∂β , contract against (η−1 )αβ , and multiply by c2 to obtain the identity c2 ((η−1 )κλ ∂κ ∂λ h) = −c2 (uκ ∂κ ∂λ uλ ) − c2 (∂κ uλ )(∂λ uκ )

(87)

− c2 uκ ∂κ (uλ ∂λ h) − c2 (∂κ uκ )(uλ∂λ h)

+ c2 q(∂κ S κ ) + c2 q;h (S κ ∂κ h) + c2 q;s S κ Sκ . Next, we use (24) and the evolution equation (26) to rewrite the first product on RHS (87) as follows: −c2 (uκ ∂κ ∂λ uλ) = c2 uκ ∂κ (c−2 uλ ∂λ h)

(88)

= uκ ∂κ (uλ ∂λ h) − 2c−1 c;h (uκ ∂κ h)(uλ ∂λ h).

Using (88) to substitute for the first product on RHS (87) and then using the resulting identity to substitute for the product c2 (η−1 )κλ ∂κ ∂λ h on RHS (86), we deduce g h = −c2 (∂κ uλ )(∂λ uκ ) − (∂κ uκ )(uλ ∂λ h) − c−1 c;h (g −1 )κλ (∂κ h)(∂λ h) − cc;s (S κ ∂κ h) + c2 q(∂κ S κ ) + c2 q;h (S κ ∂κ h) + c2 q;s S κ Sκ .

(89)

29


Finally, we use equation (24) to substitute for the factor uλ ∂λ h in the second product on RHS (89), and we use definition (23b) to express the product c2 q(∂κ S κ ) on RHS (89) as nc2 qD + (1 − c2 )q(S κ ∂κ h), which in total yields the desired equation (85).

5.3. Covariant wave equation for the rectangular four-velocity components. We now derive the covariant wave equation (39). Proposition 5.3 (Covariant wave equation for the rectangular four-velocity components). Assume that (h, s, uα) is a C 2 solution to (24)-(26) + (29). Then the rectangular velocity components uα verify the following covariant wave equations: c2 α C (90) H c2 (1 − c2 ) αβγδ (1 − c2 )q αβγδ − ǫαβγδ (∂β uγ )̟δ + ǫ uβ (∂γ h)̟δ + ǫ Sβ u γ ̟ δ H H H − (g −1)κλ (∂κ h)(∂λ uα ) − c−1 c;h (g −1)κλ (∂κ h)(∂λ uα ) + (η−1 )αλ {(∂κ uκ )(∂λ h) − (∂λ uκ )(∂κ h)} + c2 uα (∂κ uλ )(∂λ uκ ) − (∂λ uλ )(∂κ uκ )

g uα = −

− cc;s (S κ ∂κ uα ) + q(S κ ∂κ uα ) + (c2 − 1)quα(S κ uλ ∂λ uκ ) + c2 qS κ ((η−1 )αλ ∂λ uκ )

+ 2c−1 c;s S α (uκ ∂κ h) + 2c−1 c;h qS α (uκ ∂κ h) − q;h S α (uκ ∂κ h) − qS α (∂κ uκ ) + (θ − θ;h )

c2 c2 c2 α S (∂κ uκ ) + (θ − θ;h ) uα (S κ ∂κ h) + (θ;h − θ) S κ ((η−1 )αλ ∂λ uκ ). H H H

Proof. From (83) with φ := uα , we deduce g uα = (c2 − 1)uκ ∂κ (uλ∂λ uα ) + c2 ((η−1 )κλ ∂κ ∂λ uα )

(91)

+ (c2 − 1)(∂κ uκ )(uλ ∂λ uα ) + 2c−1 c;h (uκ ∂κ h)(uλ ∂λ uα ) − c−1 c;h (g −1 )κλ (∂κ h)(∂λ uα ) − cc;s (S κ ∂κ uα ).

Next, we use equations (26), (28), and (100) to rewrite the first product on RHS (91) as follows: (c2 − 1)uκ ∂κ (uλ ∂λ uα ) = (1 − c2 )(uκ ∂κ ∂α h) + (1 − c2 ) uκ ∂κ (uλ ∂λ h) uα + (1 − c2 )(uκ ∂κ uα )(uλ∂λ h) + (c2 − 1)uκ ∂κ (qSα ) = (1 − c2 )(uκ ∂κ ∂α h) + (1 − c2 ) uκ ∂κ (uλ ∂λ h) uα

(92)

+ (1 − c2 )(uκ ∂κ uα )(uλ∂λ h) + (c2 − 1)q;h (uκ ∂κ h)Sα 1 + (1 − c2 )q(S κ ∂κ uα ) + (1 − c2 )qǫαβγδ S β uγ ̟ δ + (c2 − 1)qS κ (uλ ∂λ uκ )uα . H

30

Relativistic Euler

Next, we use definition (23b), the identity (63), and the evolution equations (24), (26), and (31) to rewrite the second product on RHS (91) as follows:

c2 ((η−1 )κλ ∂κ ∂λ uα ) = c2 (∂α ∂κ uκ )

(93)

1 ǫλαγδ uγ ̟ δ − (∂λ h)uα + (∂α h)uλ + qSλ uα − qSα uλ H = (c2 − 1)(uκ ∂κ ∂α h) − (∂α uκ )(∂κ h) + 2c−1 c;h (∂α h)(uκ ∂κ h) + 2c−1 c;s Sα (uκ ∂κ h) 1 1 − c2 ǫλαγδ ((η−1 )κλ ∂κ h)uγ ̟ δ + c2 ǫλαγδ ((η−1 )κλ∂κ uγ )̟ δ H H 1 + c2 ǫλαγδ uγ ((η−1 )κλ ∂κ ̟ δ ) H 2 − c ((η−1 )κλ ∂κ ∂λ h)uα − c2 (η−1 )κλ (∂κ h)(∂λ uα ) + c2 (∂α h)(∂κ uκ ) + c2 (η−1 )κλ ∂κ

+ c2 q;h (S κ ∂κ h)uα + c2 q;s Sκ S κ uα + c2 q(∂κ S κ )uα + c2 q(S κ ∂κ uα ) − c2 q;h (uκ ∂κ h)Sα − c2 q(uκ ∂κ Sα ) − c2 q(∂κ uκ )Sα

= nc2 qDuα

+ (c2 − 1)(uκ ∂κ ∂α h) − (∂α uκ )(∂κ h) + 2c−1 c;h (∂α h)(uκ ∂κ h) 1 1 − c2 ǫλαγδ ((η−1 )κλ ∂κ h)uγ ̟ δ + c2 ǫλαγδ ((η−1 )κλ∂κ uγ )̟ δ H H 1 + c2 ǫλαγδ uγ ((η−1 )κλ ∂κ ̟ δ ) H 2 − c ((η−1 )κλ ∂κ ∂λ h)uα − c2 (η−1 )κλ (∂κ h)(∂λ uα ) + c2 (∂α h)(∂κ uκ ) + c2 q;h (S κ ∂κ h)uα + c2 q;s Sκ S κ uα + c2 q(S κ ∂κ uα ) − c2 q;h (uκ ∂κ h)Sα + c2 q(∂α uκ )Sκ − c2 q(∂κ uκ )Sα + (1 − c2 )q(S κ ∂κ h)uα + 2c−1 c;s (uκ ∂κ h)Sα .

Next, we use the identity (83) with φ := h to substitute for the term g h on LHS (85), which yields the identity

c2 ((η−1 )κλ ∂κ ∂λ h) = c2 (∂κ uκ )(∂λ uλ) − (∂λ uκ )(∂κ uλ ) 2

κ

λ

2

(94) κ

λ

+ (1 − c )u ∂κ (u ∂λ h) + (1 − c )(∂κ u )(u ∂λ h)

− 2c−1 c;h (uκ ∂κ h)(uλ ∂λ h)

+ nc2 qD

+ (1 − c2 )q(S κ ∂κ h) + c2 q;h (S κ ∂κ h) + c2 q;s Sκ S κ .


as

31

From, (94) it follows that the product −c2 ((η−1 )κλ ∂κ ∂λ h)uα on RHS (93) can be expressed −c2 ((η−1 )κλ ∂κ ∂λ h)uα = c2 (∂κ uλ)(∂λ uκ ) − (∂λ uλ )(∂κ uκ ) uα + (c2 − 1) uκ ∂κ (uλ ∂λ h) uα + (c2 − 1)(∂κ uκ )(uλ ∂λ h)uα

(95)

+ 2c−1 c;h (uκ ∂κ h)(uλ ∂λ h)uα

− nc2 qDuα

+ (c2 − 1)q(S κ ∂κ h)uα − c2 q;h (S κ ∂κ h)uα − c2 q;s Sκ S κ uα .

Using (95) to substitute for the term −c2 ((η−1 )κλ ∂κ ∂λ h)uα on RHS (93), we obtain the identity c2 ((η−1 )κλ ∂κ ∂λ uα ) = (c2 − 1)(uκ ∂κ ∂α h) − (∂α uκ )(∂κ h) + 2c−1 c;h (∂α h)(uκ ∂κ h) 1 1 − c2 ǫλαγδ ((η−1 )κλ ∂κ h)uγ ̟ δ + c2 ǫλαγδ ((η−1 )κλ ∂κ uγ )̟ δ H H 1 + c2 ǫλαγδ uγ ((η−1 )κλ ∂κ ̟ δ ) H 2 + c (∂κ uλ)(∂λ uκ ) − (∂λ uλ )(∂κ uκ ) uα + (c2 − 1) uκ ∂κ (uλ ∂λ h) uα + (c2 − 1)(∂κ uκ )(uλ ∂λ h)uα

(96)

+ 2c−1 c;h (uκ ∂κ h)(uλ ∂λ h)uα

− c2 (η−1 )κλ (∂κ h)(∂λ uα ) + c2 (∂α h)(∂κ uκ )

+ c2 q(S κ ∂κ uα )

− c2 q;h (uκ ∂κ h)Sα + c2 q(∂α uκ )Sκ − c2 q(∂κ uκ )Sα + 2c−1 c;s (uκ ∂κ h)Sα . Using (92) and (96) to substitute for the first and second products on RHS (91), and reorganizing the terms, we deduce (where we have added and subtracted (∂κ uκ )(∂α h) on the second and third lines of RHS (97)) 1 ǫλαγδ uγ ((η−1 )κλ ∂κ ̟ δ ) (97) H + (1 − c2 )(uκ ∂κ uα )(uλ ∂λ h) − c2 (η−1 )κλ (∂κ h)(∂λ uα ) + {(∂κ uκ )(∂α h) − (∂α uκ )(∂κ h)}

g uα = c2

+ (c2 − 1)(∂κ uκ )(∂α h) + (c2 − 1)(∂κ uκ )(uλ ∂λ h)uα + (c2 − 1)(∂κ uκ )(uλ∂λ uα )

+ 2c−1 c;h (∂α h)(uκ ∂κ h) + 2c−1 c;h (uκ ∂κ h)(uλ ∂λ h)uα + 2c−1 c;h (uκ ∂κ h)(uλ ∂λ uα ) 1 1 − c2 ǫλαγδ ((η−1 )κλ ∂κ h)uγ ̟ δ + c2 ǫλαγδ ((η−1 )κλ ∂κ uγ )̟ δ H H 2 λ κ λ κ + c (∂κ u )(∂λ u ) − (∂λ u )(∂κ u ) uα − c−1 c;h (g −1)κλ (∂κ h)(∂λ uα ) − cc;s (S κ ∂κ uα )

+ 2c−1 c;s (uκ ∂κ h)Sα − q;h (uκ ∂κ h)Sα + c2 q(∂α uκ )Sκ − c2 q(∂κ uκ )Sα 1 + q(S κ ∂κ uα ) + (1 − c2 )qǫαβγδ S β uγ ̟ δ + (c2 − 1)q(S κ uλ∂λ uκ )uα . H

32

Relativistic Euler

Next, using (15b), we observe the following identity for the first two terms on the second line of RHS (97): (1 − c2 )(uκ ∂κ uα )(uλ ∂λ h) − c2 (η−1 )κλ (∂κ h)(∂λ uα ) = −(g −1 )κλ (∂κ h)(∂λ uα ).

(98)

Moreover, using equation (28), we see that the terms on the third and fourth lines of RHS (97) sum to (c2 − 1)q(∂κ uκ )Sα + 2c−1c;h q(uκ ∂κ h)Sα . In addition, appealing to definition (17) with Vα := ̟α , we obtain the following identity for the first product on RHS (97): c2 H1 ǫλαγδ uγ ((η−1 )κλ ∂κ ̟ δ ) = −c2 H1 vortα (̟). From these facts, (97), and (98), we obtain the following equation: 1 vortα (̟) H 1 1 − c2 ǫλαγδ ((η−1 )κλ ∂κ h)uγ ̟ δ + c2 ǫλαγδ ((η−1 )κλ∂κ uγ )̟ δ H H + {(∂κ uκ )(∂α h) − (∂α uκ )(∂κ h)} − (g −1)κλ (∂κ h)(∂λ uα ) + c2 (∂κ uλ )(∂λ uκ ) − (∂λ uλ )(∂κ uκ ) uα − c−1 c;h (g −1)κλ (∂κ h)(∂λ uα )

g uα = −c2

(99)

− cc;s (S κ ∂κ uα )

+ 2c−1 c;s (uκ ∂κ h)Sα − q;h (uκ ∂κ h)Sα + c2 q(∂α uκ )Sκ 1 + q(S κ ∂κ uα ) + (1 − c2 )qǫαβγδ S β uγ ̟ δ + (c2 − 1)q(S κ uλ ∂λ uκ )uα H κ − q(∂κ u )Sα + 2c−1 c;h q(uκ ∂κ h)Sα . Using definition (23a) to express the product −c2 H1 vortα (̟) on RHS (99) as −c2 H1 Cα + · · · , reorganizing the terms on the RHS of the resulting identity, and raising the α index with η−1 , we arrive at the desired identity (90).

6. Transport equations for the entropy gradient and the modified divergence of the entropy In this section, with the help of the preliminary identities of Lemma 4.1, we derive equations (3b) and (43a). We start by deriving (3b) (more precisely, its η-dual). Proposition 6.1 (Transport equation for the entropy gradient). Assume that (h, s, uα) is a C 2 solution to (24)-(26) + (29). Then the rectangular components the Sα of the entropy gradient vectorfield (see Def. 2.6) verify the following transport equation: 1 ǫαβγδ S β uγ ̟ δ − (S κ ∂κ h)uα + qSκ S κ uα H 1 = −S κ ∂κ uα − ǫαβγδ S β uγ ̟ δ + S κ (uλ∂λ uκ )uα . H

uκ ∂κ Sα = −S κ ∂κ uα −

(100)


33

Proof. From equation (31), the identity (63), (27), and (29), we deduce 1 ǫαβγδ S β uγ ̟ δ + (∂α h)S κ uκ − (S κ ∂κ h)uα − q {Sα S κ uκ − S κ Sκ uα } H (101) 1 = −S κ ∂κ uα − ǫαβγδ S β uγ ̟ δ − (S κ ∂κ h)uα + qS κ Sκ uα , H which yields the first line of (100). To obtain the second line of (100) from the first, we use the identity (64). uκ ∂κ Sα = −S κ ∂κ uα −

We now derive equation (43a). Proposition 6.2 (Transport equation for the modified divergence of the entropy). Assume that (h, s, uα) is a C 2 solution to (24)-(26) + (29). Then the modified divergence of the entropy gradient D, which is defined in (23b), verifies the following transport equation: 1 2 (∂κ S κ )(∂λ uλ ) − (∂λ S κ )(∂κ uλ) + c−2 uκ (∂κ h)(∂λ S λ ) − (∂λ h)(∂κ S λ ) (102) uκ ∂κ D = n n 1 −2 κ + c S (∂κ uλ )(∂λ h) − (∂λ uλ )(∂κ h) n Sκ C κ + nH (1 − c−2 ) αβγδ 1 αβγδ + ǫ Sα uβ (∂γ h)̟δ + ǫ Sα (∂β uγ )̟δ nH nH 2q (θ − θ;h ) κ λ S (S ∂λ uκ ) − S κ (S λ ∂λ uκ ) + nH n (θ;h − θ) 2c−1 c;s c2 q;h q κ λ κ λ + Sκ S (∂λ u ) + Sκ S (∂λ u ) − Sκ S κ (∂λ uλ ) + Sκ S κ (∂λ uλ ). nH n n n Proof. We apply (η−1 )αλ ∂λ to equation (100) (where we use the first equality in (100)) and use the evolution equation (26) and the identity (64) to deduce uκ ∂κ ∂λ S λ = −uκ ∂κ (S λ ∂λ h) − 2(∂λ S κ )(∂κ uλ )

(103)

− S κ ∂κ ∂λ uλ − (S κ ∂κ h)(∂λ uλ) 1 1 1 + ǫαβγδ (∂α h)Sβ uγ ̟δ − ǫαβγδ Sβ uγ (∂α ̟δ ) − ǫαβγδ Sβ (∂α uγ )̟δ H H H κ λ κ λ κ λ + q;h Sκ S (u ∂λ h) + 2qS (u ∂λ Sκ ) + qSκ S (∂λ u ).

Next, we use the evolution equations (24) and (26) to rewrite the third product on RHS (103) as follows: −S κ ∂κ ∂λ uλ = S κ ∂κ (c−2 uλ ∂λ h)

(104)

= c−2 (S κ uλ ∂λ ∂κ h) − 2c−3 c;h (S κ ∂κ h)(uλ ∂λ h) − 2c−3 c;s Sκ S κ (uλ∂λ h) + c−2 (S κ ∂κ uλ )(∂λ h) = uκ ∂κ (c−2 S λ ∂λ h) + c−2 (S κ ∂κ uλ)(∂λ h) − c−2 (uκ ∂κ S λ )(∂λ h) − 2c−3 c;s Sκ S κ (uλ ∂λ h).

34

Relativistic Euler

Next, with the help of the evolution equation (24), we decompose the second and third products on RHS (104) as follows: c−2 (S κ ∂κ uλ )(∂λ h) = c−2 (S κ ∂κ h)(∂λ uλ ) + c−2 (S κ ∂κ uλ )(∂λ h) − (∂λ uλ )(S κ ∂κ h) ,

−c−2 (uκ ∂κ S λ )(∂λ h) = −c−2 (uκ ∂κ h)(∂λ S λ ) + c−2 (uκ ∂κ h)(∂λ S λ ) − (uκ ∂κ S λ )(∂λ h)

(105) (106)

= (∂κ uκ )(∂λ S λ ) + c−2 (uκ ∂κ h)(∂λ S λ ) − (uκ ∂κ S λ )(∂λ h) .

Using (105)-(106) to substitute for the second and third products on RHS (104) and then using the resulting identity to substitute for the third product on RHS (103), we obtain the following equation: uκ ∂κ ∂λ S λ + S λ ∂λ h − c−2 (S λ ∂λ h)

(107)

= (∂κ S κ )(∂λ uλ ) − 2(∂λ S κ )(∂κ uλ)

− (S κ ∂κ h)(∂λ uλ) + c−2 (S κ ∂κ h)(∂λ uλ ) + c−2 (S κ ∂κ uλ )(∂λ h) − (∂λ uλ )(S κ ∂κ h) + c−2 (uκ ∂κ h)(∂λ S λ ) − (uκ ∂κ S λ )(∂λ h) 1 1 1 + ǫαβγδ (∂α h)Sβ uγ ̟δ − ǫαβγδ Sβ uγ (∂α ̟δ ) − ǫαβγδ Sβ (∂α uγ )̟δ H H H + q;h Sκ S κ (uλ ∂λ h) + 2qS κ (uλ ∂λ Sκ ) + qSκ S κ (∂λ uλ ) − 2c−3 c;s Sκ S κ (uλ ∂λ h).

We now multiply both sides of (107) by 1/n, commute the factor of 1/n under the operator uκ ∂κ on LHS (107), use equation (32) (which in particular implies that uκ ∂κ (1/n) = (1/n)∂κ uκ ), and use equation (24) to replace the two factors of uλ ∂λ h on the last line of RHS (107) with −c2 ∂λ uλ , thereby obtaining the following equation:

1 1 λ 1 −2 λ λ u ∂κ (∂λ S ) + (S ∂λ h) − c (S ∂λ h) n n n 2 = (∂κ S κ )(∂λ uλ) − (∂λ S κ )(∂κ uλ) n 1 + c−2 (S κ ∂κ uλ )(∂λ h) − (∂λ uλ )(S κ ∂κ h) n 1 + c−2 (uκ ∂κ h)(∂λ S λ ) − (∂λ h)(uκ ∂κ S λ ) n 1 αβγδ 1 αβγδ 1 αβγδ ǫ (∂α h)Sβ uγ ̟δ − ǫ Sβ uγ (∂α ̟δ ) − ǫ Sβ (∂α uγ )̟δ + nH nH nH 2c−1 c;s c2 q;h q 2q Sκ S κ (∂λ uλ ) − Sκ S κ (∂λ uλ ) + Sκ S κ (∂λ uλ ). + S κ (uλ ∂λ Sκ ) + n n n n κ

(108)


35

Next, we use definitions (17) and (23a) and the identity (27) to obtain the following identity for the second product on the next-to-last line of (108):

−

1 αβγδ 1 αβγδ ǫ Sβ uγ (∂α ̟δ ) = − ǫ Sα uβ (∂γ ̟δ ) nH nH C κ Sκ 1 −2 αβγδ = − c ǫ Sα uβ (∂γ h)̟δ nH nH (θ;h − θ) (θ − θ;h ) κ λ + Sκ S κ (∂λ uλ) + S (S ∂λ uκ ). nH nH

(109)

Using (109) to substitute for the second product on the next-to-last line of (108), using (31) S κ (uλ ∂λ Sκ ) = − 2q S κ (S λ ∂λ uκ ), and to express the first product on the last line of (108) as 2q n n noting that the terms in parentheses on LHS (108) are equal to D (see (23b)), we arrive at the desired evolution equation (102).

