ON ELEMENTARY INTERACTIONS FOR HYPERBOLIC ... - UMass Math

ON ELEMENTARY INTERACTIONS FOR HYPERBOLIC CONSERVATION LAWS ROBIN YOUNG Abstract. This is a survey of interactions of weak nonlinear waves in N × N systems of hyperbolic conservation laws. Recently a variety of surprising new phenomena have been observed, including strong nonlinear instability of solutions. This implies that further assumptions must be made to develop a Glimm–Lax existence and decay theory for N ≥ 3. As a first step towards such a theory, a systematic description of local interactions in terms of naturally occurring ‘flux coefficients’ is given. These coefficients characterize the nonlinearity of the system, and each has a physical interpretation in terms of wave interactions. There are two competing nonlinear interaction effects, namely decay and wave generation, which can be quantified using the flux coefficients. The imposition of physical assumptions leads to constraints on the coefficients, which in turn prevent blowup from occurring. Relevant existence and nonexistence results are briefly surveyed, and conditions under which different nonlinear phenomena dominate are described. An example of a compactly supported unstable solution is given, illustrating this phenomenon from the point of view of local interactions. Finally, the coefficients are calculated and interpreted for the Euler equations of gas dynamics.

1. Introduction We consider the N × N system of hyperbolic conservation laws in one space dimension, given by ϕ(u) t + f (u) x = 0. Here ϕ is the vector of conserved quantitites, and f is the flux. In order to understand solutions to this system, we must know how the nonlinear waves propagate and interact. Our approach is through a study of the geometry of state space, while giving a physical interpretation to geometric constructs. We are interested in local phenomena, and so restrict our attention to weak waves. We assume that the system is strictly hyperbolic. This means that there are N families of waves, each corresponding to an eigenvalue of the system. The nonlinearity of wavespeeds leads to the formation of shocks, so that solutions must be understood in the weak sense. The existence and stability of global weak solutions for Cauchy data with small total variation was established by Glimm, using his celebrated random choice method [5]. Recently the author has shown that the Glimm existence theory applies as long as the initial total variation is bounded by an explicit constant [25]. The onset of shocks heralds decay of the solution due to cancellation between shock waves and rarefactions. This phenomenon has been analyzed in 2 × 2 systems by the 1991 Mathematics Subject Classification. 35L65, 35F20, 35-02. Key words and phrases. shock waves, interactions, decay, blowup, resonance, Cauchy problem. 1

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Glimm–Lax theory of decay, ands leads to an improved existence theory for Cauchy data with arbitrarily large total variation, as long as the sup-norm is small enough [6]. This includes the important case of periodic solutions. Decay is also observed for larger systems, as long as the initial data has small total variation [15]. It was previously thought that the Glimm–Lax theory would apply to larger systems, and that the barriers to extending the theory were merely technical. However, the theory of weakly nonlinear geometric optics for periodic solutions predicts that nonlocal ‘resonant’ interaction effects become apparent, leading to a variety of surprising new phenomena [16]. These include blowup of solutions [8], and delay in the onset of shocks [17]. Resonance effects are seen only when there are three or more equations, and reflect the effects of interactions between waves of different families. The most alarming feature of resonance is the occurrence of solutions exhibiting a strong nonlinear instability in the form of catastrophic blowup of solutions [8, 12]. The nature of the instability can be easily stated: for certain systems, there are Cauchy data with arbitrarily small oscillation ku0 k∞ that can grow arbitrarily large in finite time. That is, there is a time T∗ so that, for any K, there are solutions with arbitrarily small initial data ku0 kp , satisfying ku(·, T∗ )kp > K ku0 kp in every p-norm. We shall give an example of blowup by constructing a resonant pattern of local interactions. Each (three–wave) interaction magnifies the incident waves, and the large number of interactions leads to blowup. The solution can be periodic or compactly supported, and we give a lower bound for the total variation of the Cauchy data. Our example is elementary, using only Riemann solutions in the construction. It is natural to ask when resonant interaction effects become important, and what the obstructions to existence and stability of solutions are. Evidently, a more refined analysis of wave interactions is needed to obtain a useful theory for more than 2 equations. In this paper, we lay out a framework for establishing conditions under which existence and decay theories hold. We undertake a detailed investigation of local wave interactions as a first step towards developing an existence theory for solutions with large total variation. It turns out that all local interaction phenomena can be characterized by the ‘flux coefficients’, which are defined in terms of the Hessians of the functions ϕ and f . We shall use these coefficients to describe the different types of nonlinear phenomena, and thus give each of them a physical interpretation. For weak waves, the generalized eigenvalues of the matrix pair {Dϕ, Df } give the wave speeds, and the eigenvectors determine the approximate change in state u across a wave. Our assumption of strict hyperbolicity means that Dϕ is positive definite and the eigenvalues are real and distinct, so that there is a full set of eigenvectors. We label the eigenvalues (or wave-speeds) λ1 < · · · < λN , and the corresponding left and right generalized eigenvectors as li and ri , respectively. The nonlinear behavior of the system is characterized by the flux coefficients Πijk = li · D2 f (rj , rk ) − λi li · D2 ϕ (rj , rk ), in the sense that all leading order effects of wave interactions and changes in wave-speeds can be described in terms of them. Moreover, for any combination of numbers Πijk symmetric in

ELEMENTARY INTERACTIONS

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j and k, we can find a conservation law having these coefficients, at least at a single point. We give a brief qualitative description of the coefficients. An essential feature of nonlinearity is the dependence of wave-speed on the state of the system. This dependence is quantified by specifying the change in wave-speed across a single wave. The change in the i-th characteristic speed across a k-wave is determined directly by the quantity rk ·∇u λi = Πiik . The decay coefficient Πiii = ri ·∇u λi is of particular importance, determining the genuine nonlinearity or linear degeneracy of that family. This in turn describes the rate at which rarefactions expand and shocks form, and gives the corresponding rate of decay in the absence of other fields. The interaction of two waves gives rise to small waves in other families. These reflected waves can be described qualitatively by the flux coefficients as follows. If the incident waves are both from the k-th (genuinely nonlinear) family, the resultant i-wave is given to leading order by a multiple of Πikk . If the incident waves are from different families, say a j–k-interaction, then the reflected i-wave is determined to leading order by the interaction j k i coefficient Λjk i = ℓi · [r , r ], which is a multiple of Πjk . By a suitable normalization, we ik can assume that Λi = 0, so we take i, j and k to be distinct. Physically, this means that the incident waves do not change to leading order, and the effects of interaction must be measured in different families. In particular, to see nontrivial effects of interactions between waves of different families, we require N ≥ 3. For systems of 2 equations, interactions between waves of different families yield only third-order effects. In this case, nonlinear decay due to expansion of rarefactions and shock formation is the dominant phenomenon. Indeed, the Glimm–Lax theory tells us that periodic solutions are nonlinearly stable and decay with rate 1/t. However, if there are more than two equations, new second-order waves can be generated by interactions of waves of different families. If there are many interactions, the cumulative effects may become large and dominate, leading to potential nonlinear instabilities. In this case, we say the system is resonant. The most striking example of resonance is the blowup described above. Evidently, the most important coefficients are the decay coefficients Πiii , measuring the rate of decay in the i-th family, and the interaction coefficients Λjk i , measuring the amount of wave production by j–k-interactions. Nonlinear decay is a stabilizing feature, whereas wave generation due to interaction of waves from different families can be viewed as destabilizing. The coupling of these competing effects is not yet well understood for solutions with large variation, when resonance becomes important. We remark that whenever there is a natural ‘sound speed’ for the system (as in gas dynamics), resonant phenomena are present. If we impose physical constraints on the system, extra symmetries lead to simplified interactions. The prototype for these assumptions is the Euler equations of gas dynamics together with the Law of Thermodynamics. Firstly, the classical entropy is a Riemann coordinate for the second family, which implies that (to leading order) this family is not affected by interactions of 3-waves and 1-waves. This is equivalent to the vanishing of the 2 interaction coefficient Λ31 2 (and thus also Π31 ). More importantly, in the smooth regime, an extra conservation law, the entropy equation, can be deduced from the original system. In this case, the equations are symmetric hyperbolic, and the (convex) entropy density U can be used to pick outRappropriate weak solutions to the conservation law [4]. This is achieved by requiring that U be non-increasing with

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time. Since U is convex, this can be interpreted as a restriction on the size of the solution at later times. The symmetry of the fluxes imposes an extra condition on the coefficients, namely Λki Λij Λjk j i k + + = 0, λj − λk λk − λi λi − λj for i, j and k distinct, with an analogous condition on the Π’s. The importance of this algebraic R condition is that it rules out all known cases of blowup, and indeed the requirement that U decreases seems to proscribe large growth of the solution. In all known cases of blowup, the interaction coefficients Λjk i , j > k, have the same sign. Assuming existence of either a convex entropy function or a Riemann coordinate precludes this. We briefly outline the structure of the paper. In Section 1, we define and describe the relationships between the coefficients, and give a convenient normalization. In Section 2, we recall Lax’s solution of the Riemann problem, and calculate the difference between shock and rarefaction curves. We calculate the effects of local wave interactions in Section 3. Interactions of waves in the same family lead to shock formation and decay, while interactions between different families lead to the generation of new waves. We briefly describe the method of weakly nonlinear geometric optics, and comment on the appearance of the flux coefficients in the simplified equations. We describe physically motivated assumptions in Section 4, and examine the constraints they place on the coefficients. The assumptions are the existence of a Riemann coordinate and a convex entropy function. Note that both conditions are a consequence of the thermodynamic relation in gas dynamics. We recall Lax’s definition of Riemann invariance and briefly review the theory of symmetric hyperbolic systems. In Section 5, we give a short survey of some existence and nonexistence theorems. We describe the existence and decay theory for solutions with small sup-norm. We then mention some results on the validity of the weakly nonlinear approximations, and describe the theorems on blowup of solutions. We then give a rigorous example of an unstable solution with compact support, describing the local origin of blowup. In Section 6 we describe the important case of the Euler equations in detail, calculating the flux coefficients and deducing the qualitative effects of wave interactions. 2. Coefficients and Normalization We shall define the coefficients that can be associated to the conservation law, and describe the relations between them. We shall see that these coefficients characterize the local behavior of solutions, in the following sense. Each coefficient quantifies some aspect of local wave interactions and changes in wave-speed to leading order. Moreover, a conservation law can be associated to any self-consistent set of coefficients. We first describe a convenient local normalization for any bi-orthogonal system of vector fields. This normalization has a simple geometric interpretation, and lends clarity to the effects of interactions of weak elementary waves in systems of conservation laws. A bi-orthogonal system consists of a basis {r1 , . . . , rN } of vector fields defined in a neighborhood U ⊂ RN , together with an associated dual basis {ℓ1 , . . . , ℓN }, normalized so that ℓj · rk = δjk ,

j, k = 1, . . . , N,


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where δjk is the Kronecker delta. We assume that the vector fields are C 1 in U. The asymmetric interaction coefficients Λjk i are defined by (1) (2)

j k Λjk i = ℓi · [r , r ],

so that X jk [rj , rk ] ≡ rj ·∇rk − rk ·∇rj = Λi r i .

