Characteristic Discontinuities and Free Boundary ... - CiteSeerX

Report no. OxPDE-11/07

Characteristic Discontinuities and Free Boundary Problems for Hyperbolic Conservation Laws By Gui-Qiang Chen Mathematical Institute, University of Oxford &

Yaguang Wang Department of Mathematics, Shanghai Jiao Tong University

Oxford Centre for Nonlinear PDE Mathematical Institute, University of Oxford Radcliffe Observatory Quarter, Gibson Building Annexe Woodstock Road Oxford, England OX2 6HA Email: [email protected]

March 2011

CHARACTERISTIC DISCONTINUITIES AND FREE BOUNDARY PROBLEMS FOR HYPERBOLIC CONSERVATION LAWS GUI-QIANG CHEN AND YAGUANG WANG

Abstract. We are concerned with entropy solutions of hyperbolic systems of conservation laws in several space variables. The Euler equations of gas dynamics and magnetohydrodynamics (MHD) are prototypes of hyperbolic conservation laws. In general, there are two types of discontinuities in the entropy solutions: shock waves and characteristic discontinuities; and characteristic discontinuities can be either vortex sheets or entropy waves. In gas dynamics and MHD, across a vortex sheet, the tangential velocity field has a jump while the normal velocity is continuous; across an entropy wave, the entropy has a jump while the velocity field is continuous. A vortex sheet or entropy wave front is a part of the unknowns, which is a free boundary. Compressible vortex sheets and entropy waves, along with shock and rarefaction waves, occur ubiquitously in nature and are fundamental waves in the entropy solutions to multidimensional hyperbolic conservation laws. The local stability of shock and rarefaction waves have been relatively better understood. In this paper we discuss the stability issues for vortex sheets/entropy waves and present some recent developments and further open problems in this direction. First we discuss vortex sheets and entropy waves for the Euler equations in gas dynamics and some recent developments for a rigorous mathematical theory on their nonlinear stability/instability. Then we review our recent study and present a supplement to the proof on the nonlinear stability of compressible vortex sheets under the magnetic effect in three-dimensional MHD. The compressible vortex sheets in three dimensions are unstable in the regime of pure gas dynamics. Our main concern is whether such vortex sheets can be nonlinearly stabilized under the magnetic fields. To achieve this, we first set up the current-vortex sheet problem as a free boundary problem; then we establish high-order energy estimates of the solutions to the linearized problem, which shows that the current-vortex sheets are linearly stable when the jump of the tangential velocity is dominated by the jump of the non-paralleled tangential magnetic fields; and finally we develop a suitable iteration scheme of the Nash-Moser-H¨ ormander type to obtain the existence and nonlinear stability of compressible current-vortex sheets, locally in time. Some further open problems and several related remarks are also presented.

1. Introduction We are concerned with entropy solutions of hyperbolic systems of conservation laws in several space variables: d ∑ ∂t U + ∂xj fj (U) = 0, (1.1) ⊤

j=1 m

where U = (U1 , · · · , Um ) and fj : R → R , j = 1, · · · , d, are nonlinear smooth functions. System (1.1) consists of m quasilinear hyperbolic equations in the d-space variables x = (x1 , · · · , xd ). m

Date: March 13, 2011. 1991 Mathematics Subject Classification. Primary: 35L65,35L60,35R35,76W05,76E25,35B35,76E19; Secondary: 76N10,35L67,35A30. Key words and phrases. Stability, existence, characteristic discontinuities, compressible vortex sheets, currentvortex sheets, entropy waves, multidimension, Euler equations, gas dynamics, MHD, Mach number, free boundary problem, linearized problem, decoupled formulation, Nash-Moser-H¨ ormander iteration, energy estimates, magnetic effect. 1

2

GUI-QIANG CHEN AND YAGUANG WANG

The prototypes of hyperbolic conservation laws include the Euler equations of gas dynamics and magnetohydrodynamics (MHD). Let the level set surface Γ := {Φ(t, x) = 0} of Φ(t, x) be a discontinuity of a piecewise smooth entropy solution: { U+ (t, x), Φ(t, x) < 0, U(t, x) = (1.2) − U (t, x), Φ(t, x) > 0, where U± (t, x) are smooth solutions of (1.1) in the respective domains separated by Γ. Then U± |Γ and Φ must satisfy the Rankine-Hugoniot jump conditions across Γ: ∂t Φ[U] +

d ∑

∂xj Φ[fj (U)] = 0,

(1.3)

j=1

where the bracket [·] stands for the jump of the associated function across Γ, that is, [U] = U+ |Γ − U− |Γ , with U± |Γ as the traces of U± taken on the respective sides of Γ. In general, there are two types of discontinuities in the entropy solutions of (1.1). The first type of discontinuities is called shock waves, across whose front the strict Lax entropy inequality holds for at least one convex entropy-entropy flux pair (η, q) = (η, q1 , · · · , qd ), ∇2 η(U) ≥ 0: ∂t Φ[η(U)] +

d ∑

∂xj Φ[qj (U)] > 0.

(1.4)

j=1

The second type of discontinuities is called characteristic discontinuities, which are characteristic surfaces of the hyperbolic system (1.1). That is, for this case, the function Φ(t, x) satisfies the eikonal equation on Γ = {Φ(t, x) = 0}: ∂t Φ + λ(U± ; ∇x Φ) = 0, (1.5) ∑d where λ(U; ξ) is an eigenvalue of the matrix j=1 ξj fj′ (U) for ξ = (ξ1 , · · · , ξd ). Usually, there are two different kinds of characteristic discontinuities: vortex sheets and entropy waves. In gas dynamics and MHD, across a vortex sheet, the tangential velocity field has a jump, while the normal velocity is continuous; across an entropy wave, the entropy has a jump while the velocity field is continuous. A vortex sheet or entropy wave front Γ is a part of the unknowns, which is a free boundary. This free boundary is a characteristic with respect to either side of Γ. Compressible vortex sheets and entropy waves, along with shock and rarefaction waves, are fundamental waves in the entropy solutions to multidimensional hyperbolic systems of conservation laws. They occur ubiquitously in nature including slip-stream interfaces, lifting of aircrafts, galactic jets, tornadoes, Mach configurations in the shock reflection-diffraction patterns, and interactions among nonlinear waves; see [1, 4, 6, 7, 8, 11, 12, 21, 22, 25, 33, 34, 38, 39] and the references cited therein. The stability of shock and rarefaction waves have been studied in Majda [31], Métivier [32], and Alinhac [2]. In this paper we discuss the stability issues for vortex sheets/entropy waves and present some recent developments and further open problems in this direction. In Section 2, we discuss vortex sheets and entropy waves for the Euler equations in gas dynamics. By mode analysis, it was observed in Miles [33] that the vortex sheets in two-dimensional isentropic √ gas dynamics are linearly stable when√the Mach number is larger than 2 and violently unstable when the Mach number is less than 2, while they are always unstable in three space variables no matter how large the Mach number is. A rigorous mathematical theory on the √ nonlinear stability of the two-dimensional vortex sheets with the Mach number larger than 2 locally in