7. Transport equation for the vorticity In this section, with the help of the preliminary identities of Lemma 4.1, we derive equation (42). We also derive some preliminary identities that, in the next section, we will use when deriving equation (45b). We collect all of these results in the following proposition. Proposition 7.1 (Transport equation for the vorticity). The rectangular components ̟ α of the vorticity vectorfield defined in (18) verify the following transport equation:

uκ ∂κ ̟ α = ̟ κ ∂κ uα − (∂κ uκ )̟ α − (̟ κ ∂κ h)uα + (θ − θ;h )ǫαβγδ uβ (∂γ h)Sδ + q̟ κ Sκ uα . (110)

Moreover, the following identity holds:

(Lu ̟♭ )α = ̟ κ ∂κ uα + ̟ κ (∂α uκ ) − (∂κ uκ )̟α + (uκ ∂κ uλ)uα ̟ λ + (θ − θ;h )ǫαβγδ uβ (∂γ h)Sδ . (111)

36

Relativistic Euler In addition, the following identity holds: (dLu ̟♭ )αβ = (∂α ̟ κ )(∂κ uβ ) − (∂β ̟ κ )(∂κ uα ) + ̟ κ ∂κ ∂α uβ − ̟ κ ∂κ ∂β uα κ

(112)

κ

+ (∂α ̟ )(∂β uκ ) − (∂β ̟ )(∂α uκ ) − (∂α ∂κ uκ )̟β + (∂β ∂κ uκ )̟α

− (∂κ uκ )(∂α ̟β ) + (∂κ uκ )(∂β ̟α )

+ (∂α uβ )̟ λ(uκ ∂κ uλ ) − (∂β uα )̟ λ(uκ ∂κ uλ )

+ uβ (∂α ̟ λ)(uκ ∂κ uλ ) − uα (∂β ̟ λ )(uκ ∂κ uλ )

+ uβ ̟ λ(∂α uκ )(∂κ uλ ) − uα ̟ λ (∂β uκ )(∂κ uλ ) + uβ ̟ λ(uκ ∂κ ∂α uλ ) − uα ̟ λ (uκ ∂κ ∂β uλ )

+ (θh − θ;h;h )ǫβκγδ uκ (∂α h)(∂γ h)Sδ + (θ;h;h − θh )ǫακγδ uκ (∂β h)(∂γ h)Sδ + (θ;s − θ;h;s )ǫβκγδ uκ Sα (∂γ h)Sδ + (θ;h;s − θ;s )ǫακγδ uκ Sβ (∂γ h)Sδ + (θ − θ;h )ǫβκγδ (∂α uκ )(∂γ h)Sδ + (θ;h − θ)ǫακγδ (∂β uκ )(∂γ h)Sδ + (θ − θ;h )ǫβκγδ uκ (∂α ∂γ h)Sδ + (θ;h − θ)ǫακγδ uκ (∂β ∂γ h)Sδ

+ (θ − θ;h )ǫβκγδ uκ (∂γ h)(∂α Sδ ) + (θ;h − θ)ǫακγδ uκ (∂γ h)(∂β Sδ ).

Finally, the rectangular components vortα (̟) of the vorticity of the vorticity, which is defined by (17) and (18), verify the following transport equation: uκ ∂κ vortα (̟) = vortκ (̟)∂κ uα − (∂κ uκ )vortα (̟) + uα (uκ ∂κ uβ )vortβ (̟)

(113)

+ ǫαβγδ uβ (∂γ ∂κ uκ )̟δ − ǫαβγδ uβ (̟ κ ∂κ ∂γ uδ )

+ ǫαβγδ (uκ ∂κ uβ )(uλ∂λ ̟δ )uγ + ǫαβγδ (uκ ∂κ uβ )̟ λ(∂δ uλ )uγ − ǫαβγδ uβ (∂γ ̟ κ )(∂δ uκ ) − ǫαβγδ uβ (∂γ ̟ κ )(∂κ uδ )

+ ǫαβγδ uβ (∂κ uκ )(∂γ ̟δ )

− ǫαβγδ uβ (∂γ uδ )̟ λ (uκ ∂κ uλ )

+ (θh − θ;h;h )S α (η−1 )κλ (∂κ h)(∂λ h) + (θh − θ;h;h )S α (uκ ∂κ h)(uλ ∂λ h)

+ (θ;h;h − θh )uα (S κ ∂κ h)(uλ∂λ h) + (θ;h;h − θh )((η−1 )ακ ∂κ h)(S λ ∂λ h)

+ (θ;s − θ;h;s )S α (S κ ∂κ h) + (θ;h;s − θ;s )uα Sκ S κ (uλ∂λ h) + (θ;h;s − θ;s )Sκ S κ ((η−1 )αλ ∂λ h)

+ (θ − θ;h )S α (∂κ uκ )(uλ ∂λ h) + (θ;h − θ)(S κ ∂κ uα )(uλ ∂λ h)

+ (θ − θ;h )S α ((η−1 )κλ∂κ ∂λ h) + (θ − θ;h )S α (uκ uλ ∂κ ∂λ h)

+ (θ;h − θ)uα (S κ uλ ∂κ ∂λ h) + (θ;h − θ)(η−1 )αλ (S κ ∂κ ∂λ h)

+ (θ − θ;h )(η−1 )κλ (∂κ h)(∂λ S α ) + (θ − θ;h )(uκ ∂κ h)(uλ ∂λ S α )

+ (θ;h − θ)uα (uκ ∂κ h)(∂λ S λ ) + (θ;h − θ)((η−1 )ακ ∂κ h)(∂λ S λ )

+ (θ;h − θ)((η−1 )ακ ∂κ h)uβ (uλ ∂λ S β ) + (θ − θ;h )uα (η−1 )κλ ∂κ h)uβ (∂λ S β ).


37

Remark 7.1. Note that RHS (113) depends explicitly on two derivatives of u. For this reason, equation (113) is not suitable for obtaining top-order energy estimates for vortα (̟). To overcome this difficulty, we will derive a div-curl-transport system for ̟ that does not lose derivatives; see Prop. 8.2. Proof of Prop. 7.1. We first prove (110). From definition (18) and the Lie differentiation formula (9), we deduce that 1 uκ ∂κ ̟ α − ̟ κ ∂κ uα = Lu ̟ α = − Lu ǫαβγδ uβ (d(Hu♭))γδ 2

(114)

Using (114), the Leibniz rule for Lie derivatives, definition (18), (57), the first identity in (58), (60), and (62), we compute that uκ ∂κ ̟ α = ̟ κ ∂κ uα − (∂κ uκ )̟ α + (uκ ∂κ uλ )uα ̟ λ − (uκ ∂κ uλ )uλ̟ α − θǫαβγδ (uκ ∂κ uβ )Sγ uδ (115) − θ;h ǫαβγδ uβ (∂γ h)Sδ . Using (49), we see that the fourth product on RHS (115) vanishes. Next, we use (28) and (48) to obtain the following identity for the third product on RHS (115): (uκ ∂κ uλ )uα ̟ λ = −(̟ κ ∂κ h)uα + q̟ κ Sκ uα . Next, we use (28) to obtain the following identity for the fifth product on RHS (115): −θǫαβγδ (uκ ∂κ uβ )Sγ uδ = θǫαβγδ (∂β h)Sγ uδ = θǫαβγδ uβ (∂γ h)Sδ . Substituting these two identities for the third and fifth products on RHS (115), we arrive at the desired identity (110). Equation (111) follows from the Lie derivative identity (Lu ̟♭ )α = uκ ∂κ ̟α + ̟ κ ∂α uκ (see (9)), from using (49) to observe the vanishing of the fourth product on RHS (115), and from using the identity for the fifth product on RHS (115) proved in the previous paragraph. (112) then follows from taking the exterior derivative of equation (111) and carrying out straightforward computations. To derive (113), we first use definition (17) to deduce 1 Lu vortα (̟) = − Lu (ǫαβγδ uβ (d̟♭ )γδ ). 2

(116)

Next, we use (116), the Leibniz rule for Lie derivatives, (57), the first equality in (58), and the fact that Lu d̟♭ = dLu ̟♭ to deduce 1 1 Lu vortα (̟) = −(∂κ uκ )vortα (̟) − ǫαβγδ (uκ ∂κ uβ )(d̟♭ )γδ − ǫαβγδ uβ (dLu ̟♭ )γδ . 2 2

(117)

Next, using (71), we express the second product on RHS (117) as follows: 1 (118) − ǫαβγδ (uκ ∂κ uβ )(d̟♭ )γδ = −vortα (̟)(uκ∂κ uβ )uβ + uα (uκ ∂κ uβ )vortβ (̟) 2 − ǫαβγδ (uκ ∂κ uβ )(uλ ∂λ ̟γ )uδ + ǫαβγδ (uκ ∂κ uβ )(uλ ∂γ ̟λ )uδ . Next, using (49), we observe that the first product on RHS (118) vanishes. From this fact, the Lie derivative identity Lu vortα (̟) = uκ ∂κ vortα (̟) − vortκ (̟)∂κ uα (see (9)), (117), and

38

Relativistic Euler

(118), we deduce uκ ∂κ vortα (̟) = vortκ (̟)∂κ uα − (∂κ uκ )vortα (̟) + uα (uκ ∂κ uβ )vortβ (̟)

(119)

+ ǫαβγδ (uκ ∂κ uβ )(uλ ∂γ ̟λ )uδ − ǫαβγδ (uκ ∂κ uβ )(uλ ∂λ ̟γ )uδ 1 − ǫαβγδ uβ (dLu ̟♭ )γδ . 2

Next, we use (51) and the antisymmetry of ǫ··· to express the first product on the second line of RHS (119) as ǫαβγδ (uκ ∂κ uβ )(uλ ∂γ ̟λ )uδ = ǫαβγδ (uκ ∂κ uβ )̟ λ(∂δ uλ )uγ , use the antisymmetry of ǫ··· to express the second product on the second line of RHS (119) −ǫαβγδ (uκ ∂κ uβ )(uλ∂λ ̟γ )uδ = ǫαβγδ (uκ ∂κ uβ )(uλ ∂λ ̟δ )uγ , use (112) to substitute for the factor (dLu ̟♭ )γδ in the last product on RHS (119), and carry out straightforward computations, thereby deducing that uκ ∂κ vortα (̟) = vortκ (̟)∂κ uα − (∂κ uκ )vortα (̟) + uα (uκ ∂κ uβ )vortβ (̟) αβγδ

+ǫ

αβγδ

+ǫ

κ

αβγδ

uβ (∂γ ∂κ u )̟δ − ǫ κ

(120)

κ

uβ (̟ ∂κ ∂γ uδ )

λ

(u ∂κ uβ )(u ∂λ ̟δ )uγ + ǫαβγδ (uκ ∂κ uβ )̟ λ(∂δ uλ )uγ

− ǫαβγδ uβ (∂γ ̟ κ )(∂δ uκ ) − ǫαβγδ uβ (∂γ ̟ κ )(∂κ uδ ) + ǫαβγδ uβ (∂κ uκ )(∂γ ̟δ )

− ǫαβγδ uβ (∂γ uδ )̟ λ(uκ ∂κ uλ )

+ (θ;h;h − θh )ǫαβγδ ǫδκµν uβ uκ (∂γ h)(∂µ h)Sν + (θ;h;s − θ;s )ǫαβγδ ǫδκµν uβ uκ Sγ (∂µ h)Sν + (θ;h − θ)ǫαβγδ ǫδκµν uβ (∂γ uκ )(∂µ h)Sν

+ (θ;h − θ)ǫαβγδ ǫδκµν uβ uκ (∂γ ∂µ h)Sν

+ (θ;h − θ)ǫαβγδ ǫδκµν uβ uκ (∂µ h)(∂γ Sν ).

Finally, we use the identity ǫαβγδ ǫδκµν = (η−1 )να δκβ (η−1 )µγ −(η−1 )να δκγ (η−1 )µβ +(η−1 )νγ δκα (η−1 )µβ − (η−1 )νγ δκβ (η−1 )µα + (η−1 )νβ δκγ (η−1 )µα − (η−1 )νβ δκα (η−1 )µγ to substitute for the five products ǫαβγδ ǫδκµν on RHS (120). Also using (27), (29), and (49), we arrive at the desired identity (113). 8. The transport-div-curl system for the vorticity Our main goal in this section is to derive equations (45a) and (45b). We accomplish this in Prop. 8.2. Before proving the proposition, we will first establish some preliminary identities. 8.1. Preliminary identities. In the next lemma, we derive a large collection of identities that we will use in deriving the transport equation verified by the vectorfield C α defined in (23a). Lemma 8.1 (Identification of the null structure of some terms tied to the div-curl-transport system for the vorticity). Assume that (h, s, uα ) is a C 2 solution to (24)-(26) + (29). Then

39


the following identities hold for some of the terms on the second through sixth lines of RHS (113): ǫαβγδ uβ (∂γ ∂κ uκ )̟δ = −uκ ∂κ c−2 ǫαβγδ uβ (∂γ h)̟δ − 2(∂κ uκ )c−2 ǫαβγδ uβ (∂γ h)̟δ (121a) + c−2 ǫαβγδ (uκ ∂κ uβ )(∂γ h)̟δ

+ c−2 ǫαβγδ uβ (∂γ h)̟ κ (∂δ uκ ) + c−2 (θ − θ;h )(S κ ∂κ h)((η−1 )αλ ∂λ h) + c−2 (θ − θ;h )uα (S κ ∂κ h)(uλ ∂λ h)

+ c−2 (θ;h − θ)S α (uκ ∂κ h)(uλ ∂λ h)

+ c−2 (θ;h − θ)S α (η−1 )κλ(∂κ h)(∂λ h)

+ c−2 ǫαβγδ uβ {(∂κ uκ )(∂γ h) − (∂γ uκ )(∂κ h)} ̟δ + 2c−3 c;s (uκ ∂κ h)ǫαβγδ uβ Sγ ̟δ ,

1 1 (̟ κ ∂κ ̟ α ) − ̟ α (̟ κ ∂κ h) H H 1 α λ κ − u ̟ (̟ ∂κ uλ ) + ǫαβγδ uβ (∂γ h)̟ κ (∂δ uκ ) H − qǫαβγδ uβ Sγ ̟ κ (∂δ uκ ),

−ǫαβγδ uβ ̟ κ ∂κ ∂γ uδ =

ǫαβγδ (uκ ∂κ uβ )(uλ ∂λ ̟δ )uγ = −ǫαβγδ (∂β h)uγ ̟ κ (∂δ uκ ) + (∂κ uκ )ǫαβγδ (∂β h)uγ ̟δ

(121b)

(121c)

+ (θ − θ;h )((η−1 )ακ ∂κ h)(S λ ∂λ h) + (θ − θ;h )uα (uκ ∂κ h)(S λ ∂λ h)

+ (θ;h − θ)S α (uκ ∂κ h)(uλ ∂λ h) + (θ;h − θ)S α (η−1 )κλ (∂κ h)(∂λ h)

+ qǫαβγδ Sβ uγ ̟ κ (∂δ uκ ) − q(∂κ uκ )ǫαβγδ Sβ uγ ̟δ

+ q(θ;h − θ)((η−1 )κα ∂κ h)S λ Sλ + q(θ;h − θ)uα (uκ ∂κ h)S λ Sλ + q(θ − θ;h )S α (S κ ∂κ h),

ǫαβγδ (uκ ∂κ uβ )uγ ̟ λ (∂δ uλ ) = −ǫαβγδ (∂β h)uγ ̟ λ (∂δ uλ ) + qǫαβγδ Sβ uγ ̟ λ(∂δ uλ),

(121d)

−ǫαβγδ uβ (∂γ ̟ κ )∂κ uδ = −ǫαβγδ uβ (∂γ ̟ κ )(∂δ uκ ) − ǫαβγδ uβ (∂γ h)̟ κ (∂δ uκ ) 1 1 − (̟ κ ∂κ ̟ α ) + ̟ α (∂κ ̟ κ ) H H 1 1 α λ κ − ̟ ̟ (u ∂κ uλ) + uα ̟ λ (̟ κ ∂κ uλ ) H H αβγδ κ − qǫ uβ (∂γ u )̟κ Sδ ,

(121e)

40

Relativistic Euler

ǫαβγδ uβ (∂κ uκ )∂γ ̟δ = −(∂κ uκ )vortα (̟), −ǫαβγδ uβ (∂γ uδ )̟ λ(uκ ∂κ uλ ) =

1 α λ κ ̟ ̟ (u ∂κ uλ ). H

(121f)

(121g)

Moreover, we have (θ − θ;h )S α ((η−1 )κλ ∂κ ∂λ h) + (θ − θ;h )S α (uκ uλ ∂κ ∂λ h) = uκ ∂κ (θ;h − θ)S α (∂λ uλ)

(122a)

+ (θh − θ;h;h )S α (uκ ∂κ h)(∂λ uλ ) + (θ − θ;h )(uκ ∂κ S α )(∂λ uλ )

+ (θ;h − θ)S α (∂κ uλ )(∂λ uκ ) + (θ;h − θ)S α (uκ ∂κ uλ )(∂λ h) + (θ;h − θ)S α (∂κ uκ )(uλ ∂λ h) + (θ − θ;h )qS α (∂κ S κ ) + (θ − θ;h )q;h S α (S κ ∂κ h) + (θ − θ;h )q;s S α Sκ S κ ,

(θ;h − θ)uα (S κ uλ ∂κ ∂λ h) = uκ ∂κ (θ;h − θ)uα (S λ ∂λ h)

(122b)

+ (θh − θ;h;h )uα (uκ ∂κ h)(S λ ∂λ h) + (θ − θ;h )(uκ ∂κ uα )(S λ ∂λ h)

+ (θ − θ;h )uα (uκ ∂κ S λ )(∂λ h),

(θ;h − θ)(η−1 )αλ (S κ ∂κ ∂λ h) = uκ ∂κ (θ − θ;h )(η−1 )αλ S β (∂λ uβ )

(122c)

+ (θ;h;h − θh )(uκ ∂κ h)(η−1 )αλ S β (∂λ uβ ) + (θ;h − θ)(uκ ∂κ S β )(η−1 )αλ (∂λ uβ )

+ (θ − θ;h )S β ((η−1 )αλ ∂λ uβ )(uκ ∂κ h) + (θ − θ;h )S β ((η−1 )αλ ∂λ uκ )(∂κ uβ ) + (θ − θ;h )q((η−1)αλ ∂λ S β )Sβ + (θ;h − θ)q;h ((η−1 )αλ ∂λ h)Sκ S κ

+ (θ;h − θ)q;s S α Sκ S κ + 2(θ;h − θ)q((η−1)αλ ∂λ S κ )Sκ .

Identities that reveal null form structure and cancellations. The following identities hold: Q2 := (θ;h − θ)((η−1 )ακ ∂κ h)∂λ S λ + (θ − θ;h )(η−1 )κλ (∂κ h)(∂λ S α ) = (θ;h − θ)(η−1 )ακ (∂κ h)(∂λ S λ ) − (∂λ h)(∂κ S λ ) ,

(123a)

Q4 := (θh − θ;h;h )S α (η−1 )κλ (∂κ h)(∂λ h) + (θh − θ;h;h )S α (uκ ∂κ h)(uλ ∂λ h) + (θh − θ;h;h )S α (uκ ∂κ h)(∂λ uλ ) (123b) = c−2 (θh − θ;h;h )S α (g −1 )κλ (∂κ h)(∂λ h), Q5 := (θ;h;h − θh )((η−1 )ακ ∂κ h)(S λ ∂λ h) + (θ;h;h − θh )(uκ ∂κ h)(η−1 )αλ S β (∂λ uβ ) = (θ;h;h − θh )S β uκ (η−1 )αλ {(∂κ h)(∂λ uβ ) − (∂λ h)(∂κ uβ )} + (θ;h;h − θh )q((η−1)ακ ∂κ h)S λ Sλ ,

(123c)

41


Q6 := c−2 (θ;h − θ)S α (uκ ∂κ h)(uλ∂λ h) + c−2 (θ;h − θ)S α (η−1 )κλ (∂κ h)(∂λ h) + (θ − θ;h )S α (∂κ uκ )(∂λ uλ ) (123d) = c−4 (θ;h − θ)S α (g −1)κλ (∂κ h)(∂λ h), Q7 := (θ;h − θ)S α (uκ ∂κ h)(uλ ∂λ h) + (θ;h − θ)S α (η−1 )κλ (∂κ h)(∂λ h) + (θ;h − θ)S α (uκ ∂κ uλ )(∂λ h) (123e) = (θ;h − θ)qS α (S κ ∂κ h), Q9 := (θ;h − θ)(S κ ∂κ uα )(∂λ uλ ) + (θ;h − θ)(uκ ∂κ uα )(S λ ∂λ h) + (θ − θ;h )((η−1 )κλ ∂κ uα )S β (∂λ uβ ) = (θ;h − θ)S (∂κ uα )(∂λ uλ) − (∂λ uα )(∂κ uλ ) 1 + (θ − θ;h )ǫκβγδ (∂κ uα )Sβ uγ ̟δ + q(θ;h − θ)(uκ ∂κ uα )S λ Sλ , H

(123f)

κ

Q11 := (θ − θ;h )(uκ ∂κ S α )(∂λ uλ) + (θ − θ;h )S β ((η−1 )αλ ∂λ uκ )(∂κ uβ ) = (θ;h − θ)S β (η−1 )ακ (∂κ uβ )(∂λ uλ ) − (∂λ uβ )(∂κ uλ) , Q12 := 2(θ;h − θ)(∂κ uκ )(η−1 )αλ S β (∂λ uβ )

(123g)

(123h)

+ c−2 (θ − θ;h )(S κ ∂κ h)((η−1 )αλ ∂λ h) + (θ;h − θ)(uκ ∂κ S β )((η−1 )αλ ∂λ uβ ), = (θ;h − θ)(η−1 )ακ S β (∂κ uβ )(∂λ uλ ) − (∂λ uβ )(∂κ uλ ) + c−2 (θ − θ;h )(S κ ∂κ h)(η−1 )αλ (∂λ h) + (θ;h − θ)(uκ ∂κ S β )((η−1 )αλ ∂λ uβ ) + c−2 (θ;h − θ)(η−1 )ακ S β uλ {(∂λ uβ )(∂κ h) − (∂κ uβ )(∂λ h)} + qc−2 (θ − θ;h )Sκ S κ ((η−1 )αλ ∂λ h),

Q13 := (θ − θ;h )uα (∂κ uκ )(S λ ∂λ h) + (θ − θ;h )uα (uκ ∂κ uλ )S β (∂λ uβ ) α

β λ

κ

(123i)

κ

= (θ;h − θ)u S u {(∂κ u )(∂λ uβ ) − (∂λ u )(∂κ uβ )} + q(θ − θ;h )uα Sκ S κ (∂λ uλ ),

Q14 := (θ − θ;h )uα (∂κ uκ )(S λ ∂λ h) + (θ − θ;h )uα (uκ ∂κ h)(S λ ∂λ h) + (θ − θ;h )uα(η−1 )κλ(∂κ h)uβ (∂λ S β ) (123j) = n(θ − θ;h )uα (uκ ∂κ h)D + (θ;h − θ)uα uκ (∂κ h)(∂λ S λ ) − (∂λ h)(∂κ S λ ) ,

42

Relativistic Euler

Q15 := (∂κ uκ )ǫαβγδ (∂β h)uγ ̟δ + c−2 ǫαβγδ (uκ ∂κ uβ )(∂γ h)̟δ −2

αβγδ

= c qǫ

(123k)

Sβ (∂γ h)̟δ ,

Q16 := −c−2 uα ǫσβγδ (uκ ∂κ uσ )uβ (∂γ h)̟δ −2

α κβγδ

= −c qu ǫ

(123l)

Sκ uβ (∂γ h)̟δ ,

Q18 := (θ − θ;h )((η−1 )ακ ∂κ h)(S λ ∂λ h) + (θ;h − θ)((η−1 )ακ ∂κ h)uβ (uλ ∂λ S β )

(123m)

Q19 := (θ;h − θ)uα (uκ ∂κ uσ )S σ (∂λ uλ) + c−2 (θ − θ;h )uα(S κ ∂κ h)(uλ ∂λ h)

(123n)

Q20 := (θ − θ;h )S β ((η−1 )αλ ∂λ uβ )(uκ ∂κ h) + (θ − θ;h )(uκ ∂κ h)(uλ ∂λ S α ) = 0,

(123o)

κ

−1 αλ

= q(θ − θ;h )Sκ S ((η ) ∂λ h),

α

κ

λ

= q(θ;h − θ)u Sκ S (∂λ u ),

Q21 := (θ;h − θ)uα (uκ ∂κ uβ )uβ (S λ ∂λ h) = 0.