We shall choose a normalization of the vector fields in such a way that all the coefficients Λjk ˜ ∈ U, unless i, j, and k are distinct. i vanish at the point u To this end, suppose that for each p, we are given a function wp , with X wp (˜ u) = 0 and ∇u wp = apj ℓj . j

for some functions apj . We form the vectors r¯p = ewp rp ,

and ℓ¯p = e−wp ℓp ,

so that r¯p |u˜ = rp |u˜ , and calculate directly that rj · ∇u r¯k = ewk (akj rk + rj · ∇rk ).

(3)

¯ jk are the interaction coefficients corresponding to the r¯p ’s, Now, if Λ i ¯ jk = ℓ¯i · [¯ Λ rj , r¯k ], i we have (4)

¯ jk = ewj +wk −wi (akj δik + Λjk − ajk δji ). Λ i i

We now choose the functions wk , which determine akj , in a convenient way. We would like ¯ kj to set akj = Λkj k , so that Λk = 0, for each k 6= j. To do this throughout the neighborhood U is not generally possible, but we can do it at the origin u ˜. For example, in view of (3), we choose app = −ℓp · (rp ·∇rp ), so that throughout U ℓ¯p · (¯ rp ·∇¯ rp ) = ewp (app + ℓp · (rp ·∇rp )) = 0. This choice of the function app amounts to finding a solution wp of the first order equation rp ·∇wp = −ℓp · (rp ·∇rp ) in U, where the right hand side is known. This equation can be solved for prescribed Cauchy data on any non-characteristic hypersurface. We choose this surface and data so that the interaction coefficients vanish at the point u ˜. For example, we choose a hyperplane whose tangent space at u ˜ is the span of the vectors ℓj , where j 6= p, and take Cauchy data vp which satisfies P pj Λp u˜ ℓj , so that rj ·∇vp u˜ = Λpj u), for j 6= p. ∇vp u˜ = p (˜ j6=p

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Now, since apj = rj ·∇wp , we conclude from (4) that (5)

jk ¯ jk (˜ u) Λ i u) = Λi (˜

for i, j and k distinct, and ¯ jk (˜ u) = 0 otherwise. Λ

(6)

i

¯ jk (˜ Λ i u)

Note that the nonzero coefficients can be calculated in either basis. Henceforth, we assume that we have normalized the vectors ri in such a way that (7)

ℓi · (ri ·∇ri ) = 0 in U, and

(8)

= 0 unless i, j and k are distinct. Λjk i u ˜

Suppose that the bi-orthogonal system consists of the left and right eigenvectors of a matrix A = A(u), so that (9)

A · r j = λj r j ,

and ℓi · A = λi ℓi .

Then we can calculate the interaction coefficients as follows. Following John [10], it is useful to define the coefficients (10)

Γijk = ℓi · DAu (rk ) · rj ,

i, j, k = 1, . . . , N,

where DA is the Frech´et derivative. Upon differentiating (9) in direction rk , we get DA(rk ) · rj + A(rk ·∇rj ) = (rk ·∇λj ) rj + λj rk ·∇rj , where we write Dg(rk ) = rk ·∇g, so that Γijk = rk ·∇λj δij + (λj − λi ) ℓi · (rk ·∇rj ). From this it is immediate that Γiik = rk ·∇λi ,

that is ∇u λi =

and the gradients of the eigenvectors are given by (11)

ℓi · (rk ·∇rj ) =

Γijk

X

Γiik ℓk ,

for j 6= i.

λj − λi

Finally, the interaction coefficients (2), for i 6= j or k, are (12)

Λjk i

=

Γikj λk − λi

−

Γijk λj − λi

.

= 0, so that, using (11), According to our normalization, Λik i u ˜ ℓi · (rk ·∇ri ) u˜ = ℓi · (ri ·∇rk ) u˜ =

Γiki (˜ u) for λk − λi

k 6= i.

Thus, since ℓi · (ri ·∇ri ) = 0, all quantities rj ·∇rk are determined at the point u ˜ by the matrix A and the coefficients Γijk there. One calculates the derivatives of the ℓi by simply differentiating the relation ℓi · rj = δij , to get Dℓi (rk ) · rj = (rk ·∇ℓi ) · rj = −ℓi · (rk ·∇rj ).


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2.1. Coordinate Changes. We now regard the matrix A(u) and vector fields X(u) as geometric objects, and investigate the effects of changes of coordinates. Thus, suppose the change of coordinates is given by w = φ(u). The corresponding tangent vectors are given by Y (w) = Y (φ(u)) = Dφu · X(u), and the corresponding matrix action is defined by (13)

B(w)Y (w) = B(φ(u)) · Dφu X(u) = Dφu A(u)X(u),

so that

B(w) = Dφu A(u)Dφ−1 u ,

(14)

for u = φ−1 (w), where we have used D(φ−1 )w = Dφ−1 u . From the relation (15)

B(φ(u))Dφu = Dφu A(u),

it is clear that B has the same eigenvalues λi as A, with corresponding left and right eigenvectors li and r i given by li = ℓi · Dφ−1

and r j = Dφ rj ,

respectively. Differentiating the relation r j (φ(u)) = Dφu rj (u) in direction X, we have Dr jw Dφ(X) = D2 φu (rj , X) + Dφ Druj (X), so that the derivatives of the eigenvectors are related by (16)

2 j k li · r k ·∇w r j = li · Dr jw (r k ) = ℓi · rk ·∇rj + ℓi · Dφ−1 u D φu (r , r ).

In particular, since the last term is symmetric, we see that li · [r j , r k ] = ℓi · [rj , rk ], so that the interaction coefficients can be calculated in either basis. This is to be expected, as the Lie bracket is a geometric object, which can be defined without reference to coordinate systems. Similarly, for scalars λ, we have rj ·∇u λ = Dλu (rj ) = Dλw Dφu (rj ) = Dλw (r j ) = r j ·∇w λ. Given the coordinate change φ, and conjugate matrix B = DφA Dφ−1 , we can define the corresponding ‘conjugate coefficients’ Πijk by Πijk = li · DB(r k ) · r j . To relate the coefficients Πijk to the Γijk defined earlier, we multiply equation (15) by Y and differentiate in direction X, to get DBw (Dφ(X)) · DφY + B · D2 φ(X, Y ) = D2 φ(X, A · Y ) + Dφ DA(X) · Y, so that upon setting Y = rj and X = rk and multiplying on the left by li , we get 2 j k Πijk = Γijk + (λj − λi ) ℓi Dφ−1 u D φ(r , r ).

The symmetry of the quadratic form D2 φ now implies that, for j and k 6= i, we have Πijk − Γijk λj − λi

=

Πikj − Γikj λk − λi

,

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which, according to (12), is equivalent to the statement that the interaction coefficients Λjk i be preserved under the coordinate change, for i, j and k distinct. We are mainly concerned with the case where the matrix A(u) is given by A(u) = (Dϕu )−1 · Dfu , where ϕ and f are given functions of u, and the appropriate coordinate change is given by w = φ(u) = ϕ(u). Thus we have B(w) = B(ϕ(u)) = Dfu · Dϕ−1 = Dfu D(ϕ−1 )w = D(f ϕ−1 )w . By calculations similar to those above, we express the coefficients as (17)

Γijk = ℓi · DA(rk ) · rj = li · D2 f (rj , rk ) − λj li · D2 ϕ (rj , rk ),

and their conjugates as Πijk = li · D2 f (rj , rk ) − λi li · D2 ϕ (rj , rk ).

(18)

In particular, we see that the Πijk are symmetric, which is expected since DB = D2 (f ϕ−1 ), and we have the relations (19)

(λi − λk ) Γijk + (λj − λi ) Γikj + (λk − λj ) Πijk = 0.

Finally, substituting these expressions into (12), we express the interaction coefficients as (20)

Λjk i

=

Γikj λk − λi

−

Γijk λj − λi

= Πijk

λj − λk , (λj − λi )(λk − λi )

for i, j and k distinct (see [9]). We note that the quantities Πijk , defined in terms of D2 (f ϕ−1 ), are independent of the choice of state variable u, and so can be regarded as ‘invariants’ of the conservation law. Our goal in this paper is to give each of these coefficients a physical interpretation, and to examine the restrictions imposed on these coefficients when further physical assumptions are made. Indeed, it will be shown that rates of decay and nonlinear interaction effects can all be expressed to leading order in terms of these quantities. Thus all nonlinear effects are determined to leading order by the quadratic nonlinearities of the functions f and ϕ. Henceforth we shall refer to the coefficients Πijk as the flux coefficients of the conservation law. We remark that, given any choices of flux coefficients Πijk , symmetric in j and k, there are functions ϕ and f for which these coefficients are achieved at a particular point. This says that any self–consistent set of such invariants can be realized. To see this, suppose we are given constants Πijk = Πikj and λi , and independent vectors ri . We then choose quadratic functions f and ϕ with first and second derivatives satisfying equations (9) and (18) at u ˜. The symmetry of the Π’s ensures that this can be consistently done. Note that this construction does not determine the Hessian D2 ϕ at u ˜. This Hessian can be set to zero or specified by also choosing some of the constants Γijk , say for i 6= j ≤ k, and using the consistency condition (19) to determine the other Γ’s. This choice of undetermined coefficients amounts to a choice of state variable u. Having made these choices, we see that all quantities except li · D2 ϕ (ri , ri ) are determined. In view of our normalization (8) and equation (16), we may take this to be zero.


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3. Wave Curves We now consider the system of N conservation Laws, (21) ϕ(u) t + f (u) x = 0,

where u = u(x, t) ∈ RN represents the state of the system, ϕ is the vector of conserved quantities, and f is the flux. We assume that ϕ and f are C 3 , and that the system is strictly hyperbolic. This means that for each u, the linear map Dϕu is invertible, and the eigenvalues of the matrix A(u) = (Dϕu )−1 · Dfu are real and distinct. Thus we have (22)

A · ri = Dϕ−1 Df · ri = λi ri , i = 1, . . . , N, λ1 < · · · < λN , and ℓj · rk = δjk ,

j, k = 1, . . . , N,

where ℓi and ri are the left and right eigenvectors associated to the eigenvalue λi of the matrix A. We normalize all vectors as in (8), with direction chosen so that ri ·∇λi ≥ 0. As in the previous section, we define li = ℓi · Dϕ−1 , so that the vectors li and ri are the generalized left and right eigenvectors for (21), given by (23)

li · Df = λi li · Dϕ and Df · ri = λi Dϕ · ri ,

respectively, with the same eigenvalues as A. As in the previous section, we define the flux coefficients of the conservation law to be Πijk = li · D2 f (rj , rk ) − λi li · D2 ϕ (rj , rk ). The Riemann problem is fundamental to the study of hyperbolic conservation laws. This is the initial value problem with data consisting of two constant states, ( uL , for x < 0, u(x, 0) = (24) uR , for x > 0. The solution was constructed by Lax, and consists of constant states separated by centered elementary waves. There are N families of elementary waves, each corresponding to an eigenvalue of A, which gives the speed of the wave. They consist of Lax shocks and rarefactions when the field is genuinely nonlinear, and contact discontinuities when the field is linearly degenerate. We briefly recall Lax’s construction (see [13, 21]). A k-simple wave consists of states which lie on an integral curve of the k-th eigenvector field, each state propagating along characteristics, whose speed is given by the eigenvalue λk . Given a state u0 , the integral curve Rk (u0 ) of rk through u0 gives those states which can be connected to u0 by a k-simple wave. The curve Rk (u0 ) can be described as the solution u = u(σ) of the equation du = rk u(σ) , dσ

u(0) = u0 .