CHARACTERISTIC DISCONTINUITIES AND FREE BOUNDARY PROBLEMS

3

time was obtained recently by Coulombel-Secchi [18, 19] when the initial data is in a class of small perturbation functions of a planar vortex sheet. In Sections 3–5, we review our recent study and present a supplement to the proof on the stability of three-dimensional compressible vortex sheets under the magnetic effect, that is, the nonlinear stability of current-vortex sheets in three-dimensional MHD in Chen-Wang [15]. As we mentioned above, the compressible vortex sheets in three dimensions are unstable in the regime of pure gas dynamics; Our main concern is whether such vortex sheets can be nonlinearly stabilized under the magnetic fields. In Section 3, we first set up the current-vortex sheet problem as a free boundary problem, and state the main results. In Section 4, we establish high-order energy estimates of the solutions to the linearized problem, which shows that the current-vortex sheets are linearly stable when the jump of the tangential velocity is dominated by the jump of the non-paralleled tangential magnetic fields in the sense that λ± determined in (3.21) satisfy condition (3.20), as observed in [35]. To achieve this, our key observation is that the linearized problem (4.1), equivalently (4.3), for current-vortex sheets is endowed with a well-structured decoupled formulation so that the linear problem is decoupled into one standard initial-boundary value problem (4.8) for a symmetric hyperbolic system and another problem (4.11) for an ordinary differential equation for the front. This decoupled formulation is essential for us to establish our desired high-order energy estimates of solutions, which is one of the key ingredients for developing our nonlinear approach for the stability problem. Also see Trakhinin [36] for a different approach to make related estimates. The energy estimates of the linearized problems have loss of regularities with respect to the nonhomogeneous terms and initial data, mainly due to that the front is characteristic in the current-vortex sheets. As in [2, 19], this has inspired us to develop a suitable iteration scheme of the Nash-Moser-Hörmander type to obtain the existence and structural stability of compressible current-vortex sheets, locally in time, in the three-dimensional MHD. This is done in Section 5. In Section 6, we present further open problems and several related remarks. 2. Characteristic Discontinuities for the Euler Equations in Gas Dynamics In this section we discuss vortex sheets and entropy waves for the Euler equations in gas dynamics. 2.1. Isentropic Euler Equations. The isentropic Euler equations in gas dynamics in Rd describing the motion of inviscid gases take the following form: { ∂t ρ + ∇ · (ρv) = 0, (2.1) ∂t (ρv) + ∇ · (ρv ⊗ v) + ∇p = 0, where ρ and v = (v1 , · · · , vd ) ∈ Rd are the density and velocity, respectively; the pressure p is a function of the density ρ: p = p(ρ) (2.2) with p′ (ρ) > 0 when ρ > 0. For a piecewise smooth weak solution U(t, x) of (2.1): { U+ (t, x) for Φ(t, x) < 0, U(t, x) = − U (t, x) for Φ(t, x) > 0 on the front Γ := {Φ(t, x) = 0}, the Rankine-Hugoniot conditions must be satisfied:   [mN ] = 0, mN [vN ] + |∇x Φ|2 [p] = 0,   mN [vτ ] = 0,

(2.3)

(2.4)

4


where vN := v · ∇x Φ and vτ ∈ Rd−1 are the normal and tangential components of v on Γ, and mN = ρ(vN − ψt ) is the mass transfer flux. Suppose that mN = 0 on Γ, i.e., no mass transfer flux across the front, so (U± , Γ) is a characteristic discontinuity for (2.1). Then, on Γ, [p] = [vN ] = 0.

(2.5)

In this case, there is only one kind of characteristic discontinuities, vortex sheets, since the tangential velocity field (with respect to the interface Γ) is the only quantity that experiences a jump across Γ: [vτ ] ̸= 0,

[p] = [vN ] = 0.

(2.6)

2.2. Full Euler Equations. The full Euler equations for gas dynamics in Rd take the following form:   ∂t ρ + ∇ · (ρv) = 0, (2.7) ∂t (ρv) + ∇ · (ρv ⊗ v) + ∇p = 0,  ) ( 1 )  (1 2 2 ∂t 2 ρ|v| + e + ∇ · ( 2 ρ|v| + ρe + p)v = 0, where p = p(ρ, S) and e = e(ρ, S) are the pressure and internal energy with the entropy S, respectively. Let a piecewise smooth function U(t, x): { U+ (t, x) for Φ(t, x) < 0, U(t, x) = (2.8) U− (t, x) for Φ(t, x) > 0 be a weak solution to (2.7). Then, on the front Γ := {Φ(t, x) = 0}, U(t, x) must satisfy the Rankine-Hugoniot conditions:  [mN ] = 0,    m [v ] + |∇ Φ|2 [p] = 0, N N x (2.9)  m [v ] = 0, N τ    mN [e + 12 |v|2 ] + [pvN ] = 0, where vN := v · ∇x Φ|Γ and vτ are the normal and tangential components of v on Γ, and mN = ρ(vN − ψt ) is the mass transfer flux. As above, we consider the case that mN = 0 on Γ, i.e., no mass transfer flux across the front; so (U± , Γ) is a characteristic discontinuity for (2.7). Then, on Γ, [p] = [vN ] = 0.

(2.10)

Different from the isentropic case, there are two different kinds of characteristic discontinuities on which [p] = [vN ] = mN = 0: (i) Vortex sheets: [vτ ] ̸= 0;

(2.11)

(ii) Entropy waves: [vτ ] = 0,

[ρ] ̸= 0,

[S] ̸= 0.