(123p)

Proof. • Proof of (121a): We first use equation (24) to deduce ǫαβγδ uβ (∂γ ∂κ uκ )̟δ = −ǫαβγδ uβ ∂γ (c−2 uκ ∂κ h) ̟δ αβγδ

= −ǫ

(124)

−2 κ

uβ (c u ∂κ ∂γ h)̟δ

+ 2c−3 c;h (uκ ∂κ h)ǫαβγδ uβ (∂γ h)̟δ + 2c−3 c;s (uκ ∂κ h)ǫαβγδ uβ Sγ ̟δ

− c−2 ǫαβγδ uβ (∂γ uκ )(∂κ h)̟δ . Next, we rewrite the first term on RHS (124) as a perfect uκ ∂κ derivative plus error terms, thereby obtaining, with the help of (26), the following identity: ǫαβγδ uβ (∂γ ∂κ uκ )̟δ = −uκ ∂κ ǫαβγδ c−2 uβ (∂γ h)̟δ + c−2 ǫαβγδ (uκ ∂κ uβ )(∂γ h)̟δ (125) + c−2 ǫαβγδ uβ (∂γ h)(uκ ∂κ ̟δ ) − c−2 ǫαβγδ uβ (∂γ uκ )(∂κ h)̟δ

+ 2c−3 c;s (uκ ∂κ h)ǫαβγδ uβ Sγ ̟δ .

43


Using equation (110) to substitute for the factor uκ ∂κ ̟δ in the third product on RHS (125), we deduce ǫαβγδ uβ (∂γ ∂κ uκ )̟δ = −uκ ∂κ c−2 ǫαβγδ uβ (∂γ h)̟δ + c−2 ǫαβγδ (uκ ∂κ uβ )(∂γ h)̟δ (126) + c−2 ǫαβγδ uβ (∂γ h)̟ κ ∂κ uδ − c−2 (∂κ uκ )ǫαβγδ uβ (∂γ h)̟δ

+ c−2 (θ − θ;h )ǫαβγδ ǫδνκλ uβ uν (∂γ h)(∂κ h)Sλ − c−2 ǫαβγδ uβ (∂γ uκ )(∂κ h)̟δ

+ 2c−3 c;s (uκ ∂κ h)ǫαβγδ uβ Sγ ̟δ .

Next, using the identity (67), we express the third product on RHS (126) as follows: c−2 ǫαβγδ uβ (∂γ h)(̟ κ ∂κ uδ ) = c−2 ǫαβγδ uβ (∂γ h)̟ κ (∂δ uκ ).

(127)

Next, using the identity −ǫαβγδ ǫδνκλ = (η−1 )λβ δνα (η−1 )κγ − (η−1 )λβ δνγ (η−1 )κα

(128)

+ (η−1 )λγ δνβ (η−1 )κα − (η−1 )λγ δνα (η−1 )κβ + (η−1 )λα δνγ (η−1 )κβ − (η−1 )λα δνβ (η−1 )κγ

and equations (27), (29), and (48), we express the third-from-last product on RHS (126) as follows: c−2 (θ − θ;h )ǫαβγδ ǫδνκλ uβ uν (∂γ h)(∂κ h)Sλ = c−2 (θ − θ;h )(S κ ∂κ h)((η−1 )αλ ∂λ h)

(129)

+ c−2 (θ − θ;h )uα (S κ ∂κ h)(uλ ∂λ h)

+ c−2 (θ;h − θ)S α (uκ ∂κ h)(uλ ∂λ h)

+ c−2 (θ;h − θ)S α (η−1 )κλ (∂κ h)(∂λ h). Using (127) and (129) to substitute for the relevant products on RHS (126), adding and subtracting c−2 ǫαβγδ uβ ̟δ (∂κ uκ )(∂γ h), and reorganizing the terms, we arrive at the desired identity (121a). • Proof of (121b): We first use (68) to deduce 1 α β 1 α β αβγδ αβγδ αβγδ κ κ ̟ u − u ̟ −ǫ (∂γ h)uδ + qǫ Sγ u δ . ǫ uβ ̟ ∂κ ∂γ uδ = uβ ̟ ∂κ H H

(130)

The desired identity (121b) now follows from (19), (29), (48), (49), (51), (67), (130), and straightforward calculations. • Proof of (121c): We first use equation (28) to substitute for the factor uκ ∂κ uβ on LHS (121c), thereby obtaining the identity ǫαβγδ (uκ ∂κ uβ )(uλ ∂λ ̟δ )uγ = −ǫαβγδ (∂β h)(uλ ∂λ ̟δ )uγ + qǫαβγδ Sβ (uλ ∂λ ̟δ )uγ .

(131)

We then use equation (110) to substitute for the two factors of uλ∂λ ̟δ on RHS (131), which yields the identity ǫαβγδ (uκ ∂κ uβ )(uλ∂λ ̟δ )uγ = −ǫαβγδ (∂β h)uγ (̟ κ ∂κ uδ ) + (∂κ uκ )ǫαβγδ (∂β h)uγ ̟δ

(132)

+ (θ;h − θ)ǫαβγδ ǫδνκλ (∂β h)uν (∂κ h)Sλ uγ + qǫαβγδ Sβ uγ (̟ κ ∂κ uδ ) − q(∂κ uκ )ǫαβγδ Sβ uγ ̟δ + q(θ − θ;h )ǫαβγδ ǫδνκλ uν (∂κ h)Sλ Sβ uγ .

44

Relativistic Euler

Next, we use the identity (67) to express the first and fourth products on RHS (132) as follows: −ǫαβγδ (∂β h)(̟ λ ∂λ uδ )uγ = −ǫαβγδ (∂β h)̟ λ (∂δ uλ )uγ ,

(133)

qǫαβγδ Sβ uγ (̟ κ ∂κ uδ ) = qǫαβγδ Sβ uγ ̟ κ (∂δ uκ ).

(134)

We then use the identity ǫαβγδ ǫδνκλ = (η−1 )λβ δνγ (η−1 )κα − (η−1 )λβ δνα (η−1 )κγ +

(η−1 )λγ δνα (η−1 )κβ

−

(η−1 )λγ δνβ (η−1 )κα

(135) +

(η−1 )λα δνβ (η−1 )κγ

−

(η−1 )λα δνγ (η−1 )κβ

and equations (27) and (29) to express the third product on RHS (132) as follows: (θ;h − θ)ǫαβγδ ǫδνκλ (∂β h)uν (∂κ h)Sλ uγ = (θ − θ;h )((η−1 )ακ ∂κ h)(S λ ∂λ h)

(136)

+ (θ − θ;h )uα (uκ ∂κ h)(S λ ∂λ h)

+ (θ;h − θ)S α (uκ ∂κ h)(uλ ∂λ h)

+ (θ;h − θ)S α (η−1 )κλ (∂κ h)(∂λ h). Similarly, we express the last product on RHS (132) as follows: q(θ − θ;h )ǫαβγδ ǫδνκλ uν (∂κ h)Sλ Sβ uγ = q(θ;h − θ)((η−1 )κα ∂κ h)S λ Sλ + q(θ;h − θ)uα (uκ ∂κ h)S λ Sλ (137) + q(θ − θ;h )S α (S κ ∂κ h). Using (133)-(134) and (136)-(137) to substitute for the relevant products on RHS (132), we arrive at the desired identity (121c). • Proof of (121d): (121d) follows easily from using equation (28) to substitute for the factor uκ ∂κ uβ on the LHS. • Proof of (121e): We first use the identity (63) to deduce 1 αβγδ ǫ ǫκδθλ uθ ̟ λ uβ (∂γ ̟ κ ) H + ǫαβγδ uβ (∂γ ̟ κ )uκ (∂δ h) − qǫαβγδ uβ (∂γ ̟ κ )uκ Sδ .

ǫαβγδ uβ (∂γ ̟ κ )∂κ uδ = ǫαβγδ uβ (∂γ ̟ κ )(∂δ uκ ) +

(138)

Next, we note the identity ǫαβγδ ǫκδθλ = −ǫαβγδ ǫκλθδ = δκα δλβ δθγ − δκβ δλα δθγ

+ δκβ δλγ δθα − δκα δλγ δθβ + δκγ δλα δθβ − δκγ δλβ δθα ,

(139)

45


which, in view of (29), (48), and (51), allows us to express the second product on RHS (138) as follows: 1 1 1 αβγδ ǫ ǫκδθλ uθ ̟ λ uβ (∂γ ̟ κ ) = − ̟ αuλ (uκ ∂κ ̟ λ) + uα uλ (̟ κ ∂κ ̟ λ ) (140) H H H 1 1 + (̟ κ ∂κ ̟ α ) − ̟ α (∂κ ̟ κ ) H H 1 1 α λ κ = ̟ ̟ (u ∂κ uλ ) − uα ̟ λ (̟ κ ∂κ uλ ) H H 1 1 + (̟ κ ∂κ ̟ α ) − ̟ α (∂κ ̟ κ ). H H Using (140) to substitute for the second product on RHS (138), and using (51) to express the third product on RHS (138) as ǫαβγδ uβ (∂γ ̟ κ )uκ (∂δ h) = −ǫαβγδ uβ ̟ κ (∂γ uκ )(∂δ h) = ǫαβγδ uβ (∂γ h)̟ κ (∂δ uκ ) and the last product on RHS (138) as −qǫαβγδ uβ (∂γ ̟ κ )uκ Sδ = qǫαβγδ uβ (∂γ uκ )̟κ Sδ , we arrive at the desired identity (121e). • Proof of (121f): (121f) follows from definition (17) with Vδ := ̟δ . • Proof of (121g): (121g) is a straightforward consequence of (68), (29), and (48). • Proof of (122a): We first use (87) to express LHS (122a) as follows: (θ − θ;h )S α (η−1 )κλ ∂κ ∂λ h + (θ − θ;h )S α (uκ uλ ∂κ ∂λ h) α

λ

κ

α

λ

(141)

κ

= (θ;h − θ)S (u ∂λ ∂κ u ) + (θ;h − θ)S (∂κ u )(∂λ u )

+ (θ;h − θ)S α (uκ ∂κ uλ )(∂λ h) + (θ;h − θ)S α (∂κ uκ )(uλ∂λ h)

+ (θ − θ;h )qS α (∂κ S κ ) + (θ − θ;h )q;h S α (S κ ∂κ h) + (θ − θ;h )q;s S α Sκ S κ . Next, with the help of equation (26), we rewrite the first product on RHS (141) as follows: (θ;h − θ)S α uλ ∂λ ∂κ uκ = uκ ∂κ (θ;h − θ)S α (∂λ uλ ) (142) + (θh − θ;h;h )S α (uκ ∂κ h)(∂λ uλ ) + (θ − θ;h )(uκ ∂κ S α )(∂λ uλ ).

Using (142) to substitute for the first product on RHS (141), we arrive at the desired identity (122a). • Proof of (122b): (122b) is a straightforward consequence of equation (26). • Proof of (122c): We first differentiate equation (64) with (η−1 )αλ ∂λ and then multiply the resulting identity by (θ;h − θ) to obtain (θ;h − θ)(η−1 )αλ S κ ∂κ ∂λ h = (θ − θ;h )(η−1 )αλ S β uκ ∂κ ∂λ uβ + (θ − θ;h )((η−1 )αλ ∂λ S κ )(∂κ h)

(143)

+ (θ − θ;h )((η−1 )αλ ∂λ S β )(uκ ∂κ uβ ) + (θ − θ;h )S β ((η−1 )αλ ∂λ uκ )(∂κ uβ )

+ (θ;h − θ)q;h ((η−1 )αλ ∂λ h)Sκ S κ + (θ;h − θ)q;s S α Sκ S κ + 2(θ;h − θ)q((η−1 )αλ ∂λ S κ )Sκ .

46

Relativistic Euler

Next, with the help of equation (26), we rewrite the first product on RHS (143) as follows: (θ − θ;h )(η−1 )αλ S β uκ ∂κ ∂λ uβ = uκ ∂κ (θ − θ;h )(η−1 )αλ S β (∂λ uβ ) (144) + (θ;h;h − θh )(uκ ∂κ h)(η−1 )αλ S β (∂λ uβ )

+ (θ;h − θ)(uκ ∂κ S β )(η−1 )αλ (∂λ uβ ).

Next, we use equations (28) and (50) to express the sum of the second and third products on RHS (143) as follows: (θ − θ;h )((η−1 )αλ ∂λ S κ )(∂κ h) + (θ − θ;h )((η−1 )αλ ∂λ S β )(uκ ∂κ uβ )

(145)

= (θ;h − θ)((η−1 )αλ ∂λ S β )uβ (uκ ∂κ h) + (θ − θ;h )q((η−1)αλ ∂λ S β )Sβ

= (θ − θ;h )S β ((η−1 )αλ ∂λ uβ )(uκ ∂κ h) + (θ − θ;h )q((η−1)αλ ∂λ S β )Sβ .

Using (144) to substitute for the first product on RHS (143) and using (145) to substitute for the second and third products on RHS (143), we arrive at the desired identity (122c). • Proof of (123a): We simply use (47) to express the second product on LHS (123a) as follows: (θ − θ;h )(η−1 )κλ (∂κ h)(∂λ S α ) = (θ − θ;h )(η−1 )ακ (∂λ h)(∂κ S λ ). • Proof of (123b): We use equation (24) to substitute for the last factor ∂λ uλ on LHS (123b) and then appeal to equation (15b). • Proof of (123c): We first use (64) to express the first product on LHS (123c) as follows: (θ;h;h − θh )((η−1 )ακ ∂κ h)(S λ ∂λ h) = (θh − θ;h;h )((η−1 )ακ ∂κ h)(uλ ∂λ uβ )S β

(146)

+ (θ;h;h − θh )q((η−1 )ακ ∂κ h)S λ Sλ .

Using (146) to substitute for the first product on LHS (123c), we arrive at the desired identity. • Proof of (123d): To prove (123d), we first use equation (24) to express the last product on LHS (123d) as follows: (θ − θ;h )S α (∂κ uκ )(∂λ uλ ) = c−4 (θ − θ;h )S α (uκ ∂κ h)(uλ ∂λ h). Using this identity to substitute for the last product on LHS (123d) and appealing to equation (15b), we arrive at the desired identity. • Proof of (123e): We first use (28) to substitute for the factor uκ ∂κ uλ in the last product on LHS (123e), thereby obtaining the following identity: (θ;h − θ)S α (uκ ∂κ uλ)(∂λ h) = (θ − θ;h )S α (η−1 )κλ (∂κ h)(∂λ h) + (θ − θ;h )S α (uκ ∂κ h)(uλ ∂λ h) (147) + (θ;h − θ)qS α (S κ ∂κ h).

Using (147) to substitute for the last product on LHS (123e), we arrive at the desired identity. • Proof of (123f): We first use (47), (50), and the first equality in (100) to express the last product on LHS (123f) as follows: (θ − θ;h )((η−1 )κλ ∂κ uα )S β (∂λ uβ ) = (θ;h − θ)(∂κ uα )(uβ ∂β S κ )

(148)

1 (θ − θ;h )ǫκβγδ (∂κ uα )Sβ uγ ̟δ H + (θ − θ;h )(uκ ∂κ uα )(S λ ∂λ h) + q(θ;h − θ)(uκ ∂κ uα )S λ Sλ .

= (θ − θ;h )(∂κ uα )(S λ ∂λ uκ ) +

47


Using (148) to substitute for the last product on LHS (123f), we arrive at the desired identity. • Proof of (123g): We use (47) and (50) to express the first product on LHS (123g) as follows: (θ − θ;h )(uκ ∂κ S α )(∂λ uλ ) = (θ − θ;h )(uκ (η−1 )αβ ∂β Sκ )(∂λ uλ) β

−1 ακ

(149)

λ

= (θ;h − θ)S ((η ) ∂κ uβ )(∂λ u ).

Using (149) to substitute for the first product on LHS (123g), we conclude the desired identity. • Proof of (123h): To prove (123h), we first note the following identity, which we derive below: c−2 (θ − θ;h )(S κ ∂κ h)((η−1 )αλ ∂λ h) + (θ;h − θ)(uκ ∂κ S β )((η−1)αλ ∂λ uβ )

(150)

= (θ − θ;h )S κ (∂β uκ )((η−1 )αλ ∂λ uβ )

+ c−2 (θ;h − θ)(uλ ∂λ uβ )S β ((η−1 )ακ ∂κ h) + c−2 (θ − θ;h )((η−1 )ακ ∂κ uβ )S β (uλ∂λ h)

+ (θ − θ;h )((η−1 )ακ ∂κ uβ )S β (∂λ uλ ) + qc−2 (θ − θ;h )S β Sβ ((η−1 )ακ ∂κ h).

Using (150) to substitute for the sum of the second and third products on LHS (123h), we conclude the desired identity (123h). It remains for us to prove (150). To proceed, we first use (47) and (50) to express the second product on LHS (150) as follows: (θ;h − θ)(uκ ∂κ S β )((η−1 )αλ ∂λ uβ ) = (θ;h − θ)(uκ ∂β Sκ )((η−1)αλ ∂λ uβ )

(151)

= (θ − θ;h )(S β ∂λ uβ )((η−1 )ακ ∂κ uλ )

= (θ − θ;h )((η−1 )ακ ∂κ uβ )S β (∂λ uλ )

+ (θ;h − θ)((η−1 )ακ ∂κ uβ )S β (∂λ uλ )

+ (θ − θ;h )(S β ∂λ uβ )((η−1 )ακ ∂κ uλ ),

where to obtain the last equality, we have added and subtracted (θ;h −θ)((η−1 )ακ ∂κ uβ )S β (∂λ uλ ). Next, we use equation (24) to substitute for the factor ∂λ uλ in the first product on RHS (151), which allows us to express the product as follows: (θ − θ;h )((η−1 )ακ ∂κ uβ )S β (∂λ uλ ) = c−2 (θ;h − θ)((η−1 )ακ ∂κ uβ )S β (uλ ∂λ h)

(152)

= c−2 (θ;h − θ)(uλ ∂λ uβ )S β ((η−1 )ακ ∂κ h)

+ c−2 (θ − θ;h )(uλ ∂λ uβ )S β ((η−1 )ακ ∂κ h)

+ c−2 (θ;h − θ)((η−1 )ακ ∂κ uβ )S β (uλ ∂λ h),

where to obtain the last equality, we have added and subtracted c−2 (θ−θ;h )(uλ∂λ uβ )S β ((η−1 )ακ ∂κ h). Next, we use equation (64) to express the first product on RHS (152) as follows: c−2 (θ;h − θ)(uλ ∂λ uβ )S β ((η−1 )ακ ∂κ h) = c−2 (θ − θ;h )(S κ ∂κ h)((η−1 )αλ ∂λ h)

+ qc−2 (θ;h − θ)Sκ S κ ((η−1 )αλ ∂λ h).

(153)

48

Relativistic Euler

Combining (151)-(153), we find that c−2 (θ − θ;h )(S κ ∂κ h)((η−1 )αλ ∂λ h)

(154)

= (θ;h − θ)(uκ ∂κ S β )((η−1 )αλ ∂λ uβ )

+ (θ − θ;h )((η−1 )ακ ∂κ uβ )S β (∂λ uλ ) + (θ;h − θ)(S β ∂λ uβ )((η−1 )ακ ∂κ uλ)


+ qc−2 (θ − θ;h )Sκ S κ ((η−1 )αλ ∂λ h).

Using (154) to substitute for the first product on LHS (150), we deduce c−2 (θ − θ;h )(S κ ∂κ h)((η−1 )αλ ∂λ h) + (θ;h − θ)(uκ ∂κ S β )((η−1)αλ ∂λ uβ )

(155)

= 2(θ;h − θ)(uκ ∂κ S β )((η−1 )αλ ∂λ uβ )

+ (θ − θ;h )((η−1 )ακ ∂κ uβ )S β (∂λ uλ ) + (θ;h − θ)(S β ∂λ uβ )(η−1 )ακ (∂κ uλ)


+ qc−2 (θ − θ;h )Sκ S κ ((η−1 )αλ ∂λ h).

Next, we use (47) and (50) to express the first product on RHS (155) as 2(θ;h − θ)(uκ ∂κ S β )((η−1 )αλ ∂λ uβ ) = 2(θ;h − θ)(uκ ∂β Sκ )((η−1 )αλ ∂λ uβ )

(156)

= 2(θ − θ;h )(S κ ∂β uκ )((η−1 )αλ ∂λ uβ ).

Using (156) to substitute for the first product on RHS (155), we arrive at the desired identity (150). This completes the proof of (123h). • Proof of (123i): We use the identity (64) to substitute for the factor S λ ∂λ h on LHS (123i), thus obtaining (θ − θ;h )uα (∂κ uκ )(S λ ∂λ h) = (θ;h − θ)uα (∂κ uκ )S β (uλ ∂λ uβ ) + (θ − θ;h )quα (∂κ uκ )S λ Sλ . (157) Using (157) to substitute for the first product on LHS (123i), we arrive at the desired identity. • Proof of (123j): We first use equation (47) to express the last product on LHS (123j) as follows: (θ − θ;h )uα (η−1 )κλ (∂κ h)uβ (∂λ S β ) = (θ − θ;h )uα (∂κ h)(uβ ∂β S κ )

(158)

= (θ − θ;h )uα (uκ ∂κ h)(∂λ S λ )

+ (θ;h − θ)uα uλ {(∂λ h)(∂κ S κ ) − (∂κ h)(∂λ S κ )} ,

where to obtain the second equality in (158), we added and subtracted (θ;h −θ)uα (uκ ∂κ h)(∂λ S λ ). Next, we use definition (23b) to algebraically substitute for the factor ∂λ S λ in the first product on RHS (158), which yields the identity (θ − θ;h )uα (uκ ∂κ h)(∂λ S λ ) = n(θ − θ;h )uα (uκ ∂κ h)D + (θ;h − θ)uα (uκ ∂κ h)(S λ ∂λ h) + c−2 (θ − θ;h )uα (uκ ∂κ h)(S λ ∂λ h).