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Thus u(σ) can be joined to u0 by a k-simple wave, which we define to be of strength σ. Differentiating, we have d2 u d k k du r = Dr = Drk (rk ) = rk ·∇rk , = dσ 2 dσ dσ along Rk , and similarly (25)

(26)

du d d d3 u k k Dr (r ) = D2 rk (rk , = ) + Drk ( rk ) dσ 3 dσ dσ dσ = D2 rk (rk , rk ) + Drk (rk ·∇rk ).

A centered rarefaction is a self-similar simple wave emanating from the origin. It is clear that this must be expansive, that is the characteristic speed should increase with x for any fixed t > 0. Since dλk /dσ = rk ·∇λk > 0, those states u ∈ Rk (u0 ) which can be connected to u0 by a rarefaction correspond to those for which σ > 0. A discontinuity in a weak solution to (21) must satisfy the Rankine–Hugoniot relation (27) [f ] = f (u+ ) − f (u− ) = s ϕ(u+ ) − ϕ(u− ) = s[ϕ],

where u+ and u− are the right and left limits of the solution, s is the speed of propagation of the discontinuity, and [·] denotes the jump. Fixing the left state u0 = u− , the set of states u = u+ which can be connected to u0 by such a discontinuity form N curves Sk (u0 ) in U, each corresponding to an eigenvalue of A. By continuity, we have s(0) = λk (u0 ) and ℓk · u˙ 6= 0. We choose parameter η for Sk (u0 ) such that, at the point u0 , we have ℓp · u| ˙ 0 = ℓp · rp (u0 ) = 1, (28)

ℓp · u ¨|0 = ℓp · r˙ p (u0 ), ... ℓp · u |0 = ℓp · r¨p (u0 ),

and

where ˙ = d/dη is differentiation along Sk . The strength of the shock is defined as the value of the parameter η. In addition to (27), we must impose an extra condition to ensure that we obtain only physical discontinuities. This is the Lax Entropy condition, and is the requirement that shocks are compressive, that is, characteristics on either side of the shock impinge on the shock, λk u0 > s u(η) > λk u(η) . (29) We shall see that s˙ = 12 λ˙k > 0 at u0 , so that admissible shocks correspond to those for which the parameter η is negative. In case the k-th field is linearly degenerate, rk ·∇λk = 0 in U, the curves Rk and Sk coincide, and the solution consists of a contact discontinuity with speed λk . We again choose a parameter satisfying (28), and use this to define the strength of the wave. We construct the wave curve Wk (u0 ) consisting of those right states u which can be connected to u0 by a centered k-wave. This consists of that part of Rk for which σ > 0, and that part of Sk for which η < 0, and is a C 2 -curve when parameterized in the obvious way. The strength of the k-wave is given by the parameter, so that shocks have negative strength and rarefactions positive strength.


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The general Riemann problem is now solved by choosing N − 1 intermediate states, which are separated by centered elementary k-waves, where k increases with x, for t > 0 fixed. Existence and uniqueness of the intermediate states, and corresponding waves, is a consequence of the Implicit Function Theorem. To check that s˙ = 21 λ˙p and Wp is a C 2 -curve, one differentiates relations (27) and (23) along Sp . Similarly, we calculate the difference between the shock and simple wave curves through an arbitrary point u0 ∈ U. Differentiate the Hugoniot relation (27) along Sp , to get Df (u) ˙ = s˙ [ϕ] + s Dϕ(u), ˙ so that by (23) and (28), u| ˙ 0 = rp (u0 ). A second differentiation gives (30)

D2 f (u, ˙ u) ˙ + Df (¨ u) = s¨ [ϕ] + 2 s˙ Dϕ(u) ˙ + s D2 ϕ(u, ˙ u) ˙ + Dϕ(¨ u) ,

which becomes for η = 0,

u) . D2 f (rp , rp ) + Df (¨ u) = 2 s˙ Dϕ(rp ) + λp D2 ϕ(rp , rp ) + Dϕ(¨

On the other hand, differentiating (23) along Sp gives (31)

˙ rp ) + Dϕ(r˙ p ) . D2 f (u, ˙ rp ) + Df (r˙ p ) = λ˙p Dϕ(rp ) + λp D2 ϕ(u,

Comparing these expressions at u0 , and again using (28), we see that u ¨|0 = r˙ p = Drp (u) ˙ = rp ·∇rp (u0 ),

and

2 s˙ = λ˙p = rp ·∇λp ,

proving the claims made above. Differentiate (30) once more to get (32) (33) (34)

... D3 f (u, ˙ u, ˙ u) ˙ + 3D2 f (¨ u, u) ˙ + Df ( u ) ... s Dϕ(u) ˙ + 3s˙ D2 ϕ(u, ˙ u) ˙ + Dϕ(¨ u) = s [ϕ] + 3¨ ... +s D3 ϕ(u, ˙ u, ˙ u) ˙ + 3 D2 ϕ(¨ u, u) ˙ + Dϕ( u ) .

Similarly, differentiating (31) again yields (35) (36) (37)

D3 f (u, ˙ u, ˙ rp ) + D2 f (¨ u, rp ) + 2 D2 f (u, ˙ r˙ p ) + Df (¨ rp ) ˙ rp ) + Dϕ(r˙ p ) = λ¨p Dϕ(rp ) + 2λ˙p D2 ϕ(u,

˙ u, ˙ rp ) + D2 ϕ(¨ u, rp ) + 2 D2 ϕ(u, ˙ r˙ p ) + Dϕ(¨ rp ) . +λp D3 ϕ(u,

Now, after setting the parameter to zero and using our earlier results, subtracting these expressions yields, at the point u0 , 1 ... s − λ¨p )Dϕ(rp ) − λ˙p D2 ϕ(rp , rp ) + Dϕ(r˙ p ) . (Df − λp Dϕ)( u − r¨p ) = (3¨ 2 ... p p r , where the differentiation is along Sp (u0 ), evaluated It will be convenient to write d = u −¨ −1 at u0 . Multiplying by lp = ℓp · Dϕ and using (23) and (28) gives 1 3¨ s = λ¨p + λ˙p lp · D2 ϕ(rp , rp ) + ℓp · r˙ p , 2

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ROBIN YOUNG

and multiplying by li for i 6= p, we get ... ℓi · dp = ℓi · ( u − r¨p ) λ˙p = ℓi · r˙ p + li · D2 ϕ(rp , rp ) (38) 2(λp − λi ) =

rp ·∇λp ℓi · (rp ·∇rp ) + li · D2 ϕ(rp , rp ) . 2(λp − λi )

Note that by our choice of parameter, (28), we have ℓp · dp = 0. Using (11) and (18), we express ℓi · dp in terms of the flux coefficients as ℓi · dp =

(39)

Πppp Πipp , 2 (λp − λi )2

and this has the same sign as the coefficient Πipp . Differentiating rp twice along Sp , we have p r¨p = Dr˙ p (u) ˙ = D[u·∇r ˙ ](u) ˙ = D2 rp (u, ˙ u) ˙ + Drp (u·∇ ˙ u) ˙ = D2 rp (u, ˙ u) ˙ + Drp (¨ u),

so that at u0 , we have r¨p |u0 = D2 rp (rp , rp ) + Drp (rp ·∇rp ). Comparing to (26), we see that r¨p coincides with the third derivative of the curve Rp at u0 , so that the vector dp (u0 ) gives the leading order divergence of the curves Rp and Sp through u0 . Note that for a linearly degenerate family, we obtain dk = 0, as expected. Combining the foregoing with Taylor’s formula gives the following lemma, which is used to calculate the effects of wave interactions. Lemma 1. The difference between the curves Sp (u0 ) and Rp (u0 ) is given to leading order by the vector dp (u0 ), a third order quantity. For the state u = u(γ) ∈ Wp (u0 ), which is joined to u0 by a single admissible p-wave of strength γ, we have 1 1 1 u = u0 + γrp + γ 2 r˙ p + γ 3 r¨p + (γ − )3 dp + O(|γ|4 ), 2 6 6 all vectors being evaluated at u0 , where γ − ≤ 0 denotes the negative part of γ. We note that according to our normalization (8), ℓp (u0 ) · (u − u0 ) = γ + O(|γ|3 ), so that the strength of a wave is given to within third order by a simple projection. We remark that the appearance of terms involving the Hessian D2 ϕ of ϕ in the expressions for s¨ and dp is consistent with the fact that non-smooth solutions to hyperbolic equations in conservation form are not preserved under nonlinear transformations. This is due to the fact that the Hugoniot curve is incorrectly transformed, leading to incorrect shock speeds. It is interesting to note that although the vector dp describes the third order divergence of the shock and rarefaction curves, it can be expressed by the flux coefficients without regard for third derivatives of the maps ϕ and f .