(2.12)


5

2.3. Stability of Vortex Sheets for the Two-Dimensional Euler Equations. Choose Φ(t, x1 , x2 ) = x1 − φ(t, x2 ). Then Γ = {x1 = φ(t, x2 ) : t > 0, x2 ∈ R}. The vortex sheet Γ satisfies that (i). The Euler equations (2.1) are satisfied on either side of Γ; (ii). The Rankine-Hugoniot jump relations are satisfied on Γ: ∂t φ = v+ · (1, −∂x2 φ) = v− · (1, −∂x2 φ),

ρ− = ρ+ .

As usual, ρ± , v± denote the trace of ρ, v taken on either side of Γ. The vortex sheet Γ is part of the unknowns, which is a free boundary. This free boundary is a characteristic with respect to either side of Γ. Consider a plane vortex sheet Γ0 with constant states on either side. Then, by the Galilean invariance of frame, such a vortex sheet can always be reformulated as the following form: U± = (¯ ρ, ±¯ u, 0),

±x1 > 0,

(2.13)

where ρ¯ > 0 is a fixed density, u ¯ > 0√is a fixed tangential velocity, while the normal velocity ¯ is ρ), and the relative Mach number M vanishes. The sonic speed on Γ0 is c¯ = p′ (¯ ¯ ¯ =u M . c¯ By mode analysis, it was observed by Miles in [33] that the vortex sheets in two-dimensional √ ¯ isentropic gas √ dynamics are linearly stable when the Mach number M > 2 and violently unstable ¯ < 2. when M A rigorous √ mathematical theory on the nonlinear stability of the two-dimensional vortex sheets ¯ > 2 locally in time was obtained recently by Coulombel-Secchi [18, 19] when the initial with M data function is in a class of small perturbation functions of a planar vortex sheet Γ0 . On the other hand, the seminal work by Artola-Majda [4] indicates that the stability of √ compressible ¯ vortex sheets depends on the class of initial perturbation functions, even when M > 2. For the two-dimensional full Euler equations, as indicated in §2.2, there is an additional type of characteristic discontinuities, called entropy waves. Across an entropy wave, the velocity and pressure are continuous, though the entropy, equivalently the density, has a jump. It would be interesting to analyze the stability of entropy waves to understand fundamental features of entropy solutions. For the Euler equations in three space-dimensions, every compressible vortex sheet is violently unstable, and this violent instability is the analogue of the Kelvin-Helmholtz instability for incompressible fluids (cf. Fejer-Miles [22]). In the next sections, Sections 3–5, we analyze whether compressible vortex sheets in three dimensions, which are unstable in the regime of pure gas dynamics, become stable under the magnetic effect in three-dimensional MHD. 3. Compressible Current-Vortex Sheets in MHD: Main Theorem The Euler equations for three-dimensional MHD describing the motion of inviscid MHD fluids take the following form:  ∂t ρ + ∇ · (ρv) = 0,     ∂t (ρv) + ∇ · (ρv ⊗ v − H ⊗ H) + ∇(p + 12 |H|2 ) = 0, (3.1)  ∂t H − ∇ × (v × H) = 0,   ) ( )  (1 ∂t 2 ρ|v|2 + ρe + |H|2 + ∇ · ( 12 ρ|v|2 + ρe + p)v + H × (v × H) = 0,

6


and ∇ · H = 0,

(3.2)

where ρ, v = (v1 , v2 , v3 ), H = (H1 , H2 , H3 ), and p = p(ρ, S) are the density, velocity, magnetic field, and pressure, respectively; e = e(ρ, S) is the internal energy; and S is the entropy. For smooth solutions, the equations in (3.1) are equivalent to  (∂t + v · ∇)p + ρc2 ∇ · v = 0,    ρ(∂ + v · ∇)v + ∇p − (∇ × H) × H = 0, t (3.3)  (∂ t + v · ∇)H − (H · ∇)v + H∇ · v = 0,    (∂t + v · ∇)S = 0, √ where c = pρ (ρ, S) is the sonic speed of the fluid. The equations in (3.3) can be written as a 8 × 8 symmetric hyperbolic system for U = (p, v, H, S)⊤ of the form: B0 (U)∂t U +

3 ∑

Bj (U)∂xj U = 0.

(3.4)

j=1

Let a piecewise smooth function U(t, x): { U+ (t, x) U(t, x) = U− (t, x)

for x1 > ψ(t, x2 , x3 ), for x1 < ψ(t, x2 , x3 )

(3.5)

be a weak solution to (3.1). Then, on the front Γ := {x1 = ψ(t, x2 , x3 )}, U(t, x) must satisfy the Rankine-Hugoniot conditions:  [mN ] = 0,      [HN ] = 0,    m [v ] + (1 + ψ 2 + ψ 2 )[q] = 0, N N x2 x3 (3.6)  m [v ] = H [H ], N τ N τ     mN [ Hρτ ] = HN [vτ ],    2  mN [e + 21 (|v|2 + |H| ρ )] + [qvN − HN (H · v)] = 0, where (vN , vτ ) (resp. (HN , Hτ )) are the normal and tangential components of v (resp. H) on Γ, i.e., vN := v1 − ψx2 v2 − ψx3 v3 ,

vτ = (vτ1 , vτ2 )⊤ := (ψx2 v1 + v2 , ψx3 v1 + v3 )⊤ ,

HN := H1 − ψx2 H2 − ψx3 H3 ,

Hτ = (Hτ1 , Hτ2 )⊤ := (ψx2 H1 + H2 , ψx3 H1 + H3 )⊤ ,

mN = ρ(vN − ψt ) is the mass transfer flux, and q = p + |H| 2 is the total pressure. As in §2.2, we focus on the case that mN = 0 on Γ, i.e., no mass transfer flux across the front, so (U± , Γ) is a characteristic discontinuity for (3.1). We now focus on the case, called current-vortex sheets: + − − HN = HN = 0, H+ (3.7) τ ̸∥ Hτ . 2

Then the Rankine-Hugoniot conditions are equivalent to + − ψt = vN = vN ,

and generically ([ρ], [vτ ], [S]) ̸= 0. First, we have

[p +

|H|2 ]=0 2

on Γ

(3.8)


7

Lemma 3.1. Let (U± , ψ) be a current-vortex sheet defined as above for 0 ≤ t < T . Then, if ± HN |Γ∩{t=0} = 0,

∇ · H± (0, x) = 0,

we have ± HN |Γ = 0,

∇ · H± (t, x) = 0,

for all t ∈ [0, T ).