(159)


49

Next, we use equation (24) to substitute for the factor uκ ∂κ h in the last product on RHS (159), which yields the identity (θ − θ;h )uα (uκ ∂κ h)(∂λ S λ ) = n(θ − θ;h )uα (uκ ∂κ h)D + (θ;h − θ)uα (uκ ∂κ h)(S λ ∂λ h)

(160)

+ (θ;h − θ)uα (∂κ uκ )(S λ ∂λ h).

Substituting RHS (160) for the first product on RHS (158) and then using the resulting identity to substitute for the last product on LHS (123j), we arrive at the desired identity (123j). • Proof of (123k): We first use equation (28) to substitute for the factor of uκ ∂κ uβ in the second product on LHS (123k), which yields the identity (∂κ uκ )ǫαβγδ (∂β h)uγ ̟δ + c−2 ǫαβγδ (uκ ∂κ uβ )(∂γ h)̟δ

(161)

= (∂κ uκ )ǫαβγδ (∂β h)uγ ̟δ − c−2 (uκ ∂κ h)ǫαβγδ uβ (∂γ h)̟δ + c−2 qǫαβγδ Sβ (∂γ h)̟δ .

Using equation (24) to substitute for the factor ∂κ uκ in the first product on RHS (161) and taking into account the antisymmetry of ǫ, we see that the first and second products on RHS (161) cancel, which yields the desired identity (123k). • Proof of (123l): We simply use equation (28) to substitute for the factor uκ ∂κ uσ on LHS (123l). • Proof of (123m): We simply multiply equation (65) by (θ;h − θ)(η−1 )ακ ∂κ h. • Proof of (123n): We use equation (64) to substitute for the factor (uκ ∂κ uσ )S σ in the first product on LHS (123n) and equation (24) to substitute for the factor uλ ∂λ h in the second product on LHS (123n). • Proof of (123o): We simply use equation (31) to substitute for the factor uλ ∂λ S α in the second product on LHS (123o). • Proof of (123p): (123p) follows from (49).

8.2. The div-curl-transport system. Armed with Lemma 8.1, we now derive the main result of this section. Proposition 8.2 (The div-curl-transport system for the vorticity). Assume that (h, s, uα) is a C 2 solution to (24)-(26) + (29). Then the divergence of the vorticity vectorfield ̟ α defined in (18) verifies the following identity: ∂α ̟ α = −̟ κ ∂κ h + 2q̟ κSκ .

(162)

50

Relativistic Euler

Moreover, the rectangular components C α of the modified vorticity of the vorticity, which is defined in (23a), verify the following transport equation: uκ ∂κ C α = C κ ∂κ uα − 2(∂κ uκ )C α + uα (uκ ∂κ uλ)C λ

(163)

− 2ǫαβγδ uβ (∂γ ̟ κ )(∂δ uκ ) + (θ;h − θ) (η−1 )ακ + 2uα uκ (∂κ h)(∂λ S λ ) − (∂λ h)(∂κ S λ ) + n(θ − θ;h )uα (uκ ∂κ h)D

+ (θ − θ;h )qS α ∂κ S κ + (θ;h − θ)q((η−1 )αλ ∂λ S κ )Sκ + Q(C α ) + L(C α ) , where Q(C α ) is the linear combination of null forms defined by Q(C α ) := −c−2 ǫκβγδ (∂κ uα )uβ (∂γ h)̟δ + (c−2 + 2)ǫαβγδ uβ (∂γ h)̟ κ (∂δ uκ )

(164)

+ c−2 ǫαβγδ uβ ̟δ {(∂κ uκ )(∂γ h) − (∂γ uκ )(∂κ h)} + (θ;h;h − θh ) + c−2 (θ − θ;h ) (η−1 )αλ S β uκ {(∂κ h)(∂λ uβ ) − (∂λ h)(∂κ uβ )} + (θ;h − θ)S κ uλ {(∂κ uα )(∂λ h) − (∂λ uα )(∂κ h)} + (θ;h − θ) (η−1 )ακ + uα uκ S β (∂κ uβ )(∂λ uλ ) − (∂λ uβ )(∂κ uλ ) + (θ;h − θ)S α (∂κ uλ )(∂λ uκ ) − (∂κ uκ )(∂λ uλ ) + (θ;h − θ)S κ (∂κ uα )(∂λ uλ ) − (∂λ uα )(∂κ uλ ) + S α c−2 (θh − θ;h;h ) + c−4 (θ;h − θ) (g −1 )κλ (∂κ h)(∂λ h),

and L(C α ) , which is at most linear in the derivatives of the solution variables, is defined by 2 2q κ ̟ Sκ ̟ α − ̟ α (̟ κ ∂κ h) H H −3 κ + 2c c;s (u ∂κ h)ǫαβγδ uβ Sγ ̟δ

L(C α ) :=

(165)

− 2qǫαβγδ uβ Sγ ̟ κ (∂δ uκ ) − q(∂κ uκ )ǫαβγδ Sβ uγ ̟δ 1 + (θ − θ;h )ǫκβγδ (∂κ uα )Sβ uγ ̟δ + c−2 qǫαβγδ Sβ (∂γ h)̟δ − c−2 quα ǫκβγδ Sκ uβ (∂γ h)̟δ H + q(θ;h − θ)Sκ S κ (uλ ∂λ uα ) + q(θ;h − θ)uα Sκ S κ (uλ ∂λ h) + (θ;h;s − θ;s )uα Sκ S κ (uλ ∂λ h) + (θ;s − θ;h;s )S α (S κ ∂κ h) + (θ − θ;h )q;h S α (S κ ∂κ h)

+ q(θ;h;h − θh )Sκ S κ ((η−1 )αλ ∂λ h) + (θ;h;s − θ;s )Sκ S κ ((η−1 )αλ ∂λ h)

+ qc−2 (θ − θ;h )Sκ S κ ((η−1 )αλ ∂λ h) + (θ;h − θ)q;h Sκ S κ ((η−1 )αλ ∂λ h). Proof. • Proof of (162): First, from definition (18) and the antisymmetry of ǫκλγδ , we deduce ∂κ ̟ κ = −ǫκλγδ (∂κ uλ )∂γ (Huδ )

(166)


51

Next, using (62), we deduce that RHS (166) = ̟ λ (uκ ∂κ uλ ) − ̟ κ uλ (∂κ uλ ) − θǫκλγδ (∂κ uλ )Sγ uδ .

(167)

Using (49), we see that the second product on RHS (167) vanishes. Moreover, using equation (28) and the identity (48), we can express the first product on RHS (167) as follows: ̟ λuκ ∂κ uλ = −̟ κ ∂κ h + q̟ κ Sκ .

(168)

In addition, using definition (20) and the identity (69), we can express the last product on RHS (167) as follows: −θǫκλγδ (∂κ uλ )Sγ uδ = q̟ κ Sκ .

(169)

Combining these calculations, we arrive at the desired identity (162). • Proof of (163): The proof is a series of lengthy calculations in which we observe many cancellations. We start by using (121a)-(121g) to substitute for all of the terms on the second through sixth lines of RHS (113) except for the term −ǫαβγδ uβ (∂γ ̟ κ )(∂δ uκ ) from the fourth line, which we leave as is. We also use (162) to express the fourth product on RHS (121e) as H1 ̟ α (∂κ ̟ κ ) = − H1 ̟ α(̟ κ ∂κ h) + 2q ̟ α̟ κ Sκ , and we use (122a)-(122c) to substitute for H the four products (which depend on the second derivatives of h) on the fifth-to-last and the fourth-to-last lines of RHS (113), thereby obtaining the following equation (where at this stage in the argument, we have simply performed a term-by-term substitution and have not yet organized the terms): uκ ∂κ vortα (̟) = vortκ (̟)∂κ uα − (∂κ uκ )vortα (̟) + uα (uκ ∂κ uβ )vortβ (̟) − uκ ∂κ c−2 ǫαβγδ uβ (∂γ h)̟δ − 2(∂κ uκ )c−2 ǫαβγδ uβ (∂γ h)̟δ + c−2 ǫαβγδ (uκ ∂κ uβ )(∂γ h)̟δ

+ c−2 ǫαβγδ uβ (∂γ h)̟ κ (∂δ uκ ) + c−2 (θ − θ;h )(S κ ∂κ h)((η−1 )αλ ∂λ h)

+ c−2 (θ − θ;h )uα (S κ ∂κ h)(uλ ∂λ h)

+ c−2 (θ;h − θ)S α (uκ ∂κ h)(uλ ∂λ h)

+ c−2 (θ;h − θ)S α (η−1 )κλ (∂κ h)(∂λ h)

+ c−2 ǫαβγδ uβ {(∂κ uκ )(∂γ h) − (∂γ uκ )(∂κ h)} ̟δ

+ 2c−3 c;s (uκ ∂κ h)ǫαβγδ uβ Sγ ̟δ 1 1 + (̟ κ ∂κ ̟ α) − (̟ κ ∂κ h)̟ α H H 1 α λ κ − u ̟ (̟ ∂κ uλ ) + ǫαβγδ uβ (∂γ h)̟ κ (∂δ uκ ) H − qǫαβγδ uβ Sγ ̟ κ (∂δ uκ )

− ǫαβγδ (∂β h)uγ ̟ κ (∂δ uκ ) + (∂κ uκ )ǫαβγδ (∂β h)uγ ̟δ



(170)

52

Relativistic Euler + qǫαβγδ Sβ uγ ̟ κ (∂δ uκ ) − q(∂κ uκ )ǫαβγδ Sβ uγ ̟δ

+ q(θ;h − θ)((η−1 )κα ∂κ h)S λ Sλ + q(θ;h − θ)uα (uκ ∂κ h)S λ Sλ + q(θ − θ;h )S α (S κ ∂κ h) − ǫαβγδ (∂β h)uγ ̟ λ (∂δ uλ ) + qǫαβγδ Sβ uγ ̟ λ(∂δ uλ) − ǫαβγδ uβ (∂γ ̟ κ )(∂δ uκ )

− ǫαβγδ uβ (∂γ ̟ κ )(∂δ uκ ) − ǫαβγδ uβ (∂γ h)̟ κ (∂δ uκ ) 1 2q 1 − (̟ κ ∂κ ̟ α ) − ̟ α (̟ κ ∂κ h) + ̟ α ̟ κ Sκ H H H 1 α λ κ 1 α λ κ − ̟ ̟ (u ∂κ uλ ) + u ̟ (̟ ∂κ uλ ) H H αβγδ κ − qǫ uβ (∂γ u )̟κ Sδ

− (∂κ uκ )vortα (̟) 1 + ̟ α ̟ λ (uκ ∂κ uλ ) H + (θh − θ;h;h )S α (η−1 )κλ (∂κ h)(∂λ h) + (θh − θ;h;h )S α (uκ ∂κ h)(uλ ∂λ h)

+ (θ;h;h − θh )uα (S κ ∂κ h)(uλ ∂λ h) + (θ;h;h − θh )((η−1 )ακ ∂κ h)(S λ ∂λ h)

+ (θ;s − θ;h;s )S α (S κ ∂κ h) + (θ;h;s − θ;s )uαSκ S κ (uλ ∂λ h) + (θ;h;s − θ;s )Sκ S κ ((η−1 )αλ ∂λ h)

+ (θ − θ;h )S α (∂κ uκ )(uλ ∂λ h) + (θ;h − θ)(S κ ∂κ uα )(uλ∂λ h) + uκ ∂κ (θ;h − θ)S α (∂λ uλ )

+ (θh − θ;h;h )S α (uκ ∂κ h)(∂λ uλ) + (θ − θ;h )(uκ ∂κ S α )(∂λ uλ )

+ (θ;h − θ)S α (∂κ uλ )(∂λ uκ ) + (θ;h − θ)S α (uκ ∂κ uλ)(∂λ h) + (θ;h − θ)S α (∂κ uκ )(uλ ∂λ h) + (θ − θ;h )qS α (∂κ S κ ) + (θ − θ;h )q;h S α (S κ ∂κ h) + (θ − θ;h )q;s S α Sκ S κ + uκ ∂κ (θ;h − θ)uα (S λ ∂λ h)

+ (θh − θ;h;h )uα (uκ ∂κ h)(S λ ∂λ h) + (θ − θ;h )(uκ ∂κ uα )(S λ ∂λ h) + (θ − θ;h )uα (uκ ∂κ S λ )(∂λ h) + uκ ∂κ (θ − θ;h )(η−1 )αλ S β (∂λ uβ ) + (θ;h;h − θh )(uκ ∂κ h)S β ((η−1 )αλ ∂λ uβ ) + (θ;h − θ)(uκ ∂κ S β )((η−1 )αλ ∂λ uβ )

+ (θ − θ;h )S β ((η−1 )αλ ∂λ uβ )(uκ ∂κ h) + (θ − θ;h )S β ((η−1 )αλ ∂λ uκ )(∂κ uβ ) + (θ − θ;h )q((η−1)αλ ∂λ S β )Sβ + (θ;h − θ)q;h ((η−1 )αλ ∂λ h)Sκ S κ

+ (θ;h − θ)q;s S α Sκ S κ + 2(θ;h − θ)q((η−1 )αλ ∂λ S κ )Sκ



+ (θ;h − θ)((η−1 )ακ ∂κ h)uβ (uλ ∂λ S β ) + (θ − θ;h )uα (η−1 )κλ (∂κ h)uβ (∂λ S β ).


53

Next, we bring the four perfect-derivative terms uκ ∂κ {· · · } on RHS (170) over to the left-hand side, which yields the equation n κ u ∂κ vortα (̟) + c−2 ǫαβγδ uβ (∂γ h)̟δ + (θ − θ;h )S α (∂λ uλ ) (171) o + (θ − θ;h )uα (S λ ∂λ h) + (θ;h − θ)(η−1 )αλ S β (∂λ uβ ) = vortκ (̟)∂κ uα − 2(∂κ uκ )vortα (̟) + uα (uκ ∂κ uβ )vortβ (̟) + · · · ,

where the terms · · · do not involve vort(̟). We then use definition (23a) to algebraically substitute for the four instances of vort(̟) in equation (171) (note in particular that the terms in braces on LHS (171) are equal to C α ). In total, this yields the following equation, where we have placed the terms generated by the algebraic substitution on the first through seventh lines of RHS (172): uκ ∂κ C α = C κ ∂κ uα − 2(∂κ uκ )C α + uα (uκ ∂κ uβ )C β

− c−2 ǫκβγδ (∂κ uα )uβ (∂γ h)̟δ + (θ;h − θ)(S κ ∂κ uα )(∂λ uλ )

+ (θ;h − θ)(uκ ∂κ uα )(S λ ∂λ h) + (θ − θ;h )(η−1 )κλ (∂κ uα )S β (∂λ uβ ) + 2(∂κ uκ )c−2 ǫαβγδ uβ (∂γ h)̟δ + 2(θ − θ;h )S α (∂κ uκ )(∂λ uλ )

+ 2(θ − θ;h )uα (∂κ uκ )(S λ ∂λ h) + 2(θ;h − θ)(∂κ uκ )S β ((η−1 )αλ ∂λ uβ ) − uα (uκ ∂κ uσ )c−2 ǫσβγδ uβ (∂γ h)̟δ + (θ;h − θ)uα (uκ ∂κ uσ )S σ (∂λ uλ )

+ (θ;h − θ)uα (uκ ∂κ uβ )uβ (S λ ∂λ h) + (θ − θ;h )uα (uκ ∂κ uλ)S β (∂λ uβ ) − 2(∂κ uκ )c−2 ǫαβγδ uβ (∂γ h)̟δ + c−2 ǫαβγδ (uκ ∂κ uβ )(∂γ h)̟δ

+ c−2 ǫαβγδ uβ (∂γ h)̟ κ (∂δ uκ ) + c−2 (θ − θ;h )(S κ ∂κ h)((η−1 )αλ ∂λ h)

+ c−2 (θ − θ;h )uα (S κ ∂κ h)(uλ ∂λ h)

+ c−2 (θ;h − θ)S α (uκ ∂κ h)(uλ ∂λ h)

+ c−2 (θ;h − θ)S α (η−1 )κλ (∂κ h)(∂λ h)

+ c−2 ǫαβγδ uβ {(∂κ uκ )(∂γ h) − (∂γ uκ )(∂κ h)} ̟δ

+ 2c−3 c;s (uκ ∂κ h)ǫαβγδ uβ Sγ ̟δ 1 1 + (̟ κ ∂κ ̟ α ) − ̟ α (̟ κ ∂κ h) H H 1 α λ κ − u ̟ (̟ ∂κ uλ) + ǫαβγδ uβ (∂γ h)̟ κ (∂δ uκ ) H − qǫαβγδ uβ Sγ ̟ κ (∂δ uκ )

− ǫαβγδ (∂β h)uγ ̟ κ (∂δ uκ ) + (∂κ uκ )ǫαβγδ (∂β h)uγ ̟δ



(172)

54

Relativistic Euler + qǫαβγδ Sβ uγ ̟ κ (∂δ uκ ) − q(∂κ uκ )ǫαβγδ Sβ uγ ̟δ

+ q(θ;h − θ)((η−1 )κα ∂κ h)S λ Sλ + q(θ;h − θ)uα (uκ ∂κ h)S λ Sλ + q(θ − θ;h )S α (S κ ∂κ h) − ǫαβγδ (∂β h)uγ ̟ λ(∂δ uλ) + qǫαβγδ Sβ uγ ̟ λ (∂δ uλ ) − ǫαβγδ uβ (∂γ ̟ κ )(∂δ uκ )

− ǫαβγδ uβ (∂γ ̟ κ )(∂δ uκ ) − ǫαβγδ uβ (∂γ h)̟ κ (∂δ uκ ) 1 2q 1 − (̟ κ ∂κ ̟ α ) − ̟ α (̟ κ ∂κ h) + ̟ α ̟ κ Sκ H H H 1 α λ κ 1 α λ κ − ̟ ̟ (u ∂κ uλ) + u ̟ (̟ ∂κ uλ ) H H αβγδ κ − qǫ uβ (∂γ u )̟κ Sδ 1 + ̟ α ̟ λ(uκ ∂κ uλ ) H + (θh − θ;h;h )S α (η−1 )κλ (∂κ h)(∂λ h) + (θh − θ;h;h )S α (uκ ∂κ h)(uλ ∂λ h)

+ (θ;h;h − θh )uα (S κ ∂κ h)(uλ ∂λ h) + (θ;h;h − θh )((η−1)ακ ∂κ h)(S λ ∂λ h)


+ (θ − θ;h )S α (∂κ uκ )(uλ ∂λ h) + (θ;h − θ)(S κ ∂κ uα )(uλ ∂λ h)

+ (θh − θ;h;h )S α (uκ ∂κ h)(∂λ uλ ) + (θ − θ;h )(uκ ∂κ S α )(∂λ uλ )

+ (θ;h − θ)S α (∂κ uλ )(∂λ uκ ) + (θ;h − θ)S α (uκ ∂κ uλ )(∂λ h) + (θ;h − θ)S α (∂κ uκ )(uλ∂λ h) + (θ − θ;h )qS α (∂κ S κ ) + (θ − θ;h )q;h S α (S κ ∂κ h) + (θ − θ;h )q;s S α Sκ S κ

+ (θh − θ;h;h )uα (uκ ∂κ h)(S λ ∂λ h) + (θ − θ;h )(uκ ∂κ uα )(S λ ∂λ h) + (θ − θ;h )uα (uκ ∂κ S λ )(∂λ h) + (θ;h;h − θh )(uκ ∂κ h)S β ((η−1 )αλ ∂λ uβ ) + (θ;h − θ)(uκ ∂κ S β )((η−1 )αλ ∂λ uβ )

+ (θ − θ;h )S β ((η−1 )αλ ∂λ uβ )(uκ ∂κ h) + (θ − θ;h )S β ((η−1 )αλ ∂λ uκ )(∂κ uβ ) + (θ − θ;h )q((η−1 )αλ ∂λ S β )Sβ + (θ;h − θ)q;h ((η−1 )αλ ∂λ h)Sκ S κ + (θ;h − θ)q;s S α Sκ S κ + 2(θ;h − θ)q((η−1 )αλ ∂λ S κ )Sκ



+ (θ;h − θ)((η−1 )ακ ∂κ h)uβ (uλ ∂λ S β ) + (θ − θ;h )uα (η−1 )κλ (∂κ h)uβ (∂λ S β ).