13

4. Interactions We now consider interactions of waves, and investigate the qualitative relationship between the flux coefficients and nonlinear wave interactions. It is well known that the decay coefficient ri ·∇λi = Πiii determines the linear degeneracy or genuine nonlinearity of the i-th field, and that the size of Πiii gives a measure of the rate at which shocks form and decay occurs [6]. We shall see that the other coefficients can be given an interpretation in terms of wave interactions, and that, to leading order, the results of interactions are determined completely by them. The equations of weakly nonlinear geometric optics can be used to analyze the long-time cumulative effects of multiple interactions. We describe this method, and note the presence of the flux coefficients in the simplified equations. 4.1. Waves in a Fixed Family. We first consider interactions of waves in a fixed family. In order for the waves to interact at all, they must have different speeds, which is true as long as the family is genuinely nonlinear, Πiii = ri ·∇λi > 0. This in turn implies that the shock and rarefaction curves do not coincide. First we consider the collapse of a simple wave into a shock wave. Thus suppose that the state uL is connected to uR by a p-simple wave, and that this is a compression wave. If the strength of this wave is α, then we have uR ∈ Rp (uL ), and λp (uL ) > λp (uR ), so that this compression wave will collapse. We suppose that the collapse takes place at a single point, so that the solution of the Riemann problem represents the continuation of the solution beyond shock formation. According to Lax’s theorem, there is a unique admissible Riemann solution consisting of constant states joined by elementary waves, and our task is to calculate, to leading order, the strengths of these resultant waves. Note that we expect the emerging p-wave to be a shock, and due to the fact that Hugoniot curves coincide with integrals up to third order, we expect the waves in other families to have third order strength in α. This intuition will simplify our calculations by allowing us to expand only those quantities which are relevant to our purpose. We proceed with the details. As noted above, we have uR ∈ Rp (uL ), although the state uR is not on the curve Wp (uL ), since the simple wave we are considering is a compression wave, and so does not feature as an elementary wave in a Riemann solution. By Taylor’s theorem, we have (40)

1 1 uR = uL + αrp + α2 r˙ p + α3 r¨p + O(|α|4 ), 2 6

where ˙ denotes differentiation along Rp (uL ), and α < 0. Here and throughout this section, all vector quantities are to be evaluated at the point uL , unless explicitly indicated otherwise. Suppose that the emerging Riemann solution is resolved by waves ǫi , i = 1, . . . , N , where ǫi separates the constant states ui−1 and ui , with uL = u0 and uR = uN . As usual, we express the states ui in terms of the wave strengths ǫi and vector quantities evaluated at the point uL . This yields two expressions for uR , which can be compared to get expressions for the outgoing waves ǫi in terms of α. We then use a ‘bootstrap’ technique to get more accurate expressions. According to Lemma 2.1, for each i, ui = ui−1 + ǫi ri (ui−1 ) + O(|ǫi |2 ),

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ROBIN YOUNG

where the O(|ǫ|2 ) can be considered uniform in the neighborhood U. Thus, by induction, we have X P (41) up−1 = uL + ǫi ri + O ( |ǫi |)2 , ip

In fact, we can extend the induction to get X P ǫk rk + O ( |ǫk |)2 , uR = uL + k

so that, since ǫi = O(|α|), comparing to (40) yields

ǫi = O(|α|2 ), if i 6= p, and ǫp = α + O(|α|2 ).

(43) (44)

Now we use a bootstrap argument to get higher order accuracy for the strengths ǫi . To this end, we note that according to (44), the error terms in equations (41) and (42) become O(|α|4 ). Also, by Lemma 2.1 and (44), we have (45)

up = up−1 + ǫp rp (up−1 ) + 12 α2 r˙ p (up−1 ) + O(|α|3 ) = up−1 + ǫp rp + 21 α2 r˙ p + O(|α|3 ),

(46)

where we have used up−1 = uL + O(|α|2 ). Using this and (44) in (42) and (41) gives P (47) uR = up + j>p ǫj rj + O(|α|3 ) P = uL + k ǫk rk + 12 α2 r˙ p + O(|α|3 ). (48) Again comparing this expression to (40), we deduce that

ǫi = O(|α|3 ), if i 6= p, and ǫp = α + O(|α|3 ).

(49) (50)

Now using Lemma 2.1 and writing up−1 = uL + O(|α|3 ), we have 1 1 1 up = up−1 + ǫp rp + α2 r˙ p + α3 r¨p + (α− )3 dp + O(|α|4 ). 2 6 6 Thus we obtain, since α < 0, X 1 1 1 uR = uL + ǫk rk + α2 r˙ p + α3 r¨p + α3 dp + O(|α|4 ). 2 6 6 k

A final comparison with (40) then yields the relation X 1 ǫk rk = αrp − α3 dp + O(|α|4 ), (51) 6 k


15

where all quantities are evaluated at the point uL . Thus we have expressed the ǫi ’s in terms of α to leading order, ǫi = − 61 α3 ℓi · dp + O(|α|4 ),

(52) (53)

ǫp

=α+

if i 6= p, and

O(|α|4 ).

Since α < 0, the sign of the i-wave reflected upon collapse of a weak p-compression wave is that of ℓi · dp , which by (39) is the sign of Πipp . We use the same method to solve the interaction problem for two centered elementary waves in the same family. Thus suppose that uL is connected to uM and uM connected to uR by p-waves of strengths α and β, respectively. In order for them to interact at all, one (or both) of the waves must be a shock. According to Lemma 2.1, 1 1 1 uM = uL + αrp + α2 r˙ p + α3 r¨p + (α− )3 dp + O(|α|4 ), 2 6 6 where the vectors are evaluated at uL , and (54)

uR

= uM + βrp (uM ) + 12 β 2 r˙ p (uM ) + 16 β 3 r¨p (uM ) + 61 (β − )3 dp (uM ) + O(|α|4 ).

(55)

Again, as above, we resolve the Riemann problem with data uL and uR into increasing k-waves of strength ǫk , and we wish to calculate the strengths ǫk in terms of α and β, to leading order. We leave the details to the reader. Proceeding as above, we find that (56)

= uL + (α + β)rp + 12 (α + β)2 r˙ p + 16 (α + β)3 r¨p + 61 (α− )3 + (β − )3 dp + O (|α| + |β|)4 ,

uR

(57) and (58)

uR = uL +

(59)

1 k 2 p k ǫk r + 2 (α + β) r˙ 3 + 61 (α + β)− dp + O

P

+ 16 (α + β)3 r¨p (|α| + |β|)4 ,

where all vectors are evaluated at the point uL . We now compare these expressions, and since they are identical if α or β are zero, note that the fourth order error becomes O |α||β|(|α| + |β|)2 . Thus we have the following theorem.

Theorem 1. Upon collapse of a weak p-simple wave of strength γ < 0 into a shock, the strengths of the resulting waves are given by (60)

ǫi = − 16 γ 3 ℓi · dp + O(|γ|4 ),

(61)

ǫp

if i 6= p, and

= γ + O(|γ|4 ).

The result of the interaction of two weak p-waves α and β, at least one of which is a shock, is given by (62) ǫp = α + β + O |α||β|(|α| + |β|)2 , and, for i 6= p, 3 3 3 ǫi = 16 α− + β − − (α + β)− ℓi · dp + O |α||β|(|α| + |β|)2 . (63) Here the quantity ℓi · dp is evaluated at the left state uL .

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According to (39), we have Πppp Πipp ℓi · d = , 2 (λp − λi )2 which can be evaluated at the origin u ˜, while the error remains fourth order in the oscillation. We note that after the interaction of two shocks or collapse of a compression wave, the reflected waves have strengths of the same sign as the coefficient Πipp , and that the sign of each reflected wave changes if the sign of one of the incident waves does. We thus interpret the coefficient Πipp as a measure of the strength of reflected i-wave upon interaction of two p-waves. In the particular case where ϕ = id, we obtain the result of Smoller & Johnson that upon interaction of two shocks, the reflected i-wave is a shock if ℓi · D2 f (rp , rp ) < 0, and a rarefaction otherwise [22]. p

4.2. Waves of Different Families. The Riemann problem is the initial value problem in which the data consists of two constant states, whose solution consists of N + 1 constant states separated by elementary waves. Suppose we are given two adjacent Riemann solutions, α separating uL from uM , and β separating uM from uR . The interaction problem is to resolve the Riemann problem with constant states uL and uR , and to express the resulting waves in terms of the strengths α and β. If the resulting waves are given by ǫk , then we have P ǫi = αi + βi + j>k αj βk Λjk (64) i uL + O |α| + |β| D(α, β) P (65) ˜k D(α, β), = αi + βi + j>k αj βk Λjk i + O ku − u jk where Λjk are the interaction coefficients defined above, and D is a quadratic i = Λi u ˜ functional measuring the amount of wave interaction (see [5, 21, 26]). This result is interpreted physically as follows: the interaction coefficient Λjk i represents the amount of i-wave generated by the interaction of a j-wave and a k-wave, up to errors of third order in wave strength. This is a second-order correction, the waves simply crossing linearly to leading order. Since the system is hyperbolic, the only interactions which occur are those for which the faster wave is on the left originally, hence the restriction j > k. Once we have calculated the second-order effects, more subtle questions such as the order in which pairs of waves interact to generate new waves can be ignored, as these extra considerations contribute only third-order effects. Since the interaction coefficients are expressed in terms of the flux coefficients Πijk , for i, j and k distinct, we have the physical interpretation of these coefficients as measuring these interaction effects. If we use the normalization of Section 1, the second-order correction from a the αj − βk jk interaction to the j-th and k-th families vanishes, Λjk j = Λk = 0. In this case, interacting waves themselves suffer corrections of only third order. In other words, to second order, any interaction that a single wave undergoes can be ‘seen’ only via its effect in other families. vanish for i, j and In particular, in case we assume that the interaction coefficients Λjk i u ˜ k distinct (almost planar interactions), our estimate becomes ǫi = αi + βi + O ku − u ˜k D(α, β), and this estimate is used by Schochet to obtain an improved existence result [19]. We note that our normalizing procedure amounts to a change of the length of the eigenvectors, which


17

affects the definition of wave strengths. Indeed, Schochet obtains the above estimate by redefining wave strengths. We shall see that a family for which there exists a Riemann coordinate has Λjk p = 0, and thus is not affected to leading order by the interaction of waves of other (different) families. However, this family will certainly decay if Πppp 6= 0, and may thus affect other families. This is to be contrasted with a linearly degenerate family, for which Πiii = 0 and all waves are contact discontinuities. Waves in such families do not interact with each other and decay, although they may well be affected by interactions of waves from other families. 4.3. Weakly Nonlinear Geometric Optics. The method of weakly nonlinear geometric optics yields simplified equations for first-order approximations to solutions of systems of conservation laws [16]. This is a system of Burgers’ equations, coupled by nonlinear convolution terms, which represent the cumulative effects of wave interactions. We briefly describe the method, and comment on the appearance of the flux coefficients in the equations. We are primarily interested in periodic perturbations of constant states. We set P u(x, t) = u0 + ǫ σi (x, t; τ )ri + ǫ2 m, u0

where τ = ǫt. We are interested in t = O(1), that is τ = O(1/ǫ). We expand ϕ and f by Taylor’s formula, and substitute ∂t → ∂t + ǫ∂τ into the conservation law (21), and collect powers of ǫ. The leading order equations become (σi )t + λi (σi )x = 0, for each i, u0

so we set σi (x, t; τ ) = σi (θi ; τ ), where θi = x − λi t. The second order equations become 1X (66) (µi )t + λi (µi )x + (σi )τ + li D2 ϕ(rj , rk )∂t + li D2 f (rj , rk )∂x (σj σk ) = 0, 2 j,k

where µi = ℓi · m. The approximate equations are obtained by imposing the consistency condition ǫm = o(1) as τ → ∞. For each i, we use the change of coordinates (x, t) → (θi , η), where θi is given above, and η = t. Rewriting (66) in these coordinates, and averaging in η (with θi fixed) gives, after imposing the consistency condition and simplifying, the approximate equation for σi in conservation form, X′

i i 1 X′ i 1 (67) Πjk σj σk θ = 0, Πiij σi σj θ + (σi )τ + Πiii σi2 θ + i i i 2 2 j6=k

RK P′ indicates where h·ii is the average in η with θi fixed, hgii ≡ lim 1/K 0 gdη, and summation avoiding the index i. On the other hand, if we first differentiate the last two terms in (66) and then average in the new coordinates, we get the equations in non-conservative form X′

X′ i i Γijk (σj )θj σk = 0. (68) Γiij σj (σi )θi + (σi )τ + Γiii σi (σi )θi + j6=k

Equivalence of these systems can be verified using the relations (19) relating the coefficients. A sufficient condition for these approximate equations to make sense is that the functions σi be almost periodic.