(3.9)

± By a direct calculation, one knows that both HN and ∇ · H± satisfy a homogeneous transport equation tangential to Γ, so assertion (3.9) follows immediately if it holds initially. ± This lemma shows that both the divergence-free condition (3.2) and the condition HN |Γ = 0 are only the constraints on the initial data. Set   λ ⊤ ( ) 1 O1×3 ρc2 H ˜ D(λ, U) 0 ˜ D(λ, U) := with D(λ, U) :=  λρH I3 −ρλI3  . 0 1 O3×1 −λI3 I3

As in Trakhinin [35], we know from Lemma 3.1 that system (3.1)–(3.2) is equivalent to the following system on both sides of Γ: 3 ( ) ∑ D(λ± , U± ) B0 (U± )∂t U± + Bj (U± )∂xj U± + λ± G± ∇ · H± = 0,

(3.10)

j=1

provided ∇ · H± (0, x) = 0, where G± = −(1, 0, 0, 0, H± , 0)⊤ , and λ± = λ± (U+ , U− ) will be determined later. System (3.10) can be rewritten as the following symmetric form A0 (U± )∂t U± +

3 ∑

Aj (U± )∂xj U± = 0.

(3.11)

j=1

System (3.11) is still hyperbolic, provided that (λ± )2
0,  A0 (U )∂{t U + j=1 Aj (U )∂xj U = 0 + (3.13) U0 (x) for x1 > ψ0 (x2 , x3 ),  U|t=0 = − U0 (x) for x1 < ψ0 (x2 , x3 ) with the transmission conditions on Γ: + − ψt = vN = vN ,

[p +

|H|2 ] = 0, 2

(3.14)

± + − provided that HN |Γ∩{t=0} = 0, ∇ · H± 0 (x) = 0, and Hτ ̸∥ Hτ holds at t = 0, where ψ0 (x2 , x3 ) = ψ(0, x1 , x2 ). In the above problem, the front Γ is unknown. To deal with such a free boundary problem, it is convenient to use the following standard transformation: { t = t˜, x2 = x ˜2 , x3 = x ˜3 , (3.15) ± ˜ x1 = Ψ (t, x ˜1 , x ˜2 , x ˜3 )

8


with Ψ± satisfying

{ ±(Ψ± )x˜1 ≥ κ > 0, Ψ+ |x˜1 =0 = Ψ− |x˜1 =0 = ψ(t˜, x ˜2 , x ˜3 )

(3.16)

( ) for some constant κ > 0. Under (3.15), the domains Ω± := {± x1 − ψ(t, x2 , x3 ) > 0} are transformed into {˜ x1 > 0} and the free boundary Γ into the fixed boundary {˜ x1 = 0}. The natural candidates of Ψ± can be proper extensions of ψ(t˜, x ˜2 , x ˜3 ) in {˜ x1 > 0} satisfying the first non-degenerate condition (3.16)1 . With this in mind, we choose Ψ± to be the solutions to the following problem:  ∂t Ψ± − v1± + v2± ∂x2 Ψ± + v3± ∂x3 Ψ± = 0, t, x1 > 0 (3.17) ± Ψ± | t=0 = Ψ0 (x) := ±x1 + χ(±x1 )ψ0 (x2 , x3 ), where we drop the tildes in the formula for simplicity, χ(s) is a smooth cut-off function that is 1 for |s| ≤ 1 and 0 for |s| > 2 such that ±(Ψ± 0 )x1 ≥ κ > 0 in {x1 > 0}. Under transformation (3.15), it is easy to know that problem (3.13)–(3.14) is equivalent to that ˜ ± (t˜, x ˜ ) = U± (t, x) satisfy the following problem with a fixed boundary {x1 = 0}: U  ± ± ±  in {x1 > 0}, L(U , Ψ )U = 0 + − (3.18) B(U , U , ψ)|x1 =0 = 0,   ± ± (U , ψ)|t=0 = (U0 (x), ψ0 (x2 , x3 )), where the tildes have also been dropped, ¯ 1 (U, Ψ)∂x V + L(U, Ψ)V = A0 (U)∂t V + A 1

3 ∑

Aj (U)∂xj V

j=2

¯ 1 (U, Ψ) = with A

1 Ψx1

( ) ∑3 A1 (U ) − Ψt A0 (U ) − j=2 Ψxj Aj (U ) , and ( )⊤ ± B(U+ , U− , ψ) = ψt − Uv,N , q+ − q−

± with Uv,N = U2± − ψx2 U3± − ψx3 U4± and q = U1 + 21 |UH |2 for UH = (U5 , U6 , U7 )⊤ , under the constraints that  ± ± ± U± on {x1 = 0}, H,N = U5 − ψx2 U6 − ψx3 U7 = 0

∇ e · H± := ∂x U ± + (∂x Ψ± ∂x − ∂x Ψ± ∂x )U ± + (∂x Ψ± ∂x − ∂x Ψ± ∂x )U ± = 0 1 1 2 2 1 1 3 3 1 5 6 7

in {x1 > 0} (3.19)

hold at {t = 0}. The main feature of problem (3.18) is that the fixed boundary {x1 = 0} is characteristic of constant multiplicity. To solve (3.18), as in [2, 16, 26], it is natural to introduce the weighted anisotropic Sobolev spaces defined on ΩT := {(t, x) ∈ [0, T ] × R3 : x1 > 0}: Bµs (ΩT ) := {u ∈ L2 (ΩT ) : e−µt M α ∂xk1 u ∈ L2 (ΩT ) for |α| + 2k ≤ s} for all s ∈ IN and µ > 0, where the tangential vectors M = (M0 , M1 , M2 , M3 ) of {x1 = 0} are given by M0 = ∂t , M1 = σ(x1 )∂x1 , M2 = ∂x2 , M3 = ∂x3 , with

{ x1 σ(x1 ) := 2

for 0 ≤ x1 ≤ 1, for x1 ≥ 2.