Next, we reorganize the terms on RHS (172) to obtain the equation

uκ ∂κ C α = C κ ∂κ uα − 2(∂κ uκ )C α + uα (uκ ∂κ uβ )C β +

21 X i=1

Qi + L ,

(173)

55

M. Disconzi, J. Speck where

Q1 := −2ǫαβγδ uβ (∂γ ̟ κ )(∂δ uκ ),

(174)

−1 ακ

λ

−1 κλ

α

Q2 := (θ;h − θ)((η ) ∂κ h)∂λ S + (θ − θ;h )(η ) (∂κ h)(∂λ S ), α

κ

λ

α

κ

(175)

λ

Q3 := (θ;h − θ)u (u ∂κ h)(∂λ S ) + (θ − θ;h )u (u ∂κ (S )∂λ h),

(176)

Q4 := (θh − θ;h;h )S α (η−1 )κλ (∂κ h)(∂λ h) + (θh − θ;h;h )S α (uκ ∂κ h)(uλ∂λ h) + (θh − θ;h;h )S α (uκ ∂κ h)(∂λ uλ ), (177) Q5 := (θ;h;h − θh )(η−1 )ακ (∂κ h)(S λ ∂λ h) + (θ;h;h − θh )(uκ ∂κ h)S β ((η−1 )αλ ∂λ uβ ), −2

α

κ

λ

−2

α

−1 κλ

(178) α

Q6 := c (θ;h − θ)S (u ∂κ h)(u ∂λ h) + c (θ;h − θ)S (η ) (∂κ h)(∂λ h) + (θ − θ;h )S (∂κ uκ )(∂λ uλ), (179) Q7 := (θ;h − θ)S α (uκ ∂κ h)(uλ∂λ h) + (θ;h − θ)S α (η−1 )κλ (∂κ h)(∂λ h) + (θ;h − θ)S α (uκ ∂κ uλ )(∂λ h), (180) Q8 := (θ;h − θ)S α (∂κ uλ )(∂λ uκ ) + (θ − θ;h )S α (∂κ uκ )(∂λ uλ ),

(181)

Q9 := (θ;h − θ)(S κ ∂κ uα )(∂λ uλ ) + (θ;h − θ)(uκ ∂κ uα )(S λ ∂λ h) + (θ − θ;h )((η−1 )κλ ∂κ uα )S β (∂λ uβ ), (182)

Q10 := (θ;h − θ)(S κ ∂κ uα )(uλ∂λ h) + (θ − θ;h )(uκ ∂κ uα )(S λ ∂λ h), κ

α

λ

β

−1 αλ

(183)

κ

Q11 := (θ − θ;h )(u ∂κ S )(∂λ u ) + (θ − θ;h )S ((η ) ∂λ u )(∂κ uβ ), κ

−1 αλ

β

Q12 := 2(θ;h − θ)(∂κ u )(η ) S (∂λ uβ ) −2

κ

−1 αλ

κ

β

(184) (185)

−1 αλ

+ c (θ − θ;h )(S ∂κ h)((η ) ∂λ h) + (θ;h − θ)(u ∂κ S )((η ) ∂λ uβ ),

Q13 := (θ − θ;h )uα (∂κ uκ )(S λ ∂λ h) + (θ − θ;h )uα (uκ ∂κ uλ)S β (∂λ uβ ),

(186)

Q15 := (∂κ uκ )ǫαβγδ (∂β h)uγ ̟δ + c−2 ǫαβγδ (uκ ∂κ uβ )(∂γ h)̟δ ,

(188)

Q14 := (θ − θ;h )uα (∂κ uκ )(S λ ∂λ h) + (θ − θ;h )uα (uκ ∂κ h)(S λ ∂λ h) + (θ − θ;h )uα(η−1 )κλ(∂κ h)uβ (∂λ S β ), (187) −2 α σβγδ

Q16 := −c u ǫ

−2 κβγδ

Q17 := −c ǫ + (c

−2

κ

(u ∂κ uσ )uβ (∂γ h)̟δ , α

(∂κ u )uβ (∂γ h)̟δ αβγδ

+ 2)ǫ

(189) (190)

κ

uβ (∂γ h)̟ (∂δ uκ )

+ c−2 ǫαβγδ uβ ̟δ {(∂κ uκ )(∂γ h) − (∂γ uκ )(∂κ h)} ,

Q18 := (θ − θ;h )((η−1 )ακ ∂κ h)(S λ ∂λ h) + (θ;h − θ)((η−1 )ακ ∂κ h)uβ (uλ ∂λ S β ),

(191)

Q20 := (θ − θ;h )S β ((η−1 )αλ ∂λ uβ )(uκ ∂κ h) + (θ − θ;h )(uκ ∂κ h)(uλ ∂λ S α ),

(193)

Q19 := (θ;h − θ)uα (uκ ∂κ uσ )S σ (∂λ uλ ) + c−2 (θ − θ;h )uα (S κ ∂κ h)(uλ ∂λ h),

(192)

Q21 := (θ;h − θ)uα (uκ ∂κ uβ )(uβ S λ ∂λ h),

(194)

56

Relativistic Euler

2q 2 α κ ̟ (̟ ∂κ h) + ̟ α ̟ κ Sκ H H −3 κ αβγδ + 2c c;s (u ∂κ h)ǫ uβ Sγ ̟δ − qǫαβγδ uβ Sγ ̟ κ (∂δ uκ )

L := −

(195)

+ 2qǫαβγδ Sβ uγ ̟ κ (∂δ uκ ) − q(∂κ uκ )ǫαβγδ Sβ uγ ̟δ

+ q(θ;h − θ)((η−1 )κα ∂κ h)S λ Sλ + q(θ;h − θ)uα (uκ ∂κ h)S λ Sλ + (θ − θ;h )S α (S κ ∂κ h) − qǫαβγδ uβ (∂γ uκ )̟κ Sδ


+ (θ − θ;h )qS α (∂κ S κ ) + (θ − θ;h )q;h S α (S κ ∂κ h) + (θ − θ;h )q;s S α Sκ S κ

+ (θ − θ;h )q((η−1 )αλ ∂λ S β )Sβ + (θ;h − θ)q;h ((η−1 )αλ ∂λ h)Sκ S κ

+ (θ;h − θ)q;s S α Sκ S κ + 2(θ;h − θ)q((η−1 )αλ ∂λ S κ )Sκ .

Note that the terms on RHSs (174)-(194) are precisely quadratic in the first-order derivatives of the solution variables (h, uα , ̟ α , S α )α=0,1,2,3 while the terms on RHS (195) are at most linear in the derivatives of the solution variables. We will now show that Q1 , Q2 , · · · , Q21 can be expressed as null forms or terms that are at most linear in the derivatives of the solution variables. To this end, we simply use (123a)-(123p) to algebraically substitute for Q2 , Q4 , Q5 , Q6 , Q7 , Q9 , Q11 , Q12 , Q13 , Q14 , Q15 , Q16 , Q18 , Q19 , Q20 , and Q21 (we do not substitute for Q1 , Q3 , Q8 , Q10 , and Q17 since these terms are already manifestly linear combinations of null forms). Following this substitution, there are only two kinds of terms on RHS (173): null forms and terms that are at most linear in the derivatives of the solution variables. We now place all null forms on RHS (164) except for null forms that involve the derivatives of ̟ or S; these null forms we place directly on RHS (163). We then place all terms that are linear in C, linear in D, linear in the first-order derivatives of ̟, or linear in the first-order derivatives of S directly on RHS (163). Finally, we place all remaining terms, which are are at most linear in the derivatives of the solution variables and do not depend on the derivatives of ̟ or S, on RHS (165). This completes the proof of the proposition. 9. Well-posedness Our main goal in this section is to prove Theorem 9.2, which is a local well-posedness result for the relativistic Euler equations based on our new formulation of the equations, that is, based on the equations of Theorem 3.1. The main new feature of Theorem 9.2 compared to standard local well-posedness results for the relativistic Euler equations (see Theorem 9.1 for a statement of standard local well-posedness) is that it yields an extra degree of differentiability for the vorticity and the entropy, assuming that the initial vorticity and entropy enjoy the same extra differentiability. We stress that this gain in regularity holds even though the logarithmic enthalpy and four-velocity do not generally enjoy the same gain. As we described in Subsect. 1.2, this extra regularity for the vorticity and the entropy is essential for the study of shocks in more than one spatial dimension.


57

For convenience, instead of proving local well-posedness for the relativistic Euler equations on the standard Minkowski spacetime background, we instead consider the spacetime background (R × T3 , η), where the “spatial manifold” T3 is the standard three-dimensional torus and, relative to standard coordinates on R ×T3 , ηαβ := diag(−1, 1, 1, 1) is the standard Minkowski metric. Thus, strictly speaking, in this section, η denotes a tensor on a different manifold compared to the rest of the paper, but this minor change has no substantial bearing on the discussion. In particular, the relativistic Euler equations on (R × T3 , η) take the same form that they take in Theorem 3.1. The advantage of the compact spatial topology is that it allows for a simplified approach to some technical aspects of the proof of local wellposedness. However, the arguments that we give in this section feature all of the main ideas needed to prove local well-posedness on the standard Minkowski spacetime background. 9.1. Notation, norms, and basic tools from analysis. 9.1.1. Notation. Throughout this section, {xα }α=0,1,2,3 denote standard rectangular coordinates on R × T3 , where {xa }a=1,2,3 are standard local coordinates on T3 , and we use the alternate notation t := x0 . Note that even though {xa }a=1,2,3 are only locally defined on T3 , the coordinate partial derivative vectorfields {∂a }a=1,2,3 be extended to a smooth global frame on T3 ; by a slight abuse of notation, we will denote the globally defined “spatial” frame by {∂a }a=1,2,3 , and the corresponding globally defined “spacetime frame” by {∂α }α=0,1,2,3 . We also use the alternate partial derivative notation ∂t := ∂0 . Σt := {(t, x) | x ∈ T3 }

(196)

curli (W ) := εijk ∂j Wk

(197)

denotes the standard flat constant-time hypersurface. To each “spatial multi-index” I~ = (ι1 , ι2 , ι3 ), where the ιa are non-negative integers, we associate the spatial differential operator ∂I~ := ∂1ι1 ∂2ι2 ∂3ι3 . Note that ∂I~ is an operator of ~ := ι1 + ι2 + ι3 . order |I| If V is a spacetime vectorfield or one-form, then V denotes the η-orthogonal projection of V onto Σt . For example, ̟i := ̟ i for i = 1, 2, 3. Moreover, we use the notation (3)

to denote the standard Euclidean curl operator acting on one-forms on Σt . 9.1.2. Norms. Definition 9.1 (Lebesgue and Sobolev norms). We define the following Lebesgue norms: kf kL∞ (T3 ) := ess supx∈T3 |f (x)|, Z 1/2 2 kf kL2 (T3 ) := f (x) dx ,

(198) (199)

T3

where in the rest of this section, dx := dx1 dx2 dx3 denotes the standard volume form on T3 induced by the Euclidean metric diag(1, 1, 1). If U = (U 1 , · · · , U m ) is an array of scalar-valued functions, then for p ∈ {2, ∞}, we define kUkLp (T3 ) :=

m X a=1

kU a kLp (T3 ) .

(200)

58

Relativistic Euler

Remark 9.1 (Extending the definitions of the norms from T3 to Σt ). In our proof of local well-posedness, we will use norms in which the manifold T3 from Def. 9.1 is replaced with the constant time slice Σt = {t} × T3 , which is diffeomorphic to T3 . We will not explicitly define these norms along Σt since their definitions are obvious analogs of the ones appearing in Def. 9.1. Similar remarks apply to other norms on T3 introduced later in this subsubsection. We define the following Sobolev norms for integers r ≥ 0:  1/2 X  2 kf kH r (T3 ) := k∂I~f kL2 (R3 ) ,   ~ |I|≤r

kf kH˙ r (T3 ) :=

 X 

~ |I|=r

k∂I~f k2L2 (R3 )

1/2  

.

If r ∈ R is not an integer, then we define18 1/2  2   X , (1 + |k|2 )r fˆ(k1 , k2 , k3 ) kf kH r (T3 ) :=   3

(201a)

(201b)

(202)

(k1 ,k2 ,k3 )∈Z

P3 R a where fˆ(k1 , k2 , k3 ) := T3 f (x)e−2πi a=1 x ka dx is the spatial Fourier transform of f and P |k|2 := 3a=1 ka2 . As we alluded to in Remark 9.1, we use similar notation to denote the analogous norms R 1/2 . along Σt . For example, kf kL2 (Σt ) := T3 f 2 (t, x) dx A Sobolev norm of an array of scalar functions is defined to be the sum of the Sobolev norms of the components of the array, in analogy with (200).

Definition 9.2 (Some additional function spaces). If B is a Banach space and r ≥ 0 is an integer, then C r ([0, T ], B) denotes the space of r-times continuously differentiable functions from [0, T ] to B. We omit the superscript when r = 0. We denote the corresponding norm by k · kC r ([0,T ],B). L∞ ([0, T ], B) denotes the space of functions from [0, T ] to B that are essentially bounded over the interval [0, T ]. We denote the corresponding norm by k · kL∞ ([0,T ],B) . C r (T3 ) denotes the space of functions on T3 that are r-times continuously differentiable with uniformly bounded derivatives up to order r. We omit the superscript when r = 0. We now fix, for the rest of Sect. 9, an integer N subject to N ≥ 3.

(203)

9.1.3. Basic analytical tools. In our analysis, we will rely on the following standard results. Lemma 9.1 (Sobolev embedding, product and interpolation estimates). If r > 3/2, then H r (T3 ) continuously embeds into C(T3 ), and there exists a constant Cr > 0 such that the 18As

is well known, when r is an integer, RHS (202) defines a norm that is equivalent to the norm defined in (201a).

59

M. Disconzi, J. Speck following estimate holds for v ∈ H r (T3 ): kvkC(T3 ) ≤ Cr kvkH r (T3 ) .

(204)

Let r ≥ 0 be an integer and let v := (v 1 , · · · , v A ) be a finite-dimensional array of realvalued functions on T3 such that va ∈ H˙ r (T3 ) ∩ C(T3 ) for 1 ≤ a ≤ A. Set kvkH r (T3 ) := PA a=1 kva kH r (T3 ) , and let ( ) A X Ir := (I~1 , · · · , I~A ) | |I~a | = r . (205) a=1

Let w be any sub-array of v, and assume that w contains B elements (where B ≤ A). Assume that w(T3 ) ⊂ intK, where K is a compact subset of RB , and let f be a smooth real-valued function on an open subset of RB containing K. Then the following estimate holds:

A A

X Y Y

≤ Cf,K,r kva kH˙ r (T3 ) kvb kC(T3 ) . (206) ∂I~a va max f(w)

2 3 (I~1 ,··· ,I~A )∈Ir a=1

a=1

L (T )

b6=a

P ~ Moreover, under the same assumptions stated in the previous paragraph, if A a=1 |Ia | = r, A QA then the map v → f(w) a=1 ∂I~a va is continuous from H˙ r (T3 ) ∩ C(T3 ) to L2 (T3 ). In

particular, let δ = δw > 0 be such that the following holds:19 if d(p, w(T3 )) < δ, d(q, w(T3 )) < δ, and d(p, q) < δ, where d is the standard Euclidean distance function on RB , then the P straight line segment joining p to q is contained in intK. If A e C(T3 ) ≤ δ and if a=1 kw − wk r > 3/2, then the following estimate holds (where the function f is assumed to be the same in both appearances on LHS (207) and Ir is defined by (205)):

A A

Y Y

≤ Cf,K,kvkH r (T3 ) ,kevkH r (T3 ) ,A,r kv − vekH r (T3 ) . va e ∂I~a e ∂I~a va − f(w) max f(w)

2 3 (I~1 ,··· ,I~A )∈Ir a=1

a=1

L (T )

(207)

Furthermore, if r > 3/2 and va ∈ H r (T3 ) for a = 1, 2, then v1 v2 ∈ H r (T3 ), and there exists a constant Cr > 0 such that kv1 v2 kH r (T3 ) ≤ Cr kv1 kH r (T3 ) kv2 kH r (T3 ) ,

(208)

and function multiplication (v1 , v2 ) → v1 v2 is a continuous map from H r (T3 ) × H r (T3 ) to H r (T3 ). Finally, if 0 ≤ s ≤ r and v ∈ H r (T3 ), then there exists a constant Cr,s > 0 such that 1− s

s

r r kvkH s (T3 ) ≤ Cr,s kvkL2 (T 3 ) kvkH r (T3 ) .

(209)

Remark 9.2 (The same estimates hold along Σt ). All of the results of Lemma 9.1 hold verbatim if we replace T3 by Σt throughout. 19Such

a δ > 0 exists due to the compactness of w(T3 ) and K, where the compactness of w(T3 ) follows from the assumption that the va are continuous.

60

Relativistic Euler

9.1.4. An L2 -in-time continuity result for transport equations. We will use the following simple technical result in our proof of local well-posedness. Lemma 9.2 (An L2 -in-time continuity result for transport equations). Let F ∈ L∞ ([0, T ], L2 (T3 )) and let f be the solution to the inhomogeneous transport equation initial value problem uα ∂α f = F , f |Σ := f˚ ∈ L2 (T3 ).

(210)

f ∈ C([0, T ], L2 (T3 )).

(212)

(211)

0

Then if uα ∈ L∞ ([0, T ], C 1(T3 )) for α = 0, 1, 2, 3, we have

Proof. We will prove right continuity at t = 0; continuity at any other time t ∈ (0, T ] could be proved using similar arguments. More precisely, we will show that

= 0. (213) lim f (t, ·) − f˚ 2 3 t↓0

L (T )

∞ 3 To proceed, we let {f˚k }∞ k=1 ⊂ C (T ) be a sequence of smooth functions such that 1 kf˚− f˚k kL2 (Σ0 ) ≤ . k Note that uα ∂α (f − f˚k ) = −ua ∂a f˚k + F .

Hence, a standard integration by parts argument based on the divergence identity o n o2 ua (f − f˚k ) n a ˚ 2 ˚ ˚ (f − fk ) + 2 −u ∂a fk + F ∂t (f − fk ) = ∂a u0 u0 a u 2 ˚ − ∂a (f − fk ) u0

yields that for 0 ≤ t ≤ T , we have kf −

f˚k k2L2 (Σt )

= kf − +2

Z

f˚k k2L2 (Σ0 ) t

τ =0

Z

Στ

+

Z

t

τ =0

Z

Στ

∂a

ua u0

(f − f˚k )2 dx dτ

(214)

(215)

(216)

(217)

o (f − f˚k ) n a ˚ −u ∂ f + F dx dτ. a k u0

In particular, from (214), (217), our assumptions on F and uα , and Young’s inequality, we find that if 0 ≤ t ≤ T , then there is a constant CT that is bounded by a continuous increasing function of T such that Z t n Z t o 1 2 2 kf − f˚k kL2 (Σt ) ≤ 2 + CT 1 + kf˚k kH 1 (Σ0 ) dτ + CT kf − f˚k k2L2 (Στ ) dτ. (218) k τ =0 τ =0

In particular, from (218) and Gronwall’s inequality, we deduce (allowing CT to vary from line to line in the rest of the proof) that if 0 ≤ t ≤ T , then the following inequality holds: 1 2 2 + CT t 1 + kf˚k kH 1 (Σ0 ) exp(CT t). (219) kf − f˚k kL2 (Σt ) ≤ k2

61

M. Disconzi, J. Speck From (219), (214), and the triangle inequality, it follows that 2 lim sup kf − f˚kL2 (Σt ) ≤ . k 0≤t≤T

(220)

Finally, allowing k → ∞ in (220), we conclude (213). We have therefore proved the lemma. 9.2. The regime of hyperbolicity. Our proof of well-posedness relies on a standard assumption, namely that the solution lies in the interior of the region of state space where the equations are hyperbolic without degeneracy. This notion is precisely captured by the next definition. Definition 9.3 (Regime of hyperbolicity). We define the regime of hyperbolicity H to be the following subset of state-space: (221) H := (h, s, u1, u2 , u3 ) ∈ R5 | 0 < c(h, s) < ∞ . 9.3. Standard local well-posedness. Our main goal in this subsection is to state Theorem 9.2, which is our main local well-posedness result exhibiting the gain in regularity for the vorticity and entropy. Most aspects of the theorem are standard. In the next theorem, we summarize these standard aspects; we will use them in our proof of Theorem 9.2.

Remark 9.3 (The “fundamental” initial data). In the rest of Sect. 9, we view ˚ h := h|Σ0 , ˚ s := s|Σ0 , and ˚ ui := ui |Σ0 to be the “fundamental” initial data in the following sense: with the help of the relativistic Euler equations (24)-(26) + (29), along Σ0 , all of the other quantities that are relevant for our analysis can be expressed in term of the fundamental initial data; see Lemma 9.3. Theorem 9.1 (Standard local well-posedness). Let ˚ h := h|Σ0 , ˚ s := s|Σ0 , and ˚ ui := ui |Σ0 20 be initial data for the relativistic Euler equations (24)-(26) + (29). Assume that for some integer N > 5/2, we have ˚ h ∈ H N (Σ0 ),

˚ s ∈ H N (Σ0 ),

˚ ui ∈ H N (Σ0 ).

(222)

Assume moreover that there is a compact subset K ⊂ intH (where intH is the interior of H) such that for all p ∈ Σ0 , we have (˚ h(p),˚ s(p), ˚ u1(p), ˚ u2(p), ˚ u3 (p)) ∈ intK. Then there 21 exists a time T > 0 depending only on K, k˚ hkH 3 (Σ0 ) , k˚ skH 3 (Σ0 ) , and k˚ ui kH 3 (Σ0 ) , such α α that a unique solution (h, s, u , ̟ ) classically exists on the slab [0, T ] × T3 and satisfies (h(p), s(p), u1(p), u2 (p), u3(p)) ∈ intK for p ∈ [0, T ] × T3 . Moreover, the solution depends continuously on the initial data,22 and its components relative to the standard coordinates 20The

datum u0 |Σ0 is determined from the other data by virtue of the constraint (29). fact, using additional arguments not presented here, one can show that for any fixed real number r > 5/2, the time of existence can be controlled by a function of K, k˚ hkH r (Σ0 ) , k˚ skH r (Σ0 ) , and k˚ ui kH r (Σ0 ) . Of course, if the initial data enjoy additional Sobolev regularity, then the additional regularity persists in the solution during its classical lifespan. 22In particular, there is a H 3 (Σ ) 5 -neighborhood of (˚ h,˚ s, ˚ ui ) such that all data in the neighborhood 0 5 launch solutions that exist on the same slab [0, T ]×T3 and, assuming also that the data belong to H N (Σ0 ) , enjoy the regularity properties stated in the theorem. 21In

62

Relativistic Euler

enjoy the following regularity properties: h ∈ C([0, T ], H N (T3 )), α

N

3

s ∈ C([0, T ], H N (T3 )),

α

u ∈ C([0, T ], H (T )), S ∈ C([0, T ], H

N −1

3

(223a) α

(T )), ̟ ∈ C([0, T ], H

N −1

3

(T )). (223b)

Discussion of the proof. Theorem 9.1 is standard. Readers can consult, for example, [29] for detailed proofs in the case of the relativistic Euler equations on a family of conformally flat23 spacetimes. The main step in the proof is deriving a priori energy estimates for linearized versions of a first-order formulation of the equations, such as (24)-(26) + (29). For a firstorder formulation that is equivalent (for C 1 solutions) to (24)-(26) + (29), this step was carried out in detail in [29] using the method of energy currents, a technique that originated in the context of the relativistic Euler equations in Christodoulou’s foundational work [3] on shock formation. Remark 9.4 (C ∞ data give rise to C ∞ solutions). In view of (204), we see that Theorem 9.1 implies that C ∞ initial data give rise to (local-in-time) C ∞ solutions. We now state our main local well-posedness theorem. Its proof is located in Subsect. 9.7. Theorem 9.2 (Local well-posedness with improved regularity for the entropy and vorticity). Assume the hypotheses of Theorem 9.1, but in place of (222), instead assume that the initial vorticity and entropy enjoy one extra degree of Sobolev regularity. That is, assume that for some integer N > 5/2 and i = 1, 2, 3, we have ˚ h ∈ H N (Σ0 ), ˚ ui ∈ H N (Σ0 ), (224a) ˚ s ∈ H N +1 (Σ0 ),

̟ ˚ i ∈ H N (Σ0 ),

(224b)

where ̟ is defined in (18) and ̟ ˚ i := ̟|iΣ0 . Then the conclusions of Theorem 9.1 hold, and the solution’s components enjoy the following regularity properties for α = 0, 1, 2, 3 where T > 0 is the same time from Theorem 9.1: h ∈ C([0, T ], H N (T3 )),

s ∈ C([0, T ], H N +1(T3 )),

uα ∈ C([0, T ], H N (T3 )),

S α ∈ C([0, T ], H N (T3 )),

(225a) ̟ α ∈ C([0, T ], H N (T3 )).