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We interpret each term of the equations separately, examining the significance of the coefficients. The coefficient Πiii = Γiii of the Burgers term determines the rate at which shocks form in the i-th family, and the corresponding rate of decay of the solution. Also Πiij = Γiij , and we can loosely rewrite the next term as hσj rj ·∇λi i(σi )θi . Thus this (linear) term represents drift in the wave-speed λi due to the average perturbation of the j-th family. The last term is a convolution, and represents the effects of interactions between the j- and k-th families, as seen in the i-th family. The term Γijk h(σj )θj σk i represents the cumulative effect on the i-th family of j-waves crossing k-waves. These cumulative effects should be symmetric, and this is the case, as can be checked using (19). Similarly, in the conservation form, the term hΠijk σj σk θ represents those i-waves generated by j − k i interactions. We note that the coefficients Πipp , which represent i-waves generated by interactions of pairs of p-waves, do not appear in these equations. This is to be expected since the approximate equations are obtained from a second order expansion, while the effects of these interactions are third order. We say that nonlinear resonance occurs when the convolution terms contribute to the evolution of the system. Resonance occurs for periodic data provided the ratios of differences in wave speeds are rational. This leads to energy transfer between different fields, which yields a variety of new nonlinear phenomena. These include blow-up of solutions [8] and persistent wave structures for which shock formation can be indefinitely delayed [17, 18]. The interplay between the competing nonlinear effects of decay on the one hand, and new wave generation on the other, is not yet well understood. These effects can quantified by different flux coefficients, and it is a goal of future research to understand how changes in the coefficients lead to different properties of solutions. In the non-resonant case, the convolution terms do not contribute and we obtain a system of uncoupled Burgers equations. Although there are still local effects due to wave interactions, these are not seen globally, as they do not accumulate across any number of periods, and so vanish in the average. For data with small total variation, the convolution terms vanish due to averaging, so resonance does not occur. Finally, we shall see that if the i-th family possesses a Riemann coordinate, then Πijk = 0, so that this family decouples in the asymptotic equations, in the sense that it is not affected by the other fields, although it may act as a source for energy transfer between other fields [16]. In particular, systems for which there is a full system of Riemann coordinates are always non-resonant.

5. Remarks on Entropy The general equations considered above can have complex local interactions, which in turn could lead to very complicated behavior which cannot be easily analyzed. In most physical systems, the imposition of extra assumptions can make the analysis of interactions easier, both conceptually and technically. In this section we describe two special features which lead to such simplifications, namely Riemann invariance and symmetric hyperbolicity. We speculate that either of these assumptions is enough to prevent catastrophic blowup in finite time. Both of these assumptions, which are independent of each other, are valid for the equations of gas dynamics, as a consequence of the thermodynamic relation. These


19

assumptions, together with the fact that the ‘entropy’ family is linearly degenerate, simplify the analysis of the gas dynamics equations by allowing us to regard that family as a background field which to leading order is not affected by the flow. 5.1. Riemann invariants. The notion of Riemann invariants is classical, although several authors have used conflicting definitions. Here we wish to emphasize the distinction between Riemann invariants and Riemann coordinates, which are equivalent for 2 × 2 systems, but differ for larger systems. The definition of Riemann invariants for general systems is due to Lax [13], for gas dynamics see Courant & Friedrichs [2]. We define a k-Riemann invariant to be a function µ satisfying rk ·∇µ = 0 in U. This represents a single restriction on the function µ, so that there will be N − 1 such functions, with linearly independent gradients. From our discussion on wave curves, it is apparent that µ is constant along the rarefaction curve Rk , and the change in µ across a k-shock of strength ǫ is given by 1 1 X µ(u(ǫ)) − µ(u) = ǫ3 dk ·∇µ + O(ǫ4 ) = ǫ3 ℓi · dk ri ·∇µ + O(ǫ4 ). 6 6 As in [24], we now define a p-Riemann variable or p-Riemann coordinate to be a function which is a k-Riemann invariant for each k 6= p. This represents N −1 independent conditions, so that we do not expect Riemann coordinates to exist in general. For N = 2, the two notions are equivalent, but our notation is unfortunate in that the 1-Riemann invariant is the 2-Riemann coordinate. Indeed, this seems to be the source of conflicting terminology. We remark that by choice of an appropriate integrating factor, the p-Riemann coordinate µ can be chosen to satisfy rk ·∇µ = δkp , that is ∇µ = ℓp . In this case, the change in µ across a k-shock is Πkkk Πpkk 1 3 k 4 ǫ3 + O(ǫ4 ), µ(u(ǫ)) − µ(u) = ǫ ℓp · d + O(ǫ ) = 6 12 (λk − λp )2 which reduces to the formula given by Smoller & Johnson [22]. It is an important observation that a family for which there exists a Riemann coordinate has vanishing interaction coefficients [16]. In particular, this means that that family is not affected to second order by interactions of waves of other families. That is, up to third order errors, a family which possesses a Riemann coordinate does not see any other families, and decouples from the system, making the behavior and analysis simpler. This is also suggested by the equations of geometric optics, where the equation corresponding to the Riemann coordinate decouples from the system and can be solved separately. We present a simple proof of this observation. Thus, let µp be the Riemann coordinate. Then for j 6= p, we have rj ·∇µp = 0 = Dµp (rj ). Differentiate this in direction rk , to get D2 µp (rj , rk ) + Dµp (rk ·∇rj ) = 0. Now, observing that Dµp must be some (non-zero) multiple of ℓp , by equality of mixed partial derivatives, we must have k j j k Λkj = 0, for j, k 6= p. p = ℓp · r ·∇r − r ·∇r The converse of this result is also true.

Theorem 2. The following statements are equivalent in a neighborhood U :

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(1) (2) (3) (4)

(a) (b) (c) (d)

There is a Riemann coordinate for the i-th family. The interaction coefficients Λjk i vanish identically, for j, k 6= i. The vector αℓi is a gradient, for some integrating factor α = α(u). For some integrating factor α, we have curl(αℓi ) = 0.

Proof. We need only show that (b) implies (a). Suppose that ℓi · [rj , rk ] = 0 in U for j, k 6= i. Then the span S of the set {rj (u) : j 6= i} is closed under the Lie bracket operation, so forms an involutive N − 1–dimensional distribution [23]. Thus, by Frobenius’ Theorem, there is a function µ whose level surfaces are integral manifolds of S, i.e. have tangent spaces coinciding with those determined by S at each u ∈ U. The function µ is now clearly a Riemann coordinate for the i-th family. 5.2. Symmetric Systems. A Riemann coordinate can be regarded as a special coordinate for a single family which in some sense ‘decouples’ from the others. A different simplification comes from assuming that the equations can be transformed into a symmetric hyperbolic system. Under this assumption, the ‘total entropy’ of the system decreases, and this can be interpreted as a stabilizing condition. Also, we obtain an algebraic relation between the interaction coefficients, which disqualifies all known cases of blow-up of solutions. Thus the assumption of symmetrizability imposes conditions on detailed local interactions, which should be strict enough to avoid catastrophic blow-up. We say that the conservation law (21) is symmetric hyperbolic or symmetrizable in case there is a positive definite change of coordinates, given by u = u(v), such that if we define ψ(v) = ϕ u(v) and g(v) = f u(v) ,

then the matrices

Dψv = Dϕu Duv

and Dgv = Dfu Duv

are symmetric. The conservation law is then given by ψ(v)t + g(v)x = 0, and smooth solutions satisfy the associated quasi–linear equation (69)

Dψv · vt + Dgv · vx = 0,

with symmetric coefficient matrices. The significance of this definition lies in the observation of Friedrichs & Lax that for such systems, an additional conservation law, usually called the entropy equation, can be derived in the regime of smooth solutions [4]. We briefly sketch this derivation. Since Dψv is symmetric, we have curlv ψ = 0, and the vector ψ is (locally) the gradient of a scalar function χ, ψ(v) = (Dχv )T , so that X · ψ(v) = Dχv (X), for vectors X ∈ RN . We then calculate v · Dψv · vt = D2 χv (v, vt ) = Dχv (v) t − Dχv (vt ) = Dχv (v) − χ(v) t .

Similarly, since Dgv is symmetric, there is a scalar η = η(v) such that g(v) = (Dηv )T and v · Dgv · vx = Dηv (v) − η(v) x .


21

From these observations, we see that after multiplication of (69) on the left by v, we derive the new scalar (entropy) equation (70)

Ut + Fx = 0,

where the entropy U is given by U = Dχv (v) − χ(v), where v = v(u), and the entropy flux is F = Dηv (v) − η(v). It was noticed by Harten in [7] that this entropy is the Legendre transform of the function χ(v) with respect to the dual variable w = (Dχv )T = ψ(v), which is the vector of conserved quantities. Moreover, since both changes of coordinates u = u(v) and w = ϕ(u) are positive definite, so is the coordinate change w = ψ(v) = (Dχv )T , which implies the convexity of χ(v), and hence also U (w). Thus the entropy is properly viewed as a convex function U = U (w) of the conserved quantities. Conversely, suppose that we know that a new conservation equation (70) is derivable from smooth solutions of the original conservation law (21), and that this entropy U is a convex function of the conserved quantities w = ϕ(u). This implies that DUw · Dhw = DFw , where the vector function h is given by h(w) = f (u) = f ϕ−1 (w). Multiplying this by the vector X and differentiating in direction Y ∈ RN then yields (71)

D2 Uw (Y, Dhw X) = −DUw · D2 hw (X, Y ) + D2 Fx (X, Y ),

so that this is symmetric in X and Y . In particular, we have D2 U (r i , r j ) = 0, for i 6= j, for the eigenvectors of Dh [1]. We now show that the conservation law is symmetrizable : indeed, the appropriate coordinate v = v(u) is given by the Legendre dual variable to w, namely v = (DUw )T . Defining v thus, we must show that the matrices Dϕ(u)v = Dwv and Df (u)v = Dh(w)v = Dhw Dwv are symmetric, and Dwv is positive definite. To this end, for any vectors X and Y , we have X T · Dvw · Y = D2 Uw (X, Y ), so that Dvw is symmetric and positive definite, and therefore so is Dwv = (Dvw )−1 . It follows that X T · Dhw Dwv · Y = (Dwv X)T Dvw Dhw (Dwv Y ) = D2 Uw Dwv X, Dhw (Dwv Y ) ,

which according to (71) is also symmetric. We remark that, as above, w = w(v) can be given explicitly as w = (Dχv )T , where χ(v) is the Legendre transform of U (w). If the entropy U is given as a function of the state variables u, then we have DUu = DUw Dwu = DUw Dϕ, and after differentiating, we see that convexity of U as a function of w = ϕ(u) is equivalent to positive definiteness of the quadratic form (72)

2 D2 Uu − DUu Dϕ−1 u D ϕu .

We have seen that the existence of a convex entropy function is equivalent to the requirement that the system of conservation laws be symmetrizable. For such systems, Lax made the important observation that the entropy function can be used to distinguish the correct weak solution to the conservation law [14]. This is accomplished by requiring weak solutions of (21) to satisfy the entropy inequality Ut + Fx ≤ 0

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in the weak sense. This condition is locally equivalent to the Lax shock condition, and also arises when the solution is the limit of solutions of a range of approximate equations, including the appropriate viscous conservation laws. As a consequence of the entropy inR equality we see that the quantity U dx is non-increasing, and this can be interpreted a stability requirement for solutions. We now calculate the interaction coefficients for symmetrizable systems. Invoking the formulas of Section 1 for coordinate changes where necessary, we may assume that the conservation law is symmetric in the original coordinate system, so that Dϕu and Dfu are symmetric. Recall that the matrix A = Dϕ−1 Df has left and right eigenvectors ℓi and ri , respectively, and li = ℓi · Dϕ−1 is the corresponding left generalized eigenvector, Df · rj = λj Dϕ · rj ,

and lj · Df = λj lj · Dϕ.