9

The norms in Bµs (ΩT ) are as usual: (∫

)1/2

T

∥u(t, ·)∥2s,µ dt

∥u∥s,µ,T :=

,

0

with ∥u(t, ·)∥2s,µ :=

∑

µ2(s−|α|−2k) ∥e−µt M α ∂xk1 u(t, ·)∥2L2 .

|α|+2k≤s

We will also use a similar notation for the spaces with µ = 0, B s (ΩT ), with norm: ( ∑ )1/2 ∥u∥s,T := ∥M α ∂xk1 u∥2L2 (ΩT ) . |α|+2k≤s

Also denote by bΩT := {(t, x2 , x3 ) : t ∈ [0, T ], (x2 , x3 ) ∈ R2 }, and |u|s,T the norm of u in H s (bΩT ). ˆ ± , Sˆ± ) and ψ0 that are a small perturbation ˆ±, H Consider the initial data functions U± ρ± , v 0 = (ˆ ± ¯ ¯ ¯ ± and ψ¯ = 0 with (3.7)–(3.8) so that of a planar current-vortex sheet (U , ψ) for constant states U the following stability condition holds: ( ± )2 1 ˆ < λ . (3.20) ˆ ± |2 /(ˆ ρˆ± + |H c± )2 ˆ ± uniquely determined by for λ ( + )( ˆ −H ˆ− H 2 2 ˆ+ − H ˆ− H 3 3

ˆ+ λ ˆ λ−

)

( =

vˆ2+ − vˆ2− vˆ3+ − vˆ3−

) on {x1 = 0},

(3.21)

Then we have the following main result. Theorem 3.2 (Chen-Wang [15]). Assume that, for any fixed α ≥ 15 and s ∈ [α + 7, 2α − 5], the 2(s+2) ¯± initial data functions ψ0 ∈ H 2s+3 (R2 ) and U± (R3+ ) satisfy constraints (3.19), the 0 −U ∈ B compatibility conditions of problem (3.18) up to order s + 2, and the stability condition (3.20)– (3.21). Then there exists a solution (U± , ψ) of the initial-boundary value problem (3.18) such that ¯ ± ∈ B α (ΩT ) and ψ ∈ H α−1 (bΩT ). U± − U Remark 3.1. The stability conditions (3.20) and (3.21) for the initial data functions U± 0 = ˆ ± , Sˆ± ) and ψ0 are equivalent to ˆ±, H (ˆ ρ± , v √ + + + + ˆ + (ˆ ˆ − (ˆ ˆ + (ˆ ˆ ± |2 /(ˆ ˆ − (ˆ ρˆ± + |H c± )2 max{|H ˆ3− ) − H ˆ2− )|, |H ˆ3− ) − H ˆ2− )|} 2 v3 − v 3 v2 − v 2 v3 − v 3 v2 − v ˆ +H ˆ− − H ˆ −H ˆ + |. ≤ |H 2 3 2 3

(3.22)

Also see [35] for another equivalent form. Remark 3.2. Using the same argument as in Coulombel-Secchi [20], we conclude that the above current-vortex sheet solution to system (3.1) is also uniquely determined by its initial data. In Chen-Wang [15], some parts of the presentation of the proof were not described clearly enough. In Sections 4–5, we provide a supplement and describe the complete arguments of the proof of Theorem 3.2.

10


4. Compressible Current-Vortex Sheets in MHD: Linear Stability To study the linear stability of current-vortex sheets, we first derive a linearized problem from the nonlinear problem (3.18). By a direct calculation, we have ) d( Φ (L(U, Ψ)U)x1 , L(U + sV, Ψ + sΦ)(U + sV) |s=0 = L(U, Ψ)W + E(U, Ψ)W + ds Ψx1 where W = V −

Φ Ψx1

Ux1 is the good unknown as introduced in [2] (see also [23, 32]), and

¯ 1 (U, Ψ))Ux + E(U, Ψ)W = W · ∇U (A 1

3 ∑

W · ∇U Aj (U)Uxj .

j=2

Then we obtain the following linearized problem of (3.18):  L(U± , Ψ± )W± + E(U± , Ψ± )W± = F± in {x1 > 0},       ϕt − (W2± − ψx2 W3± − ψx3 W4± ) + U3± ϕx2 + U4± ϕx3 = h± on {x1 = 0}, 1 ∑ 7   W1+ − W1− + j=5 (Uj+ Wj+ − Uj− Wj− ) = h2 on {x1 = 0},      (W± , ϕ)|t=0 = 0,

(4.1)

± for some functions F± , h± 1 , and h2 , where Ψ (t, x) are proper extensions of ψ(t, x2 , x3 ) in {x1 > 0} satisfying (3.16). To simplify problem (4.1), we introduce J± = J(U± , Ψ± ) as an 8 × 8 regular matrix such that

X± = (J± )−1 W± satisfy

(4.2)

 ± ∑7 X1 = W1± + j=5 Uj± Wj± ,    X ± = W ± − (Ψ± ) W ± − (Ψ± ) W ± , x2 3 x3 4 2 2 ± ± ± ± ±  X5 = W5 − (Ψ )x2 W6 − (Ψ )x3 W7± ,    ± ± ± ± ± (X3 , X4 , X6 , X7 , X8 ) = (W3± , W4± , W6± , W7± , W8± ),

which means that X1 , X2 , X5 , and X8 represent the linearized total pressure, normal velocity, normal magnetic field, and the entropy respectively, while (X3 , X4 ) and (X6 , X7 ) are the associated tangential velocity and magnetic fields. Under transformation (4.2), problem (4.1) for (W± , ϕ) is equivalent to the following problem for (X± , ϕ):  ˜ ± , Ψ± )X± + E(U ˜ ± , Ψ± )X± = F ˜± L(U in {x1 > 0},       ϕt − X2± + U3± ϕx2 + U4± ϕx3 = h± on {x1 = 0}, 1 (4.3)   X1+ − X1− = h2 on {x1 = 0},      ± (X , ϕ)|t=0 = 0, ˜ ± = (J± )⊤ F± , where F ˜ ± , Ψ± ) = A ˜ 0 (U± , Ψ± )∂t + L(U