(225b)

In particular, according to (225b), the additional regularity of the entropy and vorticity featured in the initial data is propagated by the flow of the equations. Moreover, the solution depends continuously on the initial data relative the norms corresponding to (225a)-(225b). 9.4. A new inverse Riemannian metric and the classification of various combinations of solution variables. In our proof of Theorem 9.2, when controlling the top-order derivatives of the vorticity and entropy, we will rely on “geometrically sharp” elliptic estimates in which the precise details of the principal coefficients of the elliptic operators are important for our arguments. Due to the quasilinear nature of the relativistic Euler equations, these precise elliptic estimates involve the inverse Riemannian metric G−1 from the next definition. In particular, we will need to use G−1 -based norms when proving that the 23More

R

1+3

.

precisely, in [29], the spacetime metrics are scalar function multiples of the Minkowski metric on

63


top-order derivatives of ̟ and S are continuous in time with values in L2 (T3 ) (these facts are contained within the statement (225b)); the role of G−1 in our analysis will become clear in Subsect. 9.7. Definition 9.4 (A Riemannian metric on Σt ). On each Σt , we define the inverse Riemannian metric G−1 as follows: ui uj (226) (G−1 )ij := δ ij − 0 2 , (u ) where δ ij := diag(1, 1, 1) is the standard Kronecker delta. Remark 9.5. From the relation ηαβ uα uβ = −1, one can easily infer that G−1 is Riemannian, that is, of signature (+, +, +). In proving that the solution depends continuously on the initial data, we will used a modified version of Kato’s framework [16–18]. His framework was designed to handle hyperbolic systems, while our formulation of the relativistic Euler equations is elliptic-hyperbolic. For this reason, we find it convenient to divide the solution variables into various classes, which we provide in the next definition. Roughly, we will handle the “hyperbolic quantities” using Kato’s framework, and to handle the remaining quantities, we will use elliptic estimates and algebraic relationships to control them in terms of the hyperbolic quantities. Definition 9.5 (Classification of various combinations of solution variables). We define the hyperbolic quantities H, the elliptic quantities E, and the algebraic quantities AH , AH,E , and A as follows: H := (h, s, ua, ∂a h, ∂a ub , ̟ a, S a , C a , D)a,b=1,2,3 , E := (∂a ̟b , ∂a Sb )a,b=1,2,3 ,

AH := u0 − 1, ̟ 0, S 0 , C 0 , ∂t h, ∂t uα , ∂a u0 , ∂t s

(227a) (227b)

α=0,1,2,3;a=1,2,3 a (3)

∪ (G−1 )cd ∂c ̟d , (G−1 )cd ∂c Sd , (3) curla (̟),

curl (S)

AH,E := (∂t ̟α , ∂a ̟0 , ∂t Sα , ∂a S0 , ∂b ̟ b, ∂b S b )α=0,1,2,3;a=1,2,3 , A := AH ∪ AH,E .

(227c) a=1,2,3

, (227d) (227e)

Some remarks are in order. • The point of introducing the algebraic quantities A is that, by virtue of the relativistic Euler equations, they can be algebraically expressed in terms of H and E (and thus are redundant); see Lemma 9.3. We stress that in (227c), it is crucial that the inverse metric G−1 is the one from Def. 9.4; the proof of (228a) will clarify that it is essential that the inverse metric is precisely G−1 . • The elliptic quantities E can be controlled (in appropriate Sobolev norms) in terms of H via elliptic estimates; see Lemma 9.4 and its proof. • The hyperbolic quantities H solve evolution equations with source terms that depend on H and E. In view of the previous point, we see that one can bound the source terms (in appropriate Sobolev norms) in terms of H. This will allow us to derive a closed system of energy inequalities that can be used to estimate H. In view of the previous two points, the estimates for H imply corresponding estimates for E and A.

64

Relativistic Euler

Remark 9.6 (The hyperbolic quantities verify first-order hyperbolic equations). In our proof of local well-posedness, we will use the fact that the hyperbolic quantities H solve firstorder hyperbolic equations. More precisely, the elements h, s, and ua of (227a) satisfy the first-order hyperbolic system (24)-(26) + (29), the elements ∂a h and ∂a ub satisfy hyperbolic equations obtained by taking one spatial derivative of the equations (24)-(26) + (29), and S a , ̟ a , C a , and D respectively satisfy the (spatial components of the) transport equations (41), (42), (45b), and (43a); it is in this sense that we consider the variables H to be “hyperbolic.” Lemma 9.3 (Expressions for the algebraic quantities in terms of the hyperbolic and elliptic quantities). Assume that (h, s, uα ) is a C 2 solution to (24)-(26) + (29). Then we can express AH = f(H),

(228a)

AH,E = f(H, E),

(228b)

A = f(H, E),

(228c)

where f is a schematically denoted smooth function that satisfies f(0) = 0 and that varies from line to line. ~ ≥ 1. Then Moreover, let I~ be a spatial multi-index with |I| ~

−1 ab

(G ) ∂a ∂I~̟b =

|I| X

fJ~1 ,··· ,J~M (H)

~

(G ) ∂a ∂I~Sb =

|I| X

M =1 ~ |J~1 |+···+|J~M |=|I|

∂J~m H,

(229a)

M Y

∂J~m H,

(229b)

m=1

M =1 ~ |J~1 |+···+|J~M |=|I| −1 ab

M Y

fJ~1 ,··· ,J~M (H)

m=1

where fJ~1 ,··· ,J~M are schematically denoted smooth functions that vary from line to line, and QM m=1 ∂J~m H schematically denotes an order M monomial in the derivatives of the elements of H. Proof. Throughout this proof, f is a smooth function that can vary from line to line and, except in the last paragraph of the proof, satisfies f(0) = 0. Moreover, H and E are as defined in (227a) and (227b). We first prove (228a). We must show that the elements of (227c) can be written as smooth functions of the elements of (227a) that vanish at 0. We first note that by the normalization condition ηκλ uκ uλ = −1, u0 −1 is a smooth function of the spatial components of u that vanishes when u1 = u2 = u3 = 0. From this fact and the identity uκ Sκ = 0 (see (27)), we deduce that S 0 is a smooth function of the spatial components of u and S that vanishes at 0. A similar result holds for ̟ 0 by virtue of (48). Next, we note that, in view of the above discussion and the discussion surrounding equation (35), we can solve for the time derivatives of h, s, and uα in terms of their spatial derivatives. Thus far, we have shown that u0 − 1, ̟ 0, S 0 , ∂t h, ∂t uα , ∂a u0 , ∂t s can be expressed as f(H). In the rest of the proof, we will use these facts without explicitly mentioning them every time. Next, we use definitions (17) and (2.8) to deduce that uκ Cκ = f(H). Using this equation to algebraically solve for C 0 , we deduce that C 0 = f(H), as desired. We will now show that (G−1 )cd ∂c Sd = f(H). To begin, we use definition (23b) to deduce that ∂i S i = ∂α S α − ∂t S 0 =

65

M. Disconzi, J. Speck κ

i

nD − S κ ∂κ h + c−2 S κ ∂κ h − ∂t S 0 = f(H) − ∂t S 0 . Next, using the identity ∂t = uu∂0 κ − uu∂0 i and i 0 the evolution equation (41) with α = 0, we find that ∂t S 0 = f(H) − u ∂ui0S . Moreover, using (27), we find that S 0 =

Sj u j , u0

from which we deduce that

ui ∂i S 0 u0

= f(H) +

ui uj ∂i Sj . (u0 )2

Combining

ui uj ∂i Sj (u0 )2

the above calculations, we find that ∂i S i − = f(H) which, in view of definition (226), −1 cd yields the desired relation (G ) ∂c Sd = f(H) The relation (G−1 )cd ∂c ̟d = f(H) can be proved using a similar argument based on equations (42) and (45a), and we omit the details. To show that (3) curla (̟) = f(H), we first note that by definition (197), it suffices to show that ∂i ̟j − ∂j ̟i = f(H) for i, j = 1, 2, 3. To proceed, we use (56) with V := ̟ (which is applicable in view of (48)), definition (23a), and the transport equation (42) to deduce that ∂i ̟j − ∂j ̟i = ǫijγδ uγ vortδ (̟) + uj uκ ∂κ ̟i − ui uκ ∂κ ̟j + f(H) = f(H), which is the desired result. The fact that (3) curla (S) = f(H) can be proved using a similar argument based on equations (27), (41), and (43b), and we omit the details. We have therefore proved (228a). We now prove (228b). We must show that elements of (227d) can be written as smooth functions of the elements of (227a) and the elements of (227b) that vanish at 0. To handle κ uj ∂ ∂t ̟i , we use the identity ∂t = uu∂0 κ − u0 j and the transport equation (42) to deduce that κ ∂t ̟i = u ∂uκ0 ̟i + f(H, E) = f(H, E) as desired. To handle ∂t ̟0 , we simply use (48) to obtain ̟ uj

the identity ̟ 0 = uj0 , differentiate this identity with respect to ∂t , and then use the already proven facts that ̟j , uα − δ0α , and their time derivatives are equal to f(H, E). Similarly, ̟ uj by differentiating the identity ̟ 0 = uj0 with ∂a , we conclude that ∂a ̟0 = f(H, E). The relations ∂t Sα = f(H, E) and ∂a S0 = f(H, E) can be proved using a similar argument based on equations (27) and (41), and we omit the details. The facts that ∂b ̟ b = f(H, E) and ∂b S b = f(H, E) follow trivially from the definitions. We have therefore proved (228b). To prove (229a), we first note that definition (227c) and (228a) imply that (G−1 )ab ∂a ̟b = f(H). Hence, (229a) follows from the Leibniz and chain rules. (229b) follows from the same argument.

9.5. Elliptic estimates and the corresponding energies. In this subsection, we construct the energies that we will use to control the top-order derivatives of the vorticity and entropy; see Def. 9.7. The proof that energies are coercive relies on elliptic estimates; see the proof of Lemma 9.4. We start by defining a bilinear form on the relevant Hilbert space of functions. Lemma 9.4 shows that the bilinear form induces a norm on the Hilbert space.

Definition 9.6 (A new Hilbert space inner product). Let (̟, S) be (vorticity, entropy gradient) variables that have been η-orthogonally projected onto Σt (see the notation in Subsubsect. 9.1.1). Let (M −1 )ij (t, ·) be an inverse Riemannian metric on Σt . We define the

66

Relativistic Euler

following bilinear form on the corresponding Hilbert space (H N (Σt ))3 × (H N (Σt ))3 : E D X Z e (M −1 )ab ∂a ∂I~̟b (M −1 )ab ∂a ∂I~̟ (t) := e b dx e S (̟, S) , ̟, M −1

~ |I|=N −1

+

X Z

~ |I|=N −1

+

X Z

~ |I|=N −1

+

X Z

~ |I|=N −1

+

(230)

Σt

X Z

~ |I|≤N −1

Σt

Σt

Σt

o −1 ab n (M ) ∂a ∂I~Sb (M −1 )ab ∂a ∂I~Seb dx δ ab (3) curla (∂I~̟)(3) curlb (∂I~̟) e dx e dx δ ab (3) curla (∂I~S)(3) curlb (∂I~S) ab

Σt

δ (∂I~̟a )(∂I~̟ e b ) dx +

X Z

~ |I|≤N −1

where δ ab is the standard Kronecker delta.

Σt

δ ab (∂I~Sa )(∂I~Seb ) dx,

We now define the family of energies that we will use to control the top-order derivatives of the vorticity and entropy. Definition 9.7 (“Elliptic” energy). Let N > 5/2 be an integer and let (M −1 )ij (t, ·) be an inverse Riemannian metric on Σt . We define the “elliptic” energy EN ;M −1 [(̟, S)] ≥ 0 as follows: E2N ;M −1 [(̟, S)] = E2N ;M −1 [(̟, S)](t) := h(̟, S) , (̟, S)iM −1 (t).

(231)

In the next lemma, with the help of elliptic estimates, we exhibit the coercivity of EN ;M −1 [(̟, S)](t). The lemma shows in particular that the bilinear form from Def. 9.6 is a Hilbert space inner product. Lemma 9.4 (Energy-norm comparison estimate based on elliptic estimates). Let T > 0, let λ be the infimum of the eigenvalues of the 3 × 3 matrix (M −1 )ij (t, x) over (t, x) ∈ [0, T ] × T3 , and let Λ be the supremum of the eigenvalues of the 3 × 3 matrix (M −1 )ij (t, x) over (t, x) ∈ [0, T ] × T3 , and assume that 0 < λ ≤ Λ < ∞. There exists a constant C > 1, depending continuously in an increasing fashion on maxi,j=1,2,3 k(M −1 )ij k2C([0,T ],C(T3 )) , in an increasing fashion on Λ, and in an increasing fashion on λ−1 , such that the following comparison estimate holds for t ∈ [0, T ]: EN ;M −1 [(̟, S)](t) ≤ C 3 X a=1

3 X a=1

a

k̟ kH N (Σt ) + C kskH N+1 (Σt ) + C

k̟ a kH N (Σt ) + kskH N+1 (Σt ) +

3 X a=1

3 X a=1

kSa kH N (Σt ) ,

kSa kH N (Σt ) ≤ CEN ;M −1 [(̟, S)](t).

(232a) (232b)

Proof. We prove only (232b) since (232a) can be proved using similar but simpler arguments. From definitions (230) and (231) and our assumptions on the matrices M −1 (t, ·), we see that

67


there is a constant C > 1 with the properties stated in the lemma such that the following estimate holds for t ∈ [0, T ]: E2N ;M −1 [(̟, S)](t) ≥

1 C

I

L

+ (Σt )

a=1

I

L (Σt )

(233)

3 X X X

(3)

δ ab ∂a ∂~Sb 2 2

curla (∂~S) 2 2 + I I L (Σt ) L (Σt ) ~ |I|=N −1 a=1

~ |I|=N −1

3 X

3 X X

(3)

curla (∂~̟) 2 2

~ |I|=N −1 a=1

~ |I|=N −1

1 + C +

X

δ ab ∂a ∂~̟b 2 2

a

k̟ kH N−1 (Σt ) +

3 X

a,b=1

k∂a Sb kH N−1 (Σt ) .

The desired estimate (232b) now follows from applying the following “elliptic identity” for one-forms V on Σt (recall that by definition (196), Σt is diffeomorphic to T3 ) to the first four sums on RHS (233):

ab

δ ∂a Vb 2 2

L (Σt )

3 X

(3)

curla (V ) 2 2 +

L (Σt )

a=1

=

3 X

a,b=1

k∂a Vb k2L2 (Σt ) .

(234)

The identity (234) follows from integrating the following divergence identity (which is straightforward to directly check) over Σt and noting that integrals over Σt of perfect spatial derivatives vanish: 3 X (∂a Vb )2 = δ ab δ cd ∂a Vc ∂b Vd = (δ ab ∂a Vb )2 + δ ab(3) curla (V )(3) curlb (V ) (235) a,b=1

+ ∂a δ ab δ cd Vc ∂d Vb − ∂a δ ab δ cd Vb ∂c Vd .

In the next lemma, we show that Sobolev norms of the elliptic variables E can be bounded by corresponding Sobolev norms of the hyperbolic variables H. We also derive related estimates for the difference of two solutions. The main ingredients in the proofs are the elliptic estimates provided by Lemma 9.4. Lemma 9.5 (Controlling Sobolev norms of the elliptic variables in terms of Sobolev norms of the hyperbolic variables). (A). Let (h, s, u) be one of the solutions to the relativistic Euler equations (24)-(26) + (29) provided by Theorem 9.1. In particular, let N > 5/2 be an integer, let [0, T ] × T3 be the slab of existence provided by the theorem, and let K be the set featured in theorem. Assume in addition that the initial data components belong to C ∞ (T3 ) and note that by Theorem 9.1 and the Sobolev embedding result (204), the solution components belong to C ∞ ([0, T ] × T3 ). Let E and H be the corresponding elliptic and hyperbolic variables as defined in Def. 9.5. Then there exists a constant C > 0, depending only on: (1) N (2) K P P (3) khkH N (Σ0 ) , kskH N+1 (Σ0 ) , 3a=1 kua kH N (Σ0 ) , and 3a=1 k̟ akH N (Σ0 )

68

Relativistic Euler (4) khkL∞ ([0,T ],C 1 (T3 )) + kskL∞ ([0,T ],C 1 (T3 )) + +

3 X a=1

a

kS kL∞ ([0,T ],C 1 (T3 )) +

3 X a=1

3 X a=1

kua kL∞ ([0,T ],C 1 (T3 ))

(236)

k̟ akL∞ ([0,T ],C 1 (T3 )) ,

such that the following estimate holds for t ∈ [0, T ]:

kEkH N−1 (Σt ) ≤ CkHkH N−1 (Σt ) .

(237)

(B). For i = 1, 2, let (h(i) , s(i) , u(i) ) be classical solutions to the relativistic Euler equations (24)-(26) + (29) that have the properties stated in part (A). Assume that the slab of existence [0, T ] × T3 is the same for both solutions and that the set K is the same for both solutions, that is, that there exists a compact set K ⊂ intH such that for i = 1, 2, we have (h(i) , s(i) , u1(i) , u2(i) , u3(i) )([0, T ] × T3 ) ⊂ intK. Let E(i) and H(i) be the corresponding elliptic and hyperbolic variables as defined in Def. 9.5. Consider now Lemma 9.1, where in the hypotheses of the lemma, we take w := (h(1) , s(1) , u1(1) , u2(1) , u3(1) ), w e := (h(2) , s(2) , u1(2) , u2(2) , u3(2) ), the set K is the one mentioned above, the elements in the array v are defined to be w together with the remaining variables from Def. 9.5 (evaluated at the solution corresponding to i = 1), and the remaining elements in the array e v are defined to be w e together with the remaining variables from Def. 9.5 (evaluated at the solution corresponding to i = 2). Let δ > 0 be as in the statement of Lemma 9.1, and assume that kw − wk e C(Σt ) ≤ δ. Then there exists a constant C > 0, depending only on: (1) N (2) K P P (3) khkH N (Σ0 ) , kskH N+1 (Σ0 ) , 3a=1 kua kH N (Σ0 ) , and 3a=1 k̟ akH N (Σ0 ) (4) khkC([0,T ],C 1(T3 )) + kskC([0,T ],C 1 (T3 )) + +

3 X a=1

(5) δ (6)

a

kS kC([0,T ],C 1 (T3 )) +

3 X

3 X a=1

kua kC([0,T ],C 1(T3 ))

k̟ akC([0,T ],C 1 (T3 ))

a=1

3 Xn X kh(i) kC([0,T ],H N (T3 )) + kua(i) kC([0,T ],H N (T3 ))

(239)

a=1

i=1,2

ks(i) kC([0,T ],H N+1 (T3 )) +

(238)

3 X a=1

o a k̟(i) kC([0,T ],H N (T3 )) ,

such that the following estimate holds for t ∈ [0, T ]:

kE(1) − E(2) kH N−1 (Σt ) ≤ CkH(1) − H(2) kH N−1 (Σt ) .

(240)

69


Proof. Throughout this proof, C denotes a constant with the dependence properties stated in the lemma. We begin by establishing (237). Invoking definitions (230) and (231) and the estimate (232b) with M −1 := G−1 (where G−1 is defined in Def. 9.4, and we stress that the proof of (232b) relied on elliptic estimates), we find that 1/2

kEkH N−1 (Σt ) ≤ CEN ;G−1 [(̟, S)](t) X X

(G−1 )ab ∂a ∂~̟b 2

(G−1 )ab ∂a ∂~Sb 2 ≤C + C I I L (Σt ) L (Σt ) ~ |I|=N −1

~ |I|=N −1

+ CkHkH N−1 (Σt ) .

(241)

Next, using (229a)-(229b), we see that the terms (G−1 )ab ∂a ∂I~̟b and (G−1 )ab ∂a ∂I~Sb on the first line of RHS (241) are smooth functions of H and its spatial derivatives. Thus, using inequality (206) to bound RHSs (229a)-(229b) in the norm k · kL2 (Σt ) , we arrive at the desired estimate (237). We stress that RHS (206) is linear in the order r derivatives of the solution; this is the reason that RHS (206) is linear in kHkH N−1 (Σt ) . We now prove (240). For i = 1, 2, we let G−1 (i) denote the inverse Riemannian metric th corresponding to the i solution, that is, the inverse Riemannian metric whose rectangular components are formed by evaluating RHS (226) at the solution corresponding to the labeling index i. To proceed, we use definitions (230) and (231) and the comparison estimate (232b) with M −1 := G−1 (1) to deduce that 1/2

kE(1) − E(2) kH N−1 (Σt ) ≤ CEN ;G−1 [(̟(1) − ̟ (2) , S (1) − S (2) )](t) (1)

X

−1 ab

≤C

(G(1) ) ∂a ∂I~(̟(1)b − ̟(2)b )

L2 (Σt )

~ |I|=N −1

X

−1 ab

(G(1) ) ∂a ∂I~(S(1)b − S(2)b )

+C

(242)

~ |I|=N −1

L2 (Σt )

+ CkH(1) − H(2) kH N−1 (Σt ) .

Next, using the triangle inequality, we find that

X

−1 ab

−1 ab (G ) ∂ ∂ ̟ − (G RHS (242) ≤ C ) ∂ ∂ ̟

(1) a I~ (1)b a I~ (2)b (2)

L2 (Σt )

~ |I|=N −1

X

−1 ab −1 ab +C

(G(2) ) − (G(1) ) ~ |I|=N −1

+C

C(Σt )

(243)

∂a ∂~̟(2)b 2 . I L (Σt )

X

−1 ab

ab ) ∂ ∂ S )

(G(1) ) ∂a ∂I~S(1)b − (G−1 a I~ (2)b (2)

L2 (Σt )

~ |I|=N −1

X

−1 ab −1 ab +C

(G(2) ) − (G(1) ) ~ |I|=N −1

C(Σt )

∂a ∂~S(2)b 2 I L (Σt )

+ CkH(1) − H(2) kH N−1 (Σt ) .