Now, since Df and Dϕ are symmetric, we take li = (ri )T , which determines the length of the eigenvector. We then have D2 χ(ri , rj ) = ℓi · rj = δij . According to the above, we can write the functions f and ϕ as gradients, say f (u) = (Dηu )T ,

and ϕ(u) = (Dχu )T .

It follows that li · Df (rj ) = D2 η(ri , rj ), and similarly li · D2 f (rj , rk ) = D3 η(ri , rj , rk ), with analogous formulae for ϕ. In particular, the quantities li ·D2 f (rj , rk ) and li ·D2 ϕ(rj , rk ) are symmetric in all three indices i, j, and k. Thus, according to equations (17) and (18), the flux coefficients are given by (73) (74)

Πijk = D3 η(ri , rj , rk ) − λi D3 χ(ri , rj , rk ), Γijk

and

= D η(r , r , r ) − λj D χ(r , r , r ) = Πjik . 3

i

j

k

3

i

j

k

The requirement of symmetry implies that (75)

(λj − λk )Πijk + (λk − λi )Πjki + (λi − λj )Πkij = 0,

which according to (20), is equivalent to ij

(76)

Λki Λk Λjk j i + = 0, + λj − λk λk − λi λi − λj

for i, j and k distinct. Assuming for definiteness that i < j < k, we deduce that for kj ji symmetric systems, the quantities Λki j , Λi and Λk cannot all have the same sign. We shall see the significance of this observation when considering three–wave interactions and blow-up of solutions. The statement D2 χ(ri , ri ) = 1 defines the length of the eigenvectors ri in a neighborhood, so that differentiating in the direction rk and using the extra symmetries determines the interaction coefficients Λik i . Indeed, we calculate 1 1 Πiik − Πkii ℓi · (rk ·∇ri ) = − D3 χ(ri , ri , rk ) = − , 2 2 λk − λi i k so that Λik i = (Πik + Πii )/2(λk − λi ). Alternatively, we could apply the normalization of ik Section 1, to get Λi = 0 at a single point, giving up control of the length of the eigenvectors.


23

We remark that (75) is a necessary condition for symmetrizability, and is also sufficient in the sense that, given constants Πijk , symmetric in j and k and satisfying (75), there is a symmetric system realizing these values at the origin. If the vectors ri (˜ u) are to be specified, we require also that they be orthonormal with respect to D2 χ. This construction is analogous to that of Section 1, where now we choose the scalar functions χ and η to have specified first, second and third derivatives, using (75) for consistency. Note that in this case the extra symmetry requirements imply Γijk = Πjik and determine all derivatives but D3 χ(ri , ri , ri ) = − 12 ℓi · (ri ·∇ri ), which we may again take to be zero. 6. The Cauchy Problem We present a short survey of existence results for the Cauchy problem. We restrict our attention to Cauchy data with small L∞ -norm, which ensures that Riemann problems are uniquely solvable. We are primarily concerned with Cauchy data of large total variation. The problems of global existence and large–time behavior of solutions are well understood for 2 × 2 systems, but not for larger general systems for which local interactions are more complex. We describe some existence and non–existence results, and some open problems. We then describe the local origin of blow-up of solutions in terms of ‘three–wave interactions’, and obtain an instability result for a model problem. These results depend critically on the values of the interaction coefficients, and we discuss the consequences of making extra assumptions on these coefficients. 6.1. Existence Theorems. All known existence theorems for general N × N systems are based on Glimm’s celebrated Random Choice Method [5]. Glimm showed that if the total variation (T.V.) of the Cauchy data is small enough, then a solution exists for all time, and the solution is stable. By stable we mean that the T.V. of the solution at any time is bounded by a multiple of the initial T.V., and the solution operator is L1 -Lipschitz in time. Note that T.V. dominates the L∞ norm S. Glimm also showed that in case there is a full system of Riemann coordinates (which is always the case for two equations), then the requirement that T.V. be small can be replaced by the weaker requirement that S(1 + T V ) be small. For systems of two equations, Glimm & Lax showed that solutions decay in T.V., and obtained existence and large–time behavior for any data of small oscillation S [6]. In particular, this includes global existence of solutions with periodic data. The decay is due to expansion of rarefactions and formation of shocks, and occurs separately in each family due to the existence of Riemann coordinates. For larger systems with a full set of Riemann coordinates, Zumbrun has recently proved a similar existence and decay result for periodic data [28]. In these systems, there are no (second order) waves generated by interactions of waves from different families. For general systems of N × N equations, Liu has proved decay in genuinely nonlinear fields for data of small T.V. and compact support [15]. Although the fields do not decouple as for systems with Riemann coordinates, because T.V. is small the different fields separate quickly, and decay in each family separately takes over as the dominant phenomenon. For systems for which the interaction coefficients Λjk ˜ for i, j and k distinct, i vanish at the origin u Schochet has weakened the requirement that T.V. be small to needing small S(1 + T V ).

24

ROBIN YOUNG

Also, for the case of gas dynamics, he obtains existence when S(1 + T V 2 ) and the T.V. of the contact field are small [19]. The author has found an explicit constant V0 = c/Λ, where Λ = max |Λjk i | and c is a dimensional constant, such that if T V (u0 ) < V0 and S is small enough, then a global solution exists [25]. When the initial data has large total variation, then resonance effects become important and new nonlinear phenomena arise. These phenomena include blowup of solutions and indefinite delay of shock formation. These phenomena are not yet well understood, and in particular the exact conditions for large–time existence of solutions are not yet known. In particular, the Glimm–Lax decay theory is not sufficient for general systems of three or more equations, and a more refined theory of wave interactions is necessary. 6.2. Geometric Optics and Nonexistence. The equations of weakly nonlinear geometric optics describe the leading order behavior of solutions to conservation laws. For initial perturbations ofP size ǫ, the equations are derived for times of order 1/ǫ, assuming that the expression ǫ σj (x, t)rj is the first–order approximation to the solutionPu(x, t) of the conservation law. The approximation could then be said to be valid if ǫ σj rj indeed approximates u with O(ǫ2 ) errors, for times up to O(1/ǫ). The first result on validity of the approximate equations was obtained by DiPerna & Majda [3]. For Cauchy data of bounded variation and compact support, they showed that the error is given by O(ǫ2 ), uniformly in time, when measured in the L1 -norm. This is again due to the fast separation of different families. Also, for systems with a full set of Riemann coordinates and periodic data, they obtained an L1 error of O(ǫ2 t). In particular, the justification is valid beyond the time of shock formation. Joly, Metivier & Rauch showed validity of the approximation in a general setting as long as the solutions to the approximate equations remain smooth [11]. For general systems, Schochet has shown that solutions generated by Glimm’s scheme can be approximated with error o(ǫ + ǫ2 t) in L1 , where t < T /ǫ, for some positive T [20]. This restriction on the lifetime is necessary to avoid the possible blow-up of solutions due to resonance. Hunter has constructed explicit generalized solutions to the asymptotic equations which blow up in finite time [8]. Hunter sets σj (θj ; τ ) = αj (τ )S(kj θj − ξj ), where αj is a scalar amplitude and S is the sawtooth function. By a careful choice of phase kj and shifts ξj , and using the fact that ( 21 S 2 )′ = S, he effects a separation of variables, to get a system of coupled ODEs for the amplitudes αj , comparable to the equation y ′ = y 2 . For certain choices of coefficients, these amplitudes explode in finite time. This implies that the functions σj blow up in all Lp norms simultaneously. It is crucial here to have the correct resonance and phase matching conditions. Also, the T.V. of the initial data must be above a certain threshold. Recently Joly, Metivier & Rauch have given a rigorous example of a periodic solution to the conservation law which grows arbitrarily large finite time [12]. They construct a smooth solution to the approximate equations which blows up, and conclude that the corresponding solution to the conservation law must also be smooth and blow up in any Lp -norm in finite time. In particular, theirs is a rigorous result, and blow-up occurs before shock formation. Again it is necessary to carefully choose the parameters which lead to blow-up. A feature of these results is the occurrence of blow-up for all positive values of the oscillation of the initial data, however small. Indeed, given any neighborhood U, there is


25

a time T∗ , so that for any K, there are solutions with values in U for all times up to T∗ , satisfying ku(·, T∗ )kp > K ku0 kp , with initial data having arbitrarily small oscillation ku0 k∞ . We note that although the solutions to the approximate equations do become infinite in finite time, we cannot conclude that the solutions to the conservation law also become infinite, but only that they possess the strong nonlinear instability described here. This is due to the fact that the asymptotic equations will no longer be valid when the amplitude of the solution becomes large. We remark that in all known cases of blow-up of solutions in finite time, the interaction coefficients Λi = Λjk i (where i, j and k are distinct and j > k) have the same sign and dominate the decay coefficients Γiii . In particular, this excludes symmetric systems, for which (76) systems, for which the R holds. We speculate that blow-up is impossible in these R quantity U is non-increasing. Since the entropy U is convex, U can be interpreted as a ‘norm’ similar to an Lp -norm. In contrast, for all known instances of blow-up, we do not have stability in any Lp -norm. 6.3. Three–wave Interactions. We now discuss the mechanism of blow-up from a local perspective. We shall obtain an instability result for a model problem, and give a detailed description of the growth mechanism. We construct exact solutions to the conservation law for which the amplitude exhibits arbitrarily large growth in finite time. These solutions can be either periodic or compactly supported. For simplicity, we work with a 3 × 3 system, in which all three wave speeds are constant, and we assume that all interaction coefficients except Λi = Λjk i , for j > k distinct from i, vanish at the origin. This can be achieved via the normalization of Section 1. By a Galilean transformation, we can assume that the wave-speeds are given by −1, 0 and σ > 0, and we shall take σ = 1, but it is easy to see that the results hold for any rational σ. A three–wave interaction is a simultaneous interaction of three waves from different families. Note that this is not the same as the interaction problem for two Riemann solutions. In order for the waves to interact at the same point, they need to appear in slower families from right to left. Thus suppose the incident waves are γ, β and α, respectively, in decreasing families. After interaction, we resolve the Riemann solution into waves in increasing order of wave-speed. Figure 1 illustrates a three–wave interaction, showing the waves as characteristics and along wave curves in state space. According to (65), the outgoing wave strengths are given by (77) (78) (79)

α′ ≈ α + Λ1 γβ, ′ β ≈ β + Λ2 γα, and γ′ ≈ γ + Λ3 βα,

where we are ignoring third order terms. We shall say that magnification occurs in case each of the outgoing waves is stronger, and has the same sign, as the corresponding incoming wave. It is clear that this is the case if and only if the quantities αβγ and each Λi have the same sign. We suppose for definiteness that each Λi is positive. Our model problem is thus described as follows : the wave speeds are given by -1, 0 and 1, and the interaction coefficients Λi are positive. The following theorem holds for this system :

26

ROBIN YOUNG

α’

α

β’

uR β

γ’

uL

uR α γ

γ

uL

γ’ β’

r2 r3

α’

β

r1

Figure 1. A three–wave interaction Theorem 3. For any positive constant λ < min Λi , there are solutions with initial data of bounded variation which blow up in any finite time. More precisely, given positive constants τ , ǫ and K, there is some te < τ and solutions u(x, t) defined for t ∈ [0, te ) satisfying (80) (81)

ku(·, 0)k∞ < ǫ, ku(·, 0)k1 < ǫ and 8/λ ≤ T V u(·, 0) ≤ 10/λ,

while for some tb < te , we have (82) (83)

ku(·, tb )k∞ > K ku(·, 0)k∞ ,

ku(·, tb )k1 > K ku(·, 0)k1 and T V u(·, tb ) > K T V u(·, 0) .