3 ∑ j=1

˜ j (U± , Ψ± )∂x A j


11

with ˜ 1 (U± , Ψ± ) = (J± )⊤ A ¯ 1 (U± , Ψ± )J± , A ˜ j (U± , Ψ± ) = (J± )⊤ Aj (U± )J± , A j ̸= 1, ( ± ⊤ ) ( ) ± ± ± ± ± ± ˜ E(U , Ψ )X = (J ) E(U , Ψ )J X± + (J± )⊤ L(U± , Ψ± )J± X± . ˜ 1 (U± , Ψ± ) in L(U ˜ ± , Ψ± ) can be By a direct calculation, we see that the coefficient matrix A decomposed into three parts: ˜ 1 (U± , Ψ± ) = A±,0 + A±,1 + A±,2 A 1 1 1 with A±,0 = 1

[

1 (Ψ± )x1

0 a⊤

]

a O7×7

,

A±,1 = 1

± Ψ± t − Uv,N

(Ψ± )x1

˜ ±,1 , A 1

A±,2 = 1

± UH,N

(Ψ± )x1

˜ ±,2 , A 1

where a = (1, 0, 0, −λ± , 0, 0, 0), ± Uv,N = U2± − (Ψ± )x2 U3± − (Ψ± )x3 U4± ,

and ± UH,N = U5± − (Ψ± )x2 U6± − (Ψ± )x3 U7± .

When the states (U± , Ψ± ) satisfy the boundary conditions given in (3.18), i.e., ± ψt − Uv,N = 0,

± UH,N =0

on {x1 = 0},

˜ ± , Ψ± ). the boundary {x1 = 0} is a characteristic plane of constant multiplicity for the operator L(U Then, from (4.3), we obtain that, on {x1 = 0}, ( )( + ) ( + ) A1 (U+ , Ψ+ ) 0 X X ⟨ , ⟩ = 2X1± [X2 − λX5 ], (4.4) 0 A1 (U − , Ψ− ) X− X− when [X1 ] = 0 on {x1 = 0}. In order to decouple the front unknown ϕ from the boundary condition (4.3)2 , we use the ± linearization of the constraints HN |x1 =0 = 0: X5± − U6± ϕx2 − U7± ϕx3 = h± 3

on {x1 = 0}

(4.5)

to obtain [X2 − λX5 ] = ϕx2 [U3 − λU6 ] + ϕx3 [U4 − λU7 ] − [h1 + λh3 ]. H− τ

±

±

(v2± , v3± , H2± , H3± )

From the assumption ∦ on {x1 = 0}, there exist unique λ = λ that ( + ) ( − ) ( + ) ( − ) v2 v2 H2 H2 + − − = λ − λ on {x1 = 0}, H3− v3+ v3− H3+ H+ τ

(4.6) such (4.7)

that is, [U3 − λU6 ] = [U4 − λU7 ] = 0. In this case, (4.6) is simplified as [X2 − λX5 ] = −[h1 + λh3 ]. Therefore, with the aid of (4.5) and the choice of λ± in (4.7), we deduce from (4.3) that X± satisfy the following problem:  ˜ ± , Ψ± )X± + E(U ˜ ± , Ψ± )X± = F ˜± L(U in {x1 > 0},     [X2 − λX5 ] = −[h1 + λh3 ] on {x1 = 0}, (4.8) + −  X − X = h on {x1 = 0}, 2  1 1   ± X |t≤0 = 0.

12


From the discussion given in (4.4), we know that the linear problem (4.8) is maximally dissipative in the sense of Lax-Friedrichs [24]. Thus, by employing the Lax-Friedrichs theory [24] for (4.8), we conclude the following energy estimates: Theorem 4.1. For any fixed s0 > 17/2, there exist constants C0 and µ0 depending only on ˙ ∥s ,T for the coefficient functions in (4.8) such that, for any s ≥ s0 and µ ≥ µ0 , the estimate: ∥coef 0 max ∥X± (t)∥2s,µ + µ∥X± ∥2s,µ,T

0≤t≤T

≤

) C0 ( ± ± 2 2 ˙ ∥2 (∥F + |h| ) ∥F ∥s,µ,T + ∥h∥2H s+1 (bΩ ) + ∥coef ∥ s,µ,T s0 ,T s0 +1,T T µ µ

(4.9)

holds, provided that the eikonal equations: ψt = U2± − ψx2 U3± − ψx3 U4± and the constraints: ± UH,N := U5± − ψx±2 U6± − ψx3 U7± = 0

are valid for (U± , ψ) on {x1 = 0}, and λ± determined in (4.7) satisfy condition (3.12), where h = ± ⊤ s s (h± 1 , h2 , h3 ) and the norms in Hµ (bΩT ) are defined as that of Bµ (ΩT ) with functions independent ˙ (t, x) = coef (t, x) − coef (0) with coef (t, x) being the coefficient functions appeared of x1 , and coef in the equations in (4.8). By fixing µ ≫ 1 in (4.9), we conclude Corollary 4.2. For any fixed s0 > 17/2, there exists a constant C0 > 0 depending only on ˙ ∥s ,T and T such that, for any s ≥ s0 , the following estimate holds: ∥coef 0 ( ) ˙ ∥2 (∥F± ∥2 + |h|2 ∥X± ∥2s,T ≤ C0 ∥F± ∥2s,T + |h|2s+1,T + ∥coef ) . (4.10) s,T s0 ,T s0 +1,T Finally, let us study the determination of the perturbation Φ± of the front functions Ψ± . From problem (4.1), the natural idea is to solve the following problems: { ∂t Φ± − X2± + U3± ∂x2 Φ± + U4± ∂x3 Φ± = h± in {x1 > 0}, 1 (4.11) Φ± |t=0 = 0. An important question is whether we have Φ+ = Φ− on {x1 = 0}. This question is answered by the following result. Proposition 4.3. Let Φ+ (t, x) be given by problem (4.11) with the plus sign, and ϕ(t, x2 , x3 ) = Φ+ |x1 =0 . If λ± are given in (4.7), and the boundary condition: [X2 − λX5 ] = −[h1 + λh3 ]

on {x1 = 0}

(4.12)

holds as in (4.8), with h± 3 being given by X5± − U6± ϕx2 − U7± ϕx3 = h± 3

on {x1 = 0},

(4.13)

then we have ∂t ϕ − X2− + U3− ∂x2 ϕ + U4− ∂x3 ϕ = h− 1

on {x1 = 0}.

(4.14)

Proof: Notice that (4.12) and (4.13) can be rewritten as − − + + − − + − X2+ + h+ 1 − (X2 + h1 ) = λ (X5 − h3 ) − λ (X5 − h3 ),

(4.15)


and X5± − h± 3 = (∂x2 ϕ, ∂x3 ϕ)

(

U6± U7±

13

) .