P P3 P P3

Using the assumed bounds |I|=N ~ ~ a,b=1 ∂a ∂I~ S(2)b L2 (Σt ) ≤ a,b=1 ∂a ∂I~ ̟(2)b L2 (Σt ) + |I|=N −1 −1 C, (229a)-(229b), and (207) (where the hypotheses needed to invoke (207) are satisfied due to our assumptions involving δ and K), we see that the terms on the first and third lines of

70

Relativistic Euler

RHS (243) are ≤ CkH(1) − H(2) kH N−1 (Σt ) as desired. To handle the terms on the second and

P P3

fourth lines of RHS (243), we use the assumed bounds |I|=N ~ a,b=1 ∂a ∂I~̟(2)b L2 (Σt ) + −1

P P3 −1 ab −1 ab

∂ ∂ S ≤ C, the mean value theorem estimate (G ) − (G ) (2) ≤ ~ a I~ (2)b L2 (Σt ) (1) |I|=N −1 a,b=1 C H(1) − H(2) (where we are using that RHS (226) can be viewed as a smooth function of (u1 , u2 , u3)), and the Sobolev embedding result (204) to deduce that the terms on the second and fourth lines of RHS (243) are ≤ CkH(1) − H(2) kC(Σt ) ≤ CkH(1) − H(2) kH N−1 (Σt ) as desired. We have therefore proved (240). 9.6. Energies for the wave equations via the vectorfield multiplier method. In this subsection, we derive a priori estimates for our new formulation of the relativistic Euler equations. The main result is provided by the next proposition. The proposition shows in particular that the vorticity and entropy are one degree more differentiable compared to the standard estimates that follow from first-order formulations of the equations. The main analytic tools in the proof of the proposition are the elliptic estimates from Subsect. 9.5 and the vectorfield method for wave equations (see Subsubsect. 9.6.1). Proposition 9.6 (A priori estimates for solutions to the relativistic Euler equations). Let ˚ h := h|Σ0 , ˚ s := s|Σ0 , and ˚ ui := ui |Σ0 be initial data for the relativistic Euler equations (24)-(26) + (29) obeying the assumptions of Theorem 9.1 and let (h, s, u0 , u1, u2 , u3 ) be the corresponding solution. In particular, let N > 5/2 be an integer, let [0, T ] × T3 be the slab of existence provided by the theorem, and let K be the set featured in theorem. Assume in addition that the initial data components belong to C ∞ (T3 ) and note that by Remark 9.4, the solution components belong to C ∞ ([0, T ] × T3 ). Let ̟ be the vorticity (see definition 2.3), and let ̟ ˚ i := ̟ i |Σ0 be its initial spatial components. Then there exists a constant C > 0, depending only on: (1) (2) (3) (4)

N K P P ̟ akH N (Σ0 ) ua kH N (Σ0 ) , k˚ skH N+1 (Σ0 ) , and 3a=1 k˚ k˚ hkH N (Σ0 ) , 3a=1 k˚ khkL∞ ([0,T ],C 1 (T3 )) + +

3 X a=1

3 X a=1

a

kua kL∞ ([0,T ],C 1(T3 )) + kskL∞ ([0,T ],C 1 (T3 ))

kS kL∞ ([0,T ],C 1 (T3 )) +

3 X a=1

(244)

k̟ akL∞ ([0,T ],C 1 (T3 ))

such that for t ∈ [0, T ], the components of the solution with respect to the standard coordinates verify the following estimates: khkH N (Σt ) +

3 X a=0

kuα − δ0α kH N (Σt ) + kskH N+1 (Σt ) +

≤ C exp(Ct) ≤ C exp(CT ) := C∗ , where δ0α is the Kronecker delta.

3 X α=0

kS α kH N (Σt ) +

3 X α=0

k̟ αkH N (Σt ) (245)


71

The proof of Prop. 9.6 is located in Subsubsect. 9.6.4. We will first derive some preliminary results. We start by noting that we can rewrite (37), (39), (41), (42), (43a), and (45b) in concise form as follows, where f denotes a smooth function of its arguments that is free to vary from line to line and that satisfies f(0) = 0, V denotes η-orthogonal projection of V onto constant-time hypersurfaces (see Subsubsect. 9.1.1), and the hyperbolic variables H and the elliptic variables E are as in Def. 9.5: 2g h = f(H),

(246a)

2g u = f(H),

(246b)

α

u ∂α S = f(H),

(246c)

α

u ∂α ̟ = f(H),

(246d)

uα ∂α D = f(H, E),

(246e)

α

u ∂α C = f(H, E).

(246f)

The crux of the proof of Prop. 9.6 is to derive energy estimates for the covariant wave equations (246a) and (246b), energy estimates for the transport equations (246c), (246d), (246e), and (246f), and elliptic estimates to handle the terms E on RHSs (246e) and (246f). We have already derived the necessary elliptic estimates in Subsect. 9.5. In the next three subsections, we will outline the energy estimates, which are standard. 9.6.1. Energy estimates for covariant wave equations. The wave operator in (246a) and (246b) is with respect to the acoustical metric g introduced in Definition 2.1. These are covariant wave equations for the scalar quantities h and uα . Estimates for such equations can be derived by using the well-known vectorfield multiplier method24 for wave equations, which we outline in this subsubsection. Let ϕ be any element of {h, u1 , u2, u3 } (in practice, we will not need to derive separate energy estimates for u0 since estimates for u0 can be obtained from the estimates for the spatial components of u and the normalization condition ηκλ uκ uλ = −1). We start by defining the energy-momentum tensor 1 Tαβ = Tαβ [ϕ] := ∂α ϕ∂β ϕ − gαβ (g −1 )µν ∂µ ϕ∂ν ϕ. (247) 2 A crucial property of Tαβ is that it satisfies the dominant energy condition: Tαβ X α Y β ≥ 0 whenever the vectorfields X and Y are future-directed25 and timelike26 with respect to g. In practice, the dominant energy condition allows one to construct energies that are coercive along causal hypersurfaces;27 see equation (257) below for the energy that we use in deriving a priori estimates for h and u. Next, for any vectorfield X (soon to be employed in the role of a “multiplier vectorfield”), we let (X)π be its deformation tensor relative to g, which takes the following form relative to 24In

deriving a priori estimates, we will use only a trivial version of the vectorfield commutator method. Specifically, we will commute the equations only with the spatial derivative operators ∂I~. 25By a “future-directed” vectorfield X, we mean that X 0 > 0. 26 X is defined to be timelike with respect to g if gαβ X α X β < 0. 27By a “causal hypersurface,” we mean a hypersurface whose future-directed unit normal is either timelike or null at each point.

72

Relativistic Euler

arbitrary coordinates: (X)

παβ := gβµ ∇α X µ + gαµ ∇β X µ .

(248)

In (248) and in the rest of this subsubsection, ∇ is the covariant derivative associated with g. Next, we define the energy current vectorfield corresponding to X as follows: (X) α

J = (X)J α [ϕ] := (g −1)αµ Tµβ [ϕ]X β − X α ϕ2 .

(249)

From straightforward computations, we derive the following identity: 1 ∇α (X)J α = (g ϕ) X α ∂α ϕ + (g −1)αγ (g −1 )βδ Tαβ (X)πγδ + (∇α X α )ϕ2 + 2ϕ(X α∂α ϕ). 2

(250)

Applying the divergence theorem on the spacetime slab [0, T ]×T3 and using (250), we deduce the following identity: Z Z (X) α β gαβ (X)J α [ϕ]N β dµg (251) gαβ J [ϕ]N dµg = Σ0 Σt Z 1 −1 αγ −1 βδ α (X) + (g ϕ) X ∂α ϕ + (g ) (g ) Tαβ πγδ dµg 2 [0,t]×T3 Z + (∇α X α )ϕ2 + 2ϕ(X α ∂α ϕ) dµg . [0,t]×T3

In (251), dµg is the volume form that g induces on [0,t] ×T3 , N is the future-directed unit normal to Σt with respect to the metric g, and dµg is the volume form that g induces on Σt . Here g is the first fundamental form of Σt , that is, g ij := gij for 1 ≤ i, j ≤ 3. We also note p −1 )α0 that relative to the standard coordinates, N α = − √(g −1 , dµg = |detg|dx1 dx2 dx3 dx0 , 00 −(g ) p p p 1 2 3 −1 00 and dµg = detgdx dx dx = −(g ) |detg|dx1 dx2 dx3 , where the last equality is a basic linear algebraic identity. Note that N is future-directed and timelike with respect to g, and that we used the fact that (g −1 )00 < 0 (which is a simple consequence of the formula (15b)). From the above discussion, it follows that along any spacelike hypersurface with futuredirected unit norm N, we can construct a positive definite energy density gαβ (X)J α [ϕ]N β using any multiplier vectorfield X that is future-directed and timelike with respect to g. For the basic a priori estimates of interest to us, we will apply the above constructions along Σt with X := u, which is future-directed timelike with respect to g; note that by (15a) and the normalization condition ηκλ uκ uλ = −1, we have gκλuκ uλ = −1. Thus, we define the −1 )α0 following energy (where N α = − √(g −1 ): 00 −(g

)

Ewave (t) = Ewave [ϕ](t) :=

Z

gαβ (u)J α [ϕ]N β dµg .

(252)

Σt

From (251), definition (252), and the standard expansion28 of covariant derivatives in terms of partial derivatives and Christoffel symbols (which in particular can be used to derive the 28For

example, ∇α X β = ∂α X β + Γαβ γ X γ , where Γαβ γ is defined by (254).


73

identity (u)παβ = uκ ∂κ gαβ + gακ ∂β uκ + gβκ ∂α uκ ), we deduce the following energy identity relative to the standard coordinates: Ewave [ϕ](t) = Ewave [ϕ](0) (253) Z 1 −1 αγ −1 βδ κ κ −1 βδ α + (g ϕ)u ∂κ ϕ + (g ) (g ) Tαβ [ϕ]u ∂κ gγδ + (g ) Tαβ [ϕ]∂δ u dµg 2 [0,t]×T3 Z + (∂κ uκ )ϕ2 + (Γκκλ uκ )ϕ2 + 2ϕuκ ∂κ ϕ dµg . [0,t]×T3

On RHS (253),

1 Γαγ β := (g −1 )γδ {∂α gδβ + ∂β gαδ − ∂δ gαβ } 2 are the Christoffel symbols of g. Note that by (15a)-(15b) we have that Γαγ β = f(h, s, u, ∂h, S, ∂u),

(254)

(255)

where f is a smooth function (depending on α, β, and γ). Next, with the help of (15a)-(15b) and the normalization condition ηκλuκ uλ = −1, we compute that 1 gαβ (u)J α [ϕ]N β = c2 T0β [ϕ]uβ + (1 − c2 )u0 Tαβ [ϕ]uα uβ + u0 ϕ2 p (256) −(g −1 )00 1 2 0 1 2 0 ab 1 2 2 α 2 2 a c u (∂ ϕ) + c u δ (∂ ϕ)∂ ϕ + (1 − c )(u ∂ ϕ) + c (∂ ϕ)u ∂ ϕ t a b α t a 2 2 q = 2 −(g −1 )00 +q

u 0 ϕ2

−(g −1 )00

,

where δ ab is the Kronecker delta. From (252) and (256), it follows that Z dµg 1 2 0 1 2 0 ab 1 2 2 α 2 q Ewave [ϕ](t) = c u (∂t ϕ) + c u δ (∂a ϕ)∂b ϕ + (1 − c )(u ∂α ϕ) 2 2 2 Σt −(g −1 )00 (257) Z dµg 2 c (∂t ϕ)ua ∂a ϕ + u0 ϕ2 q + . Σt −(g −1 )00

The energy Ewave [ϕ](t) will yield control of ϕ and its first derivatives. In Subsubsect. 9.6.3, we will establish the coerciveness Ewave [ϕ](t). To obtain control of the higher-order spatial derivatives of ϕ, one can use energies of the form Ewave [∂I~ϕ], where I~ is a spatial multi-index.

9.6.2. Energy estimates for transport equations. One can derive energy estimates for transport equations of the form uα ∂α ϕ = f by relying on the following energy: Z ϕ2 dx, (258) Etransport [ϕ](t) := Σt

74

Relativistic Euler

as in the proof of Lemma 9.2. The analog of the energy identity (253) is the following integral identity, whose simple proof follows from the ideas featured in the proof of Lemma 9.2: Z t Z a u ∂a Etransport [ϕ](t) = Etransport [ϕ](0) + ϕ2 dx dτ (259) u0 0 Στ Z tZ uα ∂α ϕ ϕ +2 dx dτ. u0 0 Στ To control the higher-order derivatives of ϕ, one can rely on energies of the form Etransport [∂I~ϕ]. We mention that the argument we have sketched here relies on the basic fact that u0 > 0, which allows us to divide by u0 on RHS (259); for the relativistic Euler equations, this fact follows from the normalization condition ηκλ uκ uλ = −1 and the fact that u is future-directed. 9.6.3. Comparison of the energies with the Sobolev norm. The coerciveness properties of the wave equation energy Ewave [ϕ](t) constructed in Subsubsect. 9.6.1 are tied to the metric g; see (252). In order to obtain our results, we need Ewave [ϕ](t) to be uniformly comparable to a corresponding Sobolev norm along Σt . More precisely, we need to ensure the existence of a constant C > 1 such that on the slab [0, T ] × T3 of existence guaranteed by Theorem 9.1, the following estimates hold: o n X Ewave [∂I~ϕ](t) (260) C −1 kϕk2H N (Σt ) + k∂t ϕk2H N−1 (Σt ) ≤ ~ 0≤|I|≤N −1

o n ≤ C kϕk2H N (Σt ) + k∂t ϕk2H N−1 (Σt ) .

To see that such a constant C exists, we first use Young’s inequality and Cauchy–Schwarz to bound the first product in braces on the last line of RHS (257) as follows: v v u 3 u 3 u uX X 1 1 c2 (∂t ϕ)ua ∂a ϕ ≥ − c2 t (ui )2 (∂t ϕ)2 − c2 t (ui )2 δ ab (∂a ϕ)∂b ϕ (261) 2 2 i=1 i=1 1 p 1 p = − c2 (u0 )2 − 1(∂t ϕ)2 − c2 (u0 )2 − 1δ ab (∂a ϕ)∂b ϕ. 2 2

Next, we recall that Theorem 9.1 guarantees that on [0, T ] × T3 , the solution never escapes the compact subset K featured in the statement of the theorem. In view of (261), we see that this ensures that on [0, T ]×T3 , the product c2 (∂t ϕ)ua ∂a ϕ on the last line of RHS (257) can be absorbed into the sum 21 c2 u0 (∂t ϕ)2 + 21 c2 u0 δ ab (∂a ϕ)∂b ϕ from the first line of RHS (257), with room to spare. This implies that for solutions contained in K, the integrands on RHS (257) P are in total uniformly comparable to 3α=0 (∂α ϕ)2 + ϕ2 . This also ensures that on [0, T ] × T3 , dµg the volume form √ −1 00 on Σt is uniformly comparable29 to dx := dx1 dx2 dx3 . From these −g

observations, it readily follows that a C > 1 exists such that (260) holds. 29To

see this, it is helpful to note the following identity, which holds relative to the standard coordµg dinates: √ −1 00 = c−3 dx1 dx2 dx3 . This identity follows from (16a) and the linear algebraic identity −g p detg = −(g −1 )00 |detg|.

75


9.6.4. Proof of Prop. 9.6. Recall that the assumptions of the proposition guarantee that we have a smooth solution to the system (24)-(26), (29). Consider the quantities h, uα , S α , ̟ α, C α , D,

(262)

introduced in Sect. 2. According to Theorem 3.1, they satisfy the system of evolution equations given by equations (37), (39), (41), (42), (43a), and (45b). Next, we recall that the hyperbolic quantities H and the elliptic quantities E were defined in Def. 9.5. To prove the proposition, we claim that it suffices to show that the following inequality holds for t ∈ [0, T ]: Z t 2 2 kHkH N−1 (Σt ) ≤ CkHkH N−1 (Σ0 ) + C kHk2H N−1 (Στ ) dτ, (263) 0

where in (263) and in the rest of this proof, C is as in the statement of Prop. 9.6. For once we have shown (263), we can use Gronwall’s inequality to deduce (recalling that C is allowed to depend on the initial data) that the following estimate holds for t ∈ [0, T ]: kHk2H N−1 (Σt ) ≤ CkHk2H N−1 (Σ0 ) exp(Ct) ≤ C exp(Ct) ≤ C exp(CT ).

(264)

Then from (237) and (264) we conclude, in view of Def. 9.5, the desired bound (245), except for the estimates for u0 , S 0 , and ̟ 0. To obtain the desired estimate for these quantities, we first express u0 − 1, S 0 , ̟ 0 , ∂a u0 , ∂a S 0 , and ∂a ̟ 0 as f(H, E), with f smooth and satisfying f(0) = 0 (this is possible in view of definition (227e) and (228c)). We then use Lemma 9.1 to deduce that kf(H, E)kH N−1(Σt ) ≤ CkHkH N−1 (Σt ) + CkEkH N−1 (Σt ) . Finally, we use the elliptic estimate (237) and (264) to conclude that CkHkH N−1 (Σt ) + CkEkH N−1 (Σt ) ≤ RHS (245), which yields the desired estimates. It remains for us to prove (263). We start by noting that the results described in Subsubsects. 9.6.1-9.6.3 can be used to derive the following estimates, where we recall that V denotes η-orthogonal projection of V onto constant-time hypersurfaces (see Subsubsect. 9.1.1): Z t o n 2 2 2 2 kHk2H N−1 (Στ ) dτ, khkH N (Σt ) + k∂t hkH N−1 (Σt ) ≤ C khkH N (Σ0 ) + k∂t hkH N−1 (Σ0 ) + C 0

o n kuk2H N (Σt ) + k∂t uk2H N (Σt ) ≤ C kuk2H N (Σ0 ) + k∂t uk2H N (Σt ) + C kSk2H N−1 (Σt ) ≤ CkSk2H N−1 (Σ0 ) + C k̟k2H N−1 (Σt ) kDk2H N−1 (Σt ) kCk2H N−1 (Σt )

≤

Ck̟k2H N−1 (Σ0 )

≤

CkDk2H N−1 (Σ0 )

+C

Z

(265a)

t 0

kHk2H N−1 (Στ ) dτ, (265b)

t 0

Z

0

+C

Z

kHk2H N−1 (Στ ) dτ,

(265c)

kHk2H N−1 (Στ ) dτ,

(265d)

t

Z tn 0

kHk2H N−1 (Στ ) + kEk2H N−1 (Στ )

o

dτ,

(265e) Z tn o ≤ CkCk2H N−1 (Σ0 ) + C kHk2H N−1 (Στ ) + kEk2H N−1 (Στ ) dτ. 0

(265f)

76

Relativistic Euler

The estimates (265a)-(265f) are standard and can be derived by commuting the evolution equations of Theorem 3.1 (more precisely, only the evolution equations for the spatial components of u, ̟, S, and C) with spatial derivative operators ∂I~ and using the energy identities (253) and (259) (and their analogs for the ∂I~-differentiated solution variables), the coerciveness estimate (260), and the Sobolev–Moser-type estimate (206). We stress that RHS (206) is linear in the order r derivatives of the solution; this is the reason the integrands on RHS (265a)-(265f) are quadratic in kHkH N−1 (Στ ) and kEkH N−1 (Στ ) (the sup-norm factors RHS (206) can be bounded by ≤ C since those factors are among the quantities that constants C are allowed to depend on). The non-standard aspect of the remaining part of the proof is the appearance of the term kEk2H N−1 (Στ ) on RHSs (265e)-(265f); we clarify that these terms are generated by the terms ∂S and ∂̟ on RHSs (43a) and (45b) (see definition (227b)). Next, adding (265a)-(265f) and appealing to Def. 9.5, we deduce that Z tn o 2 2 (266) kHkH N−1 (Σt ) ≤ CkHkH N−1 (Σ0 ) + kHk2H N−1 (Στ ) + kEk2H N−1 (Στ ) dτ. 0

Finally, from (266) and the elliptic estimate (237), we conclude the desired bound (263).

9.7. Proof of Theorem 9.2. We now prove Theorem 9.2, which is the main result of Sect. 9. By Theorem 9.1, we need only to show that i) under the regularity assumptions on the initial data stated in Theorem 9.2, the standard local well-posedness results (223a)-(223b) can be upgraded to (225a)-(225b) and ii) that the solution depends continuously on the initial data, where continuity is measured in the norms corresponding to the function spaces featured in 5 (225a)-(225b). To proceed, we let (˚ h(m) ,˚ s(m) , ˚ ui(m) ) ⊂ (C ∞ (T3 )) be a sequence of smooth initial data such that as m → ∞, we have

i

˚ ˚

˚ → 0, u(m) − ˚ ui H N (Σ ) → 0, (267)

h(m) − h N 0 H (Σ0 )

i

˚

̟ s(m) − ˚ s H N+1 (Σ0 ) → 0, ˚(m) − ̟ ˚ i H N (Σ ) → 0, (268) 0

i denotes the initial vorticity of the mth element of the sequence and ̟ ˚ i is as in where ̟ ˚(m) α α ) denote the corresponding sethe statement of the theorem. Let (h(m) , s(m) , uα(m) , S(m) , ̟(m) quence of solution variables. Theorem 9.1 yields (see, for example, [29], for additional details) that for m sufficiently large, the element (h(m) , s(m) , uα(m) ) is a C ∞ classical solution to equations (24)-(26) + (29) on the fixed slab [0, T ]×T3 with (h(m) (p), s(m) (p), u1(m) (p), u2(m) (p), u3(m) (p)) ∈ α α intK for p ∈ [0, T ] × T3 , and that on the same slab, (h(m) , s(m) , uα(m) , S(m) , ̟(m) ) is a C ∞ solution to the equations of Theorem 3.1 (which are consequences of (24)-(26) + (29)). Moreover, Theorem 9.1 also implies that the sequence converges to the solution in the following norms as m → ∞:

α

u − uα

h(m) − h → 0, (269) → 0, N 3 (m) C([0,T ],H (T )) C([0,T ],H N (T3 ))

α

α

s(m) − s − ̟ α C([0,T ],H N−1 (T3 )) → 0. → 0, S(m) − S α C([0,T ],H N−1 (T3 )) → 0, ̟(m) C([0,T ],H N (T3 )) (270)

Next, we use the convergence results (269)-(270), Theorem 9.1, and the a priori estimates provided by Prop. 9.6 to deduce that exist a constant C > 0, depending on T and on the

77

M. Disconzi, J. Speck four types of quantities listed just above (245), and a positive integer m0 such that sup sup ks(m) kH N+1 (Στ ) ≤ C,

m≥m0 τ ∈[0,T ]

α kH N (Στ ) ≤ C. sup sup k̟(m)

α kH N (Στ ) ≤ C, sup sup kS(m)

m≥m0 τ ∈[0,T ]

m≥m0 τ ∈[0,T ]

(271) Since H r (T3 ) is a Hilbert space for r ∈ R, it follows from the norm-boundedness results α α (271) that for each τ ∈ [0, T ], there exist subsequences s(mn ) , S(m , and ̟(m that weakly n) n) N +1 N N converge in H (Στ ), H (Στ ), and H (Στ ) respectively as n → ∞. Moreover, since the norm is weakly lower semicontinuous in a Hilbert space, it follows that the limits are bounded, respectively, in the norms k · kH N+1 (Στ ) , k · kH N (Στ ) , and k · kH N (Στ ) , by ≤ C, where C is the same constant on RHS (271). From (270), it follows that the limits must be s, S α , and ̟ α respectively. We have therefore shown that sup kskH N+1 (Στ ) ≤ C,

τ ∈[0,T ]

sup kS α kH N (Στ ) ≤ C,

τ ∈[0,T ]

sup k̟ αkH N (Στ ) ≤ C.