It is remarkable that blowup occurs for data which has total variation larger than 8/λ, when contrasted with the result of the author that solutions are stable as long as the initial variation is less than 1/(3 max Λi ) and the sup-norm is small enough. In this case, although a resonating pattern can be established and rapid growth may occur, there are not waves initially to preserve the resonance, and the different families will separate, halting the growth process before catastrophic blowup. Also interesting is the fact that blowup can take place in any finite time, although this is probably an anomaly due to our assumption of total linear degeneracy. Proof. We construct a periodic resonant wave pattern in which all interactions are threewave interactions with magnification. This can be done explicitly, since all wave-speeds are constant, and so the characteristics along which waves propagate are straight lines. We first find a localized periodic configuration of waves, and then extend this by periodicity. To this end, start with eight states labeled 0–7 located as follows : starting at 1, we connect states 1–6 by waves −β, α, β, γ and −β, respectively; now we resolve the Riemann problem between 6 and 1, to get states 7 and 0, where states 6–7–0–1 are connected by waves −α′ , β ′ and −γ ′ , respectively, see Figure 2. Here the α’s, β’s and γ’s refer to 1, 2 and 3-waves, respectively. Moreover, since all waves are weak, we can take all α’s etc., to have positive nearby values. We now construct the periodic resonating wave pattern as follows. The initial 1-waves (respectively 3-waves) will emanate from the points x = 2k ∆x, and have strength α or −α′ (respectively γ or −γ ′ ) depending on whether k is even or odd. The initial 2-waves emanate from the points x = k ∆x, and have strength β, −β, β ′ or −β, according to whether k


27

mod 4 is 0, 1, 2 or 3, respectively. These waves are separated by the constant states chosen above. Now for the model problem, it is clear that all interactions of waves will be threewave interactions, and these will occur exactly at the points (x, t) = (k∆x, j∆x), for j + k even. Moreover, the solution after interaction again consists of constant states separated by waves, and a resonance pattern is set up. Figure 2 illustrates the resonance pattern, again with characteristics and in state space. We have labeled the constant states to show how they change, and the dashed lines show a single interaction. Note that although the states and wave strengths change, the general pattern is repeated. t

a

d b

e

c

0

f

3 1

5

a

7

0

g

4

2

h

6

x 6

c

c

7

3

7

3

h

d

2

g g

5 0

4 1

0

b

b 4

a

e

f

f

Figure 2. A resonant interaction pattern By our choice of initial wave strengths, the product of strengths of waves entering each interaction is positive, so that magnification occurs at each interaction. In order to see that blow-up occurs, we define ηq to be the minimum strength of all waves at time t = 2q ∆x+, that is after each wave has passed through at least q interactions. Then, assuming the waves remain weak, for some positive λ < λ′ < Λi , we have ηq+1 ≥ ηq + λ′ ηq2 ≥ ηq + λ ηq ηq+1 , for each q. This is true by the interaction estimate (65), where we observe that the error O(SD) is cubic in η, so that it is dominated by the quadratic part. Now, by comparison with the exact solution to the corresponding difference equation, we have η0 (84) for each n. ηn ≥ 1 − λ η0 n We can obtain solutions with compact support simply by cutting off the wave pattern for some |x| > L. Note that if we do this, resonance occurs in a region containing the set S = {(x, t)| − L + t < x < L − t}. Our calculation of ηn remains valid as long as ηn measures

28

ROBIN YOUNG

the strength of waves contained in S. We estimate the norms for the solution by waves contained in S. Similar estimates follow for periodic solutions. We choose the data so that all initial wave strengths are between η0 and 5η0 /4, where η0 will be chosen later. The sup-norm is measured by the maximum wave strength, and the total variation is measured by the sum of (absolute) wave strengths. We thus have η0 ≤ ku(., 0)k∞ ≤ 5η0 /4, and 4L η0 /∆x ≤ T V u(., 0) ≤ 5L η0 /∆x,

as there are 4 waves per interval of length 2∆x. The L1 -norm is given by Z 4∆x u ∼ η0 L. ku(., 0)k1 = 2L/4∆x · 0

Similarly, the norms at time t < L satisfy (85) (86) (87)

ku(., t)k∞ ≥ ηn , T V u(., t) ≥ 4(L − t) ηn /∆x, and ku(., t)k1 ≥ (L − t) ηn ,

where n is given by n = [t/2∆x]. Now, given K, our instability will hold as long as ηn > η0 · 5KL/4(L − t), or invoking (84), (88)

1 5K L > , 1 − λ η0 n 4 (L − τ )

if also t < τ < L. Thus we take L > τ , and choose η0 and ∆x such that the relations λ η0 n → 1 and t → τ are equivalent. This is accomplished by defining ρ = 2/λ τ and setting η0 = ρ ∆x. We now choose ∆x so small that (88) holds, while ηn+2 remains small enough that the estimates used above still hold. It is clear that the sup- and L1 -norms of the data can be made arbitrarily small, while the total variation is bounded below by 4Lρ > 8/λ. Also, since the waves remain a fixed distance ∆x apart while they grow in size, we have instability of the solution in all Lp -norms. The scaling relation η0 ∼ ∆x is the same as that appearing in the previous examples of blowup described above, although the blowup time was previously thought to be bounded away from zero. In our example, this choice of blowup time comes from having large waves close together in each family. Here we are implicitly using the constancy of wave-speeds. The freedom in choosing the scaling factor ρ will likely be lost when the wave-speeds are genuinely nonlinear. As in the previous examples, we make the observation that although these solutions are highly unstable, they do not necessarily become infinite in finite time. That is, although the amplitudes can be arbitrarily magnified, our construction breaks down before solutions become infinite. Indeed, once the amplitude becomes large, the global structure of the wave curves becomes important and the Riemann problem is not necessarily well-posed. It is easy to see that periodic resonance can occur for any rational value of σ, and so for any system for which the ratio (λ3 − λ2 )/(λ2 − λ1 ) is rational. For irrational σ, it may still be possible to find almost periodic solutions which blow up. As we have suggested, it appears that the assumption that each Λi have the same sign is essential for blow-up to


29

take place. We conjecture that if one of these coefficients vanish, which would be the case if there were a Riemann coordinate, then blow-up in finite time will not take place, although the solution may grow exponentially. Indeed, if in the above model problem we take Λ2 = 0, similar calculations yield exponential growth in the first and third families. Similarly, for symmetric systems, for which one Λi is negative, we expect that catastrophic blowup is not possible. On the other hand, the assumption of constant wave-speeds does not seem to be essential. Indeed, the example of Joly, Metivier & Rauch shows that blow-up can take place when all three fields are genuinely nonlinear. The main requirement appears to be that the interactions should have magnification, and that waves in a single family do not spread out and decay so fast that magnification is offset by decay. As long as this were the case, the above argument would presumably indicate that blow-up indeed takes place. We note that the initial configuration of Figure 2 can be constructed for any system, and whether or not the solution becomes unbounded should depend on the rate at which characteristics approach each other. 7. Gas Dynamics We now consider the Euler equations of gas dynamics. The thermodynamic variables are the density ρ, pressure p, internal energy e, temperature T , entropy S and fluid velocity v. These are related via the thermodynamic relation T dS = de − p/ρ2 dρ.

(89)

We assume we are given an equation of state, expressing the internal energy e as a function of two of the thermodynamic variables. If this is given as e = e(ρ, S), the pressure, temperature and sound speed c are given by p = ρ2 eρ S , T = eS ρ and c2 = pρ S , respectively. In addition, we assume that pS ρ > 0, which allows us to express the entropy S as a function of ρ and p. We shall then use these as independent variables, with the chain rule (90) fS ρ = fp ρ pS ρ = fp /Sp and fρ S = fp ρ pρ S + fρ p = fρ + c2 fp , (91) for any function f = f (ρ, p). We deduce that (92)

T (ρ, p) = ep /Sp

and c2 (ρ, p) = −Sρ /Sp =

p/ρ2 − eρ , ep

where the last equality follows from (89). The Euler equations are (93)

ρt + (ρv)x = 0

(94)

(ρv)t + (ρv 2 + p)x = 0 1 1 ( ρv 2 + ρe)t + ( ρv 3 + ρev + vp)x = 0 2 2

(95)

30

ROBIN YOUNG

representing conservation of mass, momentum and energy, respectively. We take as dependent variables the density, velocity and pressure, so that the state vector, conserved quantities, and flux are given by       ρv ρ ρ  and f =   ρv ρv 2 + p u =  v , ϕ =  1 1 2 3 p 2 ρv + ρe 2 ρv + ρev + vp

respectively. We calculate



 1 0 0 v ρ 0  Dϕ =  1 2 2 v + (ρe)ρ ρv ρep

and



v v2 Df =  1 3 2 v + (ρe)ρ v

so that



Dϕ−1 = 

which yields

 ρ 0 , 2ρv 1 3 2 2 ρv + ρe + p ρep v + v

 1 0 0 −v/ρ 1/ρ 0 , 1 2 2 v − (ρe)ρ /ρep −v/ρep 1/ρep 

 v ρ 0 A = Dϕ−1 · Df =  0 v 1/ρ  . 0 ρc2 v

The eigenvalues of the matrix A are v − c, v and v + c, and the corresponding eigenvectors are given by λ

r

ℓ

v−c 

 −ρ  c  −ρc2

v 

 ρ  0  0

v+c 

 ρ  c  ρc2

(0 1/2c − 1/2ρc2 ) (1/ρ 0 − 1/ρc2 ) (0 1/2c 1/2ρc2 ).