(4.16)

Thus, from (4.11) with the plus sign, we obtain ( + ) [ ( + ) ( − )] U3 U6 U6 − − + − ∂t ϕ + (∂x2 ϕ, ∂x3 ϕ) − (X2 + h1 ) = (∂x2 ϕ, ∂x3 ϕ) λ −λ U4+ U7+ U7− [( + ) ( − )] U3 U3 = (∂x2 ϕ, ∂x3 ϕ) − + U4 U4− (4.17) by using (4.15)–(4.16) and (4.7). From (4.17), we immediately conclude (4.14).

5. Compressible Current-Vortex Sheets in MHD: Nonlinear Stability In this section, we describe the main steps to prove Theorem 3.2, the existence of a local solution to the nonlinear problem (3.18) under constraints (3.19) for the initial data, by developing an iteration scheme of the Nash-Moser-Hörmander type. 5.1. Construction of The Zero-th Order Approximate Solutions. Suppose that the initial ¯± ¯ data (U± 0 , ψ0 ) is a perturbation of a planar current-vortex sheet (U , ψ) with the constant states ± ± s−1 2 ¯ ¯ ˙ ¯ ± ∈ B s (R3 ) for any U and ψ = 0 satisfying (3.7)–(3.8), ψ0 ∈ H (R ), and U0 = U± + 0 −U ± fixed integer s > 9/2. Suppose that (U0 , ψ0 ) satisfy the compatibility conditions of problem (3.18) up to order [ 2s ], and the constraints in (3.19) with Ψ± (0, x) being proper extensions of ψ0 (x2 , x3 ) in {x1 > 0}, satisfying ±∂x1 Ψ± |t=0 ≥ κ > 0. In a classical way, one can construct the ± ± [s/2]+1 ˙± ¯± zero-th order approximate solutions (U± (R+ × R3+ ), a , Ψa ) such that Ua = Ua − U ∈ B [s/2]+2 (R+ × R3+ ) with ±∂x1 Ψ± |t=0 ≥ κ/2 > 0 satisfying Ψ± a ∓ x1 ∈ B ( ) s ± ± ∂tj L(U± for 0 ≤ j ≤ [ ] − 1 (5.1) a , Ψa )Ua ) |t=0 = 0 2 and − B(U+ a , Ua , ψa ) = 0,

± ± ± Ua,5 − (ψa )x2 Ua,6 − (ψa )x3 Ua,7 =0

on {x1 = 0}

(5.2)

with Ψ± a |x1 =0 = ψa (t, x2 , x3 ). Set V± = U± − U± a,

Φ± = Ψ± − Ψ± a.

(5.3)

Then it follows from (5.1) and (5.2) that problem (3.18) is equivalent to the following problem for (V± , Φ± ):  ± ± ± ±  in {t > 0, x1 > 0}, L(V , Φ )V = fa + − (5.4) B(V , V , ϕ) = 0 on {x1 = 0},   ± V |t≤0 = 0, ϕ|t≤0 = 0, ± ± where ϕ(t, x2 , x3 ) = Φ± |x1 =0 and fa± = −L(U± a , Ψa )Ua , ± ± ± ± ± ± ± ± L(V± , Φ± )V± = L(U± a + V , Ψa + Φ )(Ua + V ) − L(Ua , Ψa )Ua ,

and + − − B(V+ , V− , ϕ) = B(U+ a + V , Ua + V , ψa + ϕ).

14


5.2. Iteration Scheme. From the linear stability estimate established in Theorem 4.1, we observe that there exists a loss of regularity for the linearized problem (4.9). This inspires us to use a suitable iteration scheme of the Nash-Moser-Hörmander type (cf. [28]) to study the nonlinear problem (5.4). To do this, we first recall a standard family of smoothing operators (cf. [2, 19]): {Sθ }θ>0 : Bµ0 (ΩT ) −→ ∩s≥0 Bµs (ΩT ) satisfying

and

 (s−α)+  ∥u∥α,T ∥Sθ u∥s,T ≤ Cθ s−α ∥Sθ u − u∥s,T ≤ Cθ ∥u∥α,T   d ∥ dθ Sθ u∥s,T ≤ Cθs−α−1 ∥u∥α,T (Sθ u+ − Sθ u− )|x1 =0

s,T

(5.5)

for all s, α ≥ 0, for all s ∈ [0, α], for all s, α ≥ 0,

≤ Cθ(s+1−α)+ (u+ − u− )|x1 =0 α,T

(5.6)

for all s, α ≥ 0.

(5.7)

Similarly, one has a family of smoothing operators (still denoted by) {Sθ }θ>0 acting on H (bΩT ), satisfying also (5.6) for the norms of H s (bΩT ) (cf. [2, 19]). s

Now we construct the iteration scheme for solving the nonlinear problem (5.4) in R+ × R3+ . Let V±,0 = 0 and Φ±,0 = 0. Assume that (V±,k , Φ±,k ) have been known for k = 0, . . . , n, and satisfy Φ+,k = Φ−,k on {x1 = 0}, (V±,k , Φ±,k ) = 0

in {t ≤ 0}.

(5.8)

Denote the (n + 1)th approximate solutions to (5.4) in R+ × R3+ by V±,n+1 = V±,n + δV±,n , Φ±,n+1 = Φ±,n + δΦ±,n . (5.9) √ 2 Let θ0 ≥ 1 and θn = θ0 + n for any n ≥ 1. Let Sθn be the associated smoothing operator defined as above. Denote by ˙ ±,n L′ ± ±,n+ 1 ± δV 2 ,Ψa +Sθ Φ±,n ) e,(Ua +V n (5.10) ±,n+ 12 ± ±,n ˙ ±,n + E(U± + V±,n+ 12 , Ψ± + Sθ Φ±,n )δ V ˙ ±,n = L(U± + V , Ψ + S Φ )δ V θ a a a a n n the effective linearized operator, and ± ±,n+ 2 )x1 ˙ ±,n = δV±,n − δΦ±,n (Ua + V δV ± ±,n (Ψa + Sθn Φ )x1 1