(272)

τ ∈[0,T ]

To complete the proof of (225b), we must show that for each spatial multi-index I~ with ~ = N, the map t → ∂~S α (t, ·) is a continuous map from [0, T ] into L2 (T3 ), and similarly |I| I for ̟ α (the desired time-continuity results for s then follow from the relation ∂i s = Si ). To keep the presentation short, we illustrate only the right-continuity of these maps at t = 0; the general statement can be proved by making minor modifications to the argument that we give. That is, we will show that ˚α (·)kL2 (T3 ) = 0, ~ = N, lim k∂~S α (t, ·) − ∂~S |I| (273a) t↓0

I

I

lim k∂I~̟ α(t, ·) − ∂I~̟ ˚ α (·)kL2 (T3 ) = 0, t↓0

~ = N, |I|

(273b)

˚α (·) := S α (0, ·). The rest of our proof is based on Lemmas 9.2 and 9.4, but to apply where S the lemmas, we first have to derive some preliminary results. We will use the estimates provided by Lemma 9.1 without giving complete details each time we use them; we will refer to these estimates as the “standard Sobolev calculus.” As a first step in proving (273a)-(273b), we will show that H, AH ∈ C([0, T ], H N −1(T3 )),

(274)

where H and AH are defined in (227a) and (227c). Note that by (228a) and the standard Sobolev calculus, the desired result AH ∈ C([0, T ], H N −1(T3 )) would follow from H ∈ C([0, T ], H N −1(T3 )). The latter statement is equivalent to showing that ∂I~H ∈ ~ ≤ N − 1. All of these results, except in the case of the top-order C([0, T ], L2 (T3 )) for |I| (i.e., order N − 1) derivatives of C i and D, follow from the standard local well-posedness time-continuity results (223a), and the standard Sobolev calculus. Thus, to complete the proof of (274), we need only to show that ∂I~C i , ∂I~D ∈ C([0, T ], L2(T3 )),

~ = N − 1. |I|

(275)

The desired result (275) follows from using equations (43a) and (45b) (more precisely, we need only to consider the spatial components of (45b)), the boundedness result (272), the standard local well-posedness time-continuity results (223a)-(223b), and the standard Sobolev calculus to deduce that ∂I~C i and ∂I~D solve transport equations that satisfy the hypotheses

78

Relativistic Euler

of Lemma 9.2; put succinctly, we can apply Lemma 9.2 with f = ∂I~C i and f = ∂I~D. We have therefore proved (274). In particular, it follows from (274) and the definition of AH that (3)

curli (̟),

(3)

curli (S) ∈ C([0, T ], H N −1(T3 )).

(276)

Next, we note that in view of Def. 9.5, Lemma 9.3 (in particular the relation (228b) for S 0 and ̟ 0), (274), and the standard Sobolev calculus, the desired results (273a)-(273b) would follow from proving the following convergence result: ~ = N − 1. lim k∂~E(t, ·) − ∂~E(0, ·)kL2(T3 ) = 0, |I| (277) t↓0

I

I

To establish (277), we first use (274), (229a)-(229b), and the standard Sobolev calculus to deduce that ~ = N − 1, (G−1 )ab ∂a ∂~Sb ∈ C([0, T ], L2 (T3 )), |I| (278) I

~ = N − 1. |I|

−1 ab

(G ) ∂a ∂I~̟b ∈ C([0, T ], L2 (T3 )),

Next, setting

˚−1 )ij (·) := (G−1 )ij (0, ·), (G

(279) (280)

˚−1 )ij , and appealing to definition (227b), we see applying Lemma 9.4 with (M −1 )ij := (G that in order to prove (277), it suffices to show that lim EN ;G˚−1 [(̟, S) − (˚ ̟, ˚ S)](t) = 0. (281) t↓0

To initiate the proof of (281), we let ϕ ∈ H −N (T3 ) be any element of the dual space of H (T3 ). From the below-top-order continuity result (223b), the boundedness result (272), and the density of C ∞ functions in H −N (T3 ), it is straightforward to deduce that the following “weak continuity” result holds for i = 1, 2, 3: Z Z i ˚i ϕ dx. lim S (t, x)ϕ dx = S (282) N

t↓0

T3

T3

˚i in H N (T3 ) as t ↓ 0. Since ϕ was arbitrary, we conclude that S i (t, ·) weakly converges to S i i N 3 Similarly, ̟ (t, ·) weakly converges to ̟ ˚ in H (T ) as t ↓ 0. We now let h·, ·iG˚−1 denote the inner product (230) on the Hilbert space (H N (T3 ))3 × (H N (T3 ))3 , and we let h·, ·i denote the standard inner product on the same Hilbert space (obtained by keeping only the two sums on the last line of RHS (230) and replacing N − 1 with N in the summation bounds). By Lemma 9.4, the two corresponding norms (i.e., the norms on the left- and right-hand sides of (232a)-(232b)) are equivalent. It is a basic result of functional analysis that given these two inner products with equivalent norms, a sequence weakly convergences relative to h·, ·iG˚−1 if and only if it weakly converges relative to h·, ·i. In particular, in view of the above weak convergence results, we infer (see Def. 9.6 regarding the notation) that (̟(t, ·), S(t, ·)) ˚ weakly converges to (˚ ̟(·), S(·)) relative to the inner product h·, ·iG˚−1 as t ↓ 0. Moreover, it is another basic result of functional analysis that based on this weak convergence and Lemma 9.4, in order to prove the result (281), it suffices to show that ̟, ˚ S)]. (283) lim sup E ˚−1 [(̟, S)](t) ≤ E ˚−1 [(˚ t↓0

N ;G

N ;G

79


Moreover, since the standard local well-posedness time-continuity results (223a) and (204)

−1 ij ˚−1 )ij = 0, it follows from (i.e., Sobolev embedding) imply that limt↓0 (G ) (t, ·) − (G

C(T3 )

(223a), (223b), and the high-norm boundedness result (272) that in order to prove (283), it suffices to show that lim sup EN ;G−1 [(̟, S)](t) ≤ EN ;G˚−1 [(˚ ̟, ˚ S)], (284) t↓0

where we stress that the inverse metric G−1 on LHS (284) depends on t (which is different compared to (283)). Moreover, from definitions (230) and (231) and the standard local well-posedness time-continuity results (223a), we see that all terms in the definition of EN ;G−1 [(̟, S)](t) have been shown to have the desired continuous time dependence except for the ones depending on the order N derivatives of ̟ or S (i.e., the ones corresponding to the terms on the first four lines of RHS (230)), whose continuous time dependence we established in (276) and (278)-(279). We have therefore proved (284), which finishes the proof of the desired result (225b). To complete our proof of Theorem 9.2, we need to show continuous dependence on the initial data. To proceed, we let (˚ h(m) , ˚ ui(m) ,˚ s(m) ) be a sequence of initial data (not necessarily ∞ α α C now) such that as m → ∞, we have (267)-(268). We again let (h(m) , s(m) , uα(m) , S(m) , ̟(m) ) ∞ denote the corresponding sequence of solution variables (not necessarily C now). We aim to show that the sequence converges to the limiting solution (h, uα, s, S α , ̟ α) in the norm supτ ∈[0,T ] k · kH N (Στ ) as m → ∞. To proceed, we first note that Theorem 9.1 yields α α that for m sufficiently large, the element (h(m) , s(m) , uα(m) , S(m) , ̟(m) ) is a classical solution ∞ (not necessarily C now) to equations (24)-(26) + (29) on the fixed slab [0, T ] × T3 with (h(m) (p), s(m) (p), u1(m) (p), u2(m) (p), u3(m) (p)) ∈ intK for p ∈ [0, T ] × T3 , that it also is a strong solution30 to the equations of Theorem 3.1, that there exists an integer m0 such that sup ks(m) kC([0,T ],H N+1 (T3 )) ≤ C,

m≥m0

α sup kS(m) kC([0,T ],H N (T3 )) ≤ C,

m≥m0

α sup k̟(m) kC([0,T ],H N (T3 )) ≤ C,

m≥m0

(285) and that the following convergence results (which are below top-order for S and ̟) hold as m → ∞:

α

s − s(m)

u − uα

h − h(m) → 0, → 0, → 0, N 3 (m) C([0,T ],H N (T3 )) C([0,T ],H (T )) C([0,T ],H N (T3 )) (286a)

α

S − S α → 0. (286b) → 0, ̟ α − ̟ α (m) C([0,T ],H N−1 (T3 ))

(m) C([0,T ],H N−1 (T3 ))

In view of (286a)-(286b), to complete our proof of Theorem 9.2 we need only to show ~ = N, then as m → ∞, continuity in the top-order norms. That is, we must show that if |I| we have

∂~S α − ∂~S α

∂~̟ α − ∂~̟ α → 0, → 0. (287) I

I

(m) C([0,T ],L2 (T3 ))

I

I

(m) C([0,T ],L2 (T3 ))

To proceed, we first review an approach to proving the standard estimates (286a)-(286b). These estimates can be proved by applying Kato’s abstract framework [16–18], which is 30By

“strong solution,” we mean in particular that at each fixed t ∈ [0, T ], the equations of Theorem 3.1 are satisfied for almost every x ∈ T3 .

80

Relativistic Euler

designed to handle first-order hyperbolic systems in a rather general Banach space setting. In particular, one can apply Kato’s framework to the first-order system (24)-(26); this is described in detail, for example, in [29]. To prove (287), we will modify Kato’s framework so that it applies to the hyperbolic variables H and the elliptic variables E from Def. 9.5. To employ Kato’s framework, one relies on the propagators U(t, τ ) := U(t, τ ; H) for the linear homogeneous hyperbolic system corresponding to the (nonlinear) first-order hyperbolic system that H satisfies; see Remark 9.6 for clarification on the first-order hyperbolic system satisfied by H. By definition, U(t, τ ; H) maps initial data given at time τ to the solution of the linear homogeneous hyperbolic system (whose principal coefficients depend on H) at time t. Similarly, one relies on the operators U(m) (t, τ ) := U(t, τ ; H(m) ) corresponding to the homogeneous linear system whose principal coefficients depend on H(m) . By Duhamel’s principle, we have Z t ˚ H(t) = U(t, 0)H + U(t, τ )f(H(τ ), E(τ )) dτ, (288a) τ =0 Z t ˚ H(m) (t) = U(m) (t, 0)H(m) + U(m) (t, τ )f(H(m) (τ ), E(m) (τ )) dτ, (288b) τ =0

˚ and H ˚ (m) respectively denote the initial data of H and H(m) , and on RHSs (288a)where H (288b), f denotes the inhomogeneous term in the first-order hyperbolic system satisfied by the elements of H and H(m) . In particular f is a smooth function of its arguments satisfying f(0) = 0, and the same f appears on RHSs (288a)-(288b). The strategy behind Kato’s framework is to control the difference H(t, ·)−H(m) (t, ·) in the norm k · kH N−1 (T3 ) by subtracting (288a)-(288b), splitting the right-hand side of the resulting equation into various pieces, and bounding each piece by exploiting some standard properties of the propagators U(t, τ ) and U(m) (t, τ ). This is explained in detail in [29], and most of the arguments given there for controlling kH(t, ·) − H(m) (t, ·)kH N−1 (T3 ) go through without any substantial changes. The one part of the argument that does require substantial changes is that in order to obtain a closed inequality for kH(t, ·) − H(m) (t, ·)kH N−1 (T3 ) , one needs to show that the difference of the inhomogeneous terms on RHSs (288a)-(288b) satisfies the following estimate for t ∈ [0, T ]:

f(H, E) − f(H(m) , E(m) ) N−1 ≤ CkH − H(m) kH N−1 (Σt ) , (289) H (Σt )

where the key point is that the quantity kE − E(m) kH N−1 (Σt ) does not appear on RHS (289). The estimate (289) can be obtained with the help of elliptic estimates, as we now explain. First, we note that the top-order norm-boundedness results (285) and the convergence results (286a)-(286b) imply that n o

lim H − H(m) C([0,T ],L2(T3 )) + E − E(m) C([0,T ],L2(T3 )) = 0, (290) m→∞

and that there exists an integer m0 and a constant C > 0 such that

kHkC([0,T ],H N−1 (T3 )) + kEkC([0,T ],H N−1 (T3 )) ≤ C, n o

sup H(m) C([0,T ],H N−1 (T3 )) + E(m) C([0,T ],H N−1 (T3 )) ≤ C.

m≥m0

(291a) (291b)

81


From (290), (291a)-(291b), and the Sobolev interpolation result (209), we deduce that if N ′ < N − 1, then n o

lim (292) H − H(m) C([0,T ],H N ′ (T3 )) + E − E(m) C([0,T ],H N ′ (T3 )) = 0. m→∞

From (292) and the Sobolev embedding result (204), we deduce that o n

lim H − H(m) C([0,T ]×T3 ) + E − E(m) C([0,T ]×T3 ) = 0. m→∞

(293)

Next, we use (293) and (207) to deduce that there is a constant C > 0 such that if m is sufficiently large, then for t ∈ [0, T ], the following estimate holds for the function f appearing on RHSs (288a)-(288b):

f(H, E) − f(H(m) , E(m) ) N−1 ≤ CkH − H(m) kH N−1 (Σt ) + CkE − E(m) kH N−1 (Σt ) . (294) H (Σt ) Next, we use (240) to deduce that if m is sufficiently large, then for t ∈ [0, T ], the last term on RHS (294) obeys the following bound: kE − E(m) kH N−1 (Σt ) ≤ CkH − H(m) kH N−1 (Σt ) .

(295)

The desired bound (289) follows from (294) and (295), and Kato’s framework then allows one to conclude that

(296) lim H − H(m) C([0,T ],H N−1 (T3 )) = 0. m→∞

We clarify that (295) and (296) imply that

lim E − E(m) C([0,T ],H N−1 (T3 )) = 0, m→∞

(297)

and that, in view of definition (227b), (297) yields the desired convergence result (287).

References [1] Serge Alinhac, Blowup of small data solutions for a quasilinear wave equation in two space dimensions, Ann. of Math. (2) 149 (1999), no. 1, 97–127. MR1680539 (2000d:35147) [2] , The null condition for quasilinear wave equations in two space dimensions. II, Amer. J. Math. 123 (2001), no. 6, 1071–1101. MR1867312 (2003e:35193) [3] Demetrios Christodoulou, The formation of shocks in 3-dimensional fluids, EMS Monographs in Mathematics, European Mathematical Society (EMS), Z¨ urich, 2007. MR2284927 (2008e:76104) , The shock development problem, ArXiv e-prints (May 2017), available at 1705.00828. [4] [5] Demetrios Christodoulou and Sergiu Klainerman, The global nonlinear stability of the Minkowski space, Princeton Mathematical Series, vol. 41, Princeton University Press, Princeton, NJ, 1993. MR1316662 (95k:83006) [6] Demetrios Christodoulou and André Lisibach, Shock development in spherical symmetry, Annals of PDE 2 (2016), no. 1, 1–246. [7] Demetrios Christodoulou and Shuang Miao, Compressible flow and Euler’s equations, Surveys of Modern Mathematics, vol. 9, International Press, Somerville, MA; Higher Education Press, Beijing, 2014. MR3288725 [8] Daniel Coutand, Hans Lindblad, and Steve Shkoller, A priori estimates for the free-boundary 3D compressible Euler equations in physical vacuum, Comm. Math. Phys. 296 (2010), no. 2, 559–587. MR2608125 (2011c:35629)

82

Relativistic Euler

[9] Daniel Coutand and Steve Shkoller, Well-posedness in smooth function spaces for moving-boundary 1-D compressible Euler equations in physical vacuum, Comm. Pure Appl. Math. 64 (2011), no. 3, 328–366. MR2779087 (2012d:76103) , Well-posedness in smooth function spaces for the moving-boundary three-dimensional com[10] pressible Euler equations in physical vacuum, Arch. Ration. Mech. Anal. 206 (2012), no. 2, 515–616. MR2980528 [11] Marcelo M. Disconzi and David G. Ebin, Motion of slightly compressible fluids in a bounded domain, II, Commun. Contemp. Math. 19 (2017), no. 4, 1650054, 57. MR3665357 [12] Mahir Hadˇzić, Steve Shkoller, and Jared Speck, A priori estimates for solutions to the relativistic Euler equations with a moving vacuum boundary, ArXiv e-prints (November 2015), available at https://arxiv.org/abs/1511.07467. [13] Gustav Holzegel, Sergiu Klainerman, Jared Speck, and Willie Wai-Yeung Wong, Smalldata shock formation in solutions to 3d quasilinear wave equations: An overview, Journal of Hyperbolic Differential Equations 13 (2016), no. 01, 1–105, available at http://www.worldscientific.com/doi/pdf/10.1142/S0219891616500016. [14] Juhi Jang, Philippe G. LeFloch, and Nader Masmoudi, Lagrangian formulation and a priori estimates for relativistic fluid flows with vacuum, Journal of Differential Equations 260 (20163), no. 6, 5481–5509. [15] Fritz John, Formation of singularities in one-dimensional nonlinear wave propagation, Comm. Pure Appl. Math. 27 (1974), 377–405. MR0369934 (51 #6163) [16] Tosio Kato, Linear evolution equations of “Hyperbolic” type, Journal of the Faculty of Science Section, The University of Tokyo, I 17 (1970), 241–258. [17] , Linear evolution equations of “Hyperbolic” type ii, Journal of the Mathematical Society of Japan 25 (1973), 648–666. , The Cauchy problem for quasi-linear symmetric hyperbolic systems, Archive for Rational Me[18] chanics and Analysis 58 (1975), no. 3, 181–205. [19] Sergiu Klainerman and Igor Rodnianski, Improved local well-posedness for quasilinear wave equations in dimension three, Duke Math. J. 117 (2003), no. 1, 1–124. MR1962783 (2004b:35233) [20] Sergiu Klainerman, Igor Rodnianski, and Jeremie Szeftel, The bounded L2 curvature conjecture, Invent. Math. 202 (2015), no. 1, 91–216. MR3402797 [21] Jonathan Luk and Jared Speck, The hidden null structure of the compressible Euler equations and a prelude to applications, To appear in Journal of Hyperbolic Differential Equations; preprint available (October 2016), available at https://arxiv.org/abs/1610.00743. [22] , Shock formation in solutions to the 2D compressible Euler equations in the presence of non-zero vorticity, Inventiones mathematicae (2018Jun). [23] Shuang Miao, On the formation of shock for quasilinear wave equations with weak intensity pulse, Ann. PDE 4 (2018), no. 1, Art. 10, 140. MR3780823 [24] Shuang Miao and Pin Yu, On the formation of shocks for quasilinear wave equations, Inventiones mathematicae 207 (2017), no. 2, 697–831. [25] Alan D. Rendall, The initial value problem for a class of general relativistic fluid bodies, J. Math. Phys. 33 (1992), no. 3, 1047–1053. MR1149225 [26] Luciano Rezzolla and Olindo Zanotti, Relativistic hydrodynamics, OUP Oxford, 2013. [27] Jan Sbierski, On the existence of a maximal Cauchy development for the Einstein equations: a dezornification, Ann. Henri Poincaré 17 (2016), no. 2, 301–329. MR3447847 [28] Hart F. Smith and Daniel Tataru, Sharp local well-posedness results for the nonlinear wave equation, Ann. of Math. (2) 162 (2005), no. 1, 291–366. MR2178963 (2006k:35193) [29] Jared Speck, The non-relativistic limit of the Euler-Nordstr¨ om system with cosmological constant, Rev. Math. Phys. 21 (2009), no. 7, 821–876. MR2553428 , Shock formation in small-data solutions to 3D quasilinear wave equations, Mathematical Sur[30] veys and Monographs, 2016. , A new formulation of the 3D compressible Euler equations with dynamic entropy: Re[31] markable null structures and regularity properties, ArXiv e-prints (January 2017), available at https://arxiv.org/abs/1701.06626.

M. Disconzi, J. Speck [32] [33]

[34] [35] [36] [37]

83

, Multidimensional nonlinear geometric optics for transport operators with applications to stable shock formation, ArXiv e-prints (September 2017), available at https://arxiv.org/abs/1709.04509. , Shock formation for 2D quasilinear wave systems featuring multiple speeds: blowup for the fastest wave, with non-trivial interactions up to the singularity, Ann. PDE 4 (2018), no. 1, Art. 6, 131. MR3740634 Jared Speck, Gustav Holzegel, Jonathan Luk, and Willie Wong, Stable shock formation for nearly simple outgoing plane symmetric waves, Annals of PDE 2 (2016), no. 2, 1–198. Qian Wang, A geometric approach for sharp local well-posedness of quasilinear wave equations, Annals of PDE 3 (2017May), no. 1, 12. Steven Weinberg, Cosmology, Oxford University Press, Oxford, 2008. MR2410479 (2009g:83182) Willie Wai-Yeung Wong, A comment on the construction of the maximal globally hyperbolic Cauchy development, J. Math. Phys. 54 (2013), no. 11, 113511, 8. MR3154377

The relativistic Euler equations: Remarkable null structures and

The relativistic Euler equations: Remarkable null structures and

Suggest Documents

Blowup of smooth solutions for relativistic Euler equations

Blowup for the Euler and Euler-Poisson Equations with Repulsive

Invariant Euler-Lagrange Equations and the

Relativistic effects and quasipotential equations

PARALLELIZATION OF THE EULER EQUATIONS ON ... - CiteSeerX

Euler Lagrange Equations. - Google Sites

Remarks on the Perturbed Euler equations 1

Dynamics of the incompressible Euler equations at

Dense Oscillations for the Euler Equations II

Lagrange coordinates for the Einstein-Euler equations

NON-RELATIVISTIC CONFORMAL STRUCTURES ()

New forms of non-relativistic and relativistic hydrodynamic equations

On fractional Euler-Lagrange and Hamilton equations and the ...

Newtonian hydrodynamic equations with relativistic pressure and ...

Relativistic Burgers and Nonlinear SchrÃ¶dinger Equations

Relativistic five-quark equations and negative parity

Euler equations on the general linear group, cubic curves, and ...

G-STRUCTURES AND DIFFERENTIAL EQUATIONS

RELATIVISTIC PARTIAL WAVE INTEGRAL EQUATIONS FOR THE ...

1538 Relativistic Wave Equations without the Velo

On the Dynamics of Navier-Stokes and Euler Equations

on the metrics and euler-lagrange equations of ... - Semantic Scholar

Ghost cell boundary conditions for the Euler equations and their ...

Euler Method for Solving Ordinary Differential Equations