A Riemann coordinate can be found for the second family, by finding a function w = w(ρ, p) satisfying c2 wp + wρ = 0, so that by (92), the classical entropy S = S(ρ, p) is an appropriate Riemann coordinate. Clearly the second family is also linearly degenerate. We remark that since there is a natural sound speed c, the ratio of differences of wave–speeds is always rational, so that long–term interaction effects are always present. It is well known that for smooth solutions, the Euler equations together with the thermodynamic relation yield an equation for conservation of entropy, namely St + vSx = 0, which in turn can be written in conservation form as (−ρS)t + (−ρvS)x = 0.


31

We remark that the convex entropy function here is U = −ρS, rather than simply the classical entropy S. The Euler equations can thus be symmetrized, and according to Section 3, the coordinates in which the equations are symmetric are given by D(−ρS)w T , where w = ϕ(u). Using (92), we calculate (96) D(−ρS)w = D(−ρS)u · Duw = − (ρS)ρ 0 ρSp · Dϕ−1 1 2 = − (ρS)ρ + 2 v − (ρe)ρ Sp /ep − vSp /ep Sp /ep (97) (98) = 1/T G − 12 v 2 v − 1 ,

where G is the Gibbs free energy, given by G = e + pρ−1 − T S. According to (72), convexity of U = −ρS as a function of the conserved variables is equivalent to definiteness of the quadratic form D2 Uu − DUu · Dϕ−1 D2 ϕ. Using (89), we represent this form as the matrix   (ρe)ρρ − T (ρS)ρρ 0 (ρe)pρ − T (ρS)pρ . 0 ρ 0 1/T  (ρe)pρ − T (ρS)pρ 0 (ρe)pp − T (ρS)pp After differentiating the thermodynamic relations T Sp = ep and T Sρ = eρ − p/ρ2 , this matrix simplifies to   Tρ Sρ 0 Tρ Sp 1 0 . ρ/T  0 Tρ Sp 0 Tp Sp

Now since Tρ Sp = Tp Sρ + 1/ρ2 and Sρ = −c2 Sp < 0, we conclude that the equations are symmetrizable with a convex entropy as long as Tρ < 0. When expressed in terms of ρ 2 and S, this is equivalent to the condition c2 TS ρ − ρTρ S > 0. We remark that since the entropy has the special properties described earlier, it follows from Theorem 3.1 and (76) that the interaction coefficients satisfy Λ31 2 =0

and

32 Λ21 3 = −Λ1 .

To calculate the various coefficients, we write DA as a vector–valued matrix, whose elements are simply the gradients of the elements of A. The matrix DA(v) is then the matrix obtained by multiplying each vector by v. We have   (0 1 0) (1 0 0) 0 2  0 (0 1 0) DA =  (−1/ρ 0 0) , 2 2 (0 1 0) 0 (ρc )ρ 0 (ρc )p and similarly,



0 0 (0 1 0) (1 0 0) D2 ϕ =  (ρe)ρρ v (ρe)ρp (v ρ 0)

(ρe)ρp

 0  0 . 0 (ρe)pp

All coefficients Γijk and Πijk can now be calculated according to the formulas of Section 1. We now assume we are dealing with an ideal polytropic γ-law gas, whose constitutive relation is given by pρ−1 e = e(ρ, p) = . γ−1

32

ROBIN YOUNG

Thus we have ρe = p/(γ − 1), so that ρc2 = γp,

that is c =

r

γp , ρ

which can be substituted into the above expressions. It is a routine but tedious calculation to calculate B = Df · Dϕ−1 , and after differentiating, we obtain   0 0 0 . 0 (0 3 − γ 0) 0 Dϕ−1 · DB · Dϕ =  2 2 (0 − c 0) (−c 0 γ) (0 γ 0)

As we have noted, the classical entropy S = cV log(pρ−γ ) is a Riemann coordinate for the second family. Using indices −, 0 and + in place of 1, 2 and 3, respectively, we have r0 = (ρ 0 0)T and ± r = (±ρ c ± γp)T . The flux coefficients Γijk = ℓi · DA(rk ) rj

and

Πijk = ℓi · Dϕ−1 DB(rk ) Dϕ rj

are now easily calculated. For example, we have Π000 = Π0+− = 0,

(99)

+ Π− +0 = −Π0− = c/2,

(100) (101) (102)

+ Π− −− = Π++ = (γ + 1)c/2

Π+ −−

=

Π− ++

and

= (5 − 3γ)c/2.

Alternatively, by observing that pcp = 12 c = −ρcρ , we calculate directly that (103)

r+ ·∇r0 = r0 ·∇r0 = −r− ·∇r0 = r0 ,

r0 ·∇r± = −14(r+ + r− ) ± r0 and 1 1 (105) r+ ·∇r± = −r− ·∇r± = (γ − 1 ± 2γ)r+ + (γ − 1 ∓ 2γ)r− ∓ (γ − 1)r0 . 4 4 From these we get γ−1 + (106) (r + r− ), [r+ , r− ] = 2 1 (107) [r+ , r0 ] = (r+ + r− ) and 4 1 (108) [r0 , r− ] = − (r+ + r− ), 4 +− +0 1 so that in particular, we have Λ0 = 0 and Λ− = −Λ0− + = 4 , as can be deduced from the j Πijk s. Similarly, we calculate the quantities rk ·∇λj = Γjk . Finally, using (39), we read off the values of the differences di ,

(104)

(109) (110) (111)

ℓ+ · d0 = ℓ− · d0 = 0, 1 ℓ0 · d− = −ℓ0 · d+ = (γ + 1)(γ − 1) and 2 γ+1 ℓ− · d+ = ℓ+ · d− = (5 − 3γ). 32


33

We remark that a change in the length of the eigenvectors changes the coefficients by a corresponding scalar factor. Thus, once these coefficients have been calculated from DA and D2 ϕ, we could normalize the eigenvectors as appropriate, while modifying each coefficient by the corresponding scalar factor. In particular, we could carry out the normalization vanish unless i, j described in Section 1 to arrange that the interaction coefficients Λjk i u ˜ and k are distinct, without changing the values of the calculated coefficients at u ˜. Of course, j k in that case the direct calculation of the vectors [r , r ] would no longer apply. Following our earlier remarks, once the above quantities have been calculated, we can deduce the type of waves which are reflected after an elementary interaction. Since the second family is linearly degenerate, 0-waves are contact discontinuities. According to our choice of r0 , a positive wave strength corresponds to an increase in density from left to right. We can now read off directly the effects of interactions. For example, the interaction of a fast shock α < 0 with a positive contact β > 0 will reflect a slow wave of approximate strength αβΛ+0 − = αβ/4 < 0, which is a shock. Similarly, according to Theorem 3.1, the reflected i-wave upon interaction of two weak p-shocks has the sign of Πipp , and we obtain the well known result that the reflected wave of the opposite family is a rarefaction in case γ < 5/3, and a shock if γ > 5/3. Treating the other cases similarly, we obtain the qualitative effects of weak interactions (see [2, 27]). References 1. J.G. Conlon and Tai-Ping Liu, Admissibility criteria for hyperbolic conservation laws, Ind. Univ. Math. Jour. 30 (1981), 641–652. 2. R. Courant and K.O. Friedrichs, Supersonic flow and shock waves, Wiley, New York, 1948. 3. R. DiPerna and A. Majda, The validity of nonlinear geometric optics for weak solutions of conservation laws, Comm. Math. Phys. 98 (1985), 313–347. 4. K.O. Friedrichs and P.D. Lax, Systems of conservation laws with a convex extension, Proc. Nat. Acad. Sci. 68 (1971), 1686–1688. 5. J. Glimm, Solutions in the large for nonlinear hyperbolic systems of equations, Comm. Pure Appl. Math. 18 (1965), 697–715. 6. J. Glimm and P.D. Lax, Decay of solutions of systems of nonlinear hyperbolic conservation laws, Memoirs Amer. Math. Soc. 101 (1970). 7. Amiram Harten, On the symmetric form of systems of conservation laws with entropy, J. Comp. Phys. 49 (1983), 151–164. 8. John Hunter, Strongly nonlinear hyperbolic waves, Nonlinear Hyperbolic Equations – Theory, Computation Methods, and Applications (J. Ballmann & R.Jeltsch, ed.), Viewig, 1989, pp. 257–268. 9. , No total variation bounds for large data, Unpublished note, 1991. 10. Fritz John, Formation of singularities in one-dimensional nonlinear wave propagation, Comm. Pure Appl. Math. 27 (1974), 377–405. 11. J.-L. Joly, G. Metivier, and J. Rauch, Resonant one dimensional nonlinear geometric optics, J. Fun. Anal. (1992). , A nonlinear instability for 3 × 3 systems of conservation laws, Commun. Math. Physics 162 12. (1994), 47–59. 13. P.D. Lax, Hyperbolic systems of conservation laws, II, Comm. Pure Appl. Math. 10 (1957), 537–566. 14. Peter Lax, Shock waves and entropy, Contributions to Nonlinear Functional Analysis (E. Zarantonello, ed.), Acad. Press, 1971, pp. 603–634. 15. Tai-Ping Liu, Decay to N -waves of solutions of general systems of nonlinear hyperbolic conservation laws, Comm. Pure Appl. Math. 30 (1977), 585–610. 16. A. Majda and R. Rosales, Resonantly interacting weakly nonlinear hyperbolic waves I. A single space variable, Stud. Appl. Math. 71 (1984), 149–179.

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17. A. Majda, R. Rosales, and M. Schonbeck, A canonical system of integrodifferential equations arising in resonant nonlinear acoustics, Stud. Appl. Math. 79 (1988), 205–262. 18. R. L. Pego, Some explicit resonating waves in weakly nonlinear gas dynamics, Studies in Appl. Math. 79 (1988), 263–270. 19. S. Schochet, Glimm’s scheme for systems with almost–planar interactions, Commun. PDE 16 (1991), 1423–1440. 20. , Resonant nonlinear geometric optics for weak solutions of conservation laws, Preprint, 1992. 21. J. Smoller, Shock waves and reaction-diffusion equations, Springer-Verlag, New York, 1982. 22. J.A. Smoller and J.L. Johnson, Global solutions for an extended class of hyperbolic systems of conservation laws, Arch. Rational Mech. Anal. 32 (1969), 169–189. 23. F.W. Warner, Foundations of differentiable manifolds and lie groups, Springer–Verlag, New York, 1983. 24. G.B. Whitham, Linear and nonlinear waves, Wiley, New York, 1974. 25. R.C. Young, An extension of Glimm’s method to third order in wave interactions, Ph.D. thesis, U.C. Davis, 1991. 26. Robin Young, Sup-norm stability for Glimm’s scheme, Comm. Pure Appl. Math. 46 (1993), 903–948. 27. Tong Zhang and Ling Hsiao, The Riemann problem and interaction of waves in gas dynamics, Longman, New York, 1989. 28. Kevin Zumbrun, In preparation. Department of Mathematics & Statistics, University of Massachusetts