(5.11)

the good unknown. By a direct computation, we have L(V

±,n+1

,Φ

±,n+1

)V

±,n+1

n ∑ ( ′ = L j=0

1

±,j+ ±,j ) 2 ,Ψ± e,(U± a +V a +Sθj Φ

) ˙ ±,j + e±,j , δV

(5.12)

where the modified states V±,j+ 2 will be chosen such that the boundary {x1 = 0} is uniformly characteristic of constant multiplicity for the operator L′ ± ±,j+ 1 ± for all j ≥ 0, ±,j 1

e,(Ua +V

and e±,j =

4 ∑

(k)

e±,j

2

,Ψa +Sθj Φ

)

(5.13)

k=1

with ±,j+1 ±,j+1 ±,j+1 ±,j ±,j ±,j e±,j =L(U± , Ψ± )(U± ) − L(U± , Ψ± )(U± ) a +V a +Φ a +V a +V a +Φ a +V (1)

− L′(U± +V±,j ,Ψ± +Φ±,j ) (δV±,j , δΦ±,j ), a

a

(5.14)


e±,j = L′(U± +V±,j ,Ψ± +Φ±,j ) (δV±,j , δΦ±,j ) − L′(U± +S (2)

a

e±,j = L′(U± +S (3)

a

a

a

θj V

±,j ,Ψ± +S ±,j ) θj Φ a

θj V

±,j ) ±,j ,Ψ± +S θj Φ a

(δV±,j , δΦ±,j ) − L′

(δV±,j , δΦ±,j ),

1

±,j+ ±,j ) 2 ,Ψ± (U± a +V a +Sθj Φ

15

(5.15)

(δV±,j , δΦ±,j ), (5.16)

and (4)

e±,j =

(Ψ± a

) ( δΦ±,j ±,j ±,j+ 21 ±,j+ 12 , Ψ± )(U± ) x1 . L(U± a + Sθ j Φ a +V a +V ±,j + Sθj Φ )x1

(5.17)

For the boundary condition given in (5.4), we have B(V± , Φ± ) = (B1+ (V+ , Φ+ ), B1− (V− , Φ− ), B2 (V+ , V− ))⊤ , with  ± ± ± ± ± ± ± B1± (V± , Φ± ) = (∂t + Ua,3 ∂x2 + Ua,4 ∂x3 )Φ± − V2± + (Ψ± a + Φ )x2 V3 + (Ψa + Φ )x3 V4 , B (V+ , V− ) = V + − V − + 1 (|V+ |2 − |V− |2 ) + ⟨U+ , V+ ⟩ − ⟨U− , V− ⟩, 2 1 1 H H H a,H a,H H 2 (5.18)

± ± ± ⊤ ± ± ± ± ⊤ and U± a,H = (Ua,5 , Ua,6 , Ua,7 ) and VH = (V5 , V6 , V7 ) . ± Associated with the constraints HN |x1 =0 = 0, denote by

± ± ± ± ± ± ± ± ± B3± (V± , Φ± ) = V5± − (Ψ± a + Φ )x2 V6 − (Ψa + Φ )x3 V7 − (Φ )x2 Ua,6 − (Φ )x3 Ua,7 .

By a direct calculation, for i = 1, 3, we have Bi± (V±,n+1 , Φ±,n+1 ) − Bi± (V±,n , Φ±,n ) = B ±′

1

i,(V±,n+ 2 ,Sθn Φ±,n )

± ˙ ±,n , δΦ±,n ) + eg (δ V i,n ,

(5.19)

where ± eg i,n =

4 ∑ g e±,k i,n

(5.20)

k=1

with the errors:  g ±,1 ± ±′  ±,n+1 ±,n  , Φ±,n+1 ) − Bi± (V±,n , Φ±,n ) − Bi,(V , δΦ±,n ) ±,n ,Φ±,n ) (δV ei,n = Bi (V       (δΦ±,n )x2 δV3±,n + (δΦ±,n )x3 δV4±,n ,  i = 1,    =   −(δΦ±,n ) δV ±,n − (δΦ±,n ) δV ±,n ,  i = 3,  x2 x3 6 7     g ±,2 ±,n ei,n = B ±′ ±,n ±,n (δV±,n , δΦ±,n ) − B ±′ , δΦ±,n ) i,(V ,Φ ) i,(Sθn V±,n ,Sθn Φ±,n ) (δV  ( ) ( )   (I − Sθn )Φ±,n x δV3±,n + (I − Sθn )Φ±,n x δV4±,n     2 3        ±,n ±,n ±,n    +(δΦ )x2 (I − Sθn )V3 + (δΦ )x3 (I − Sθn )V4±,n , i = 1,    =  ( ) ( )  ±,n ±,n − (I − S )Φ±,n δV   − (I − Sθn )Φ±,n x3 δV7  θn  6  x2         −(δΦ±,n ) (I − S )V ±,n − (δΦ±,n ) (I − S )V ±,n , i = 3, x2

θn

6

x3

θn

7

(5.21)

16


and  g ±,3 ±′ ±′ ±,n ±,n ±,n ±,n   ei,n = Bi,(Sθn V±,n ,Sθn Φ±,n ) (δV , δΦ ) − Bi,(V±,n+ 12 ,Sθ Φ±,n ) (δV , δΦ )  n    1 1   (Sθn V3±,n − V3±,n+ 2 )(δΦ±,n )x2 + (Sθn V4±,n − V4±,n+ 2 )(δΦ±,n )x3 ,  i = 1,    =   ±,n+ 12  ±,n+ 21  (V6 − Sθn V6±,n )(δΦ±,n )x2 + (V7 − Sθn V7±,n )(δΦ±,n )x3 , i = 3, g ±,4 ±′ ±′ ±,n ±,n ±,n ±,n  ˙  ei,n = B (δV , δΦ ) − B (δ V , δΦ ) 1 1   i,(V±,n+ 2 ,Sθn Φ±,n ) i,(V±,n+ 2 ,Sθn Φ±,n )    ) ( ±,n 1    ± δΦ ±,n ET ∂x1 B1± (V±,n+ 2 , Sθn ϕn ) , i = 1,   (Ψa +Sθn Φ )x1   =  ) ( ±,n 1   ± δΦ  i = 3, ET ∂x1 B3± (V±,n+ 2 , Sθn ϕn ) , ±,n (Ψa +Sθn Φ

)x1

(5.22) for ET (·) being a proper bounded extension from H s (bΩT ) to B s+1 (ΩT ). Using (5.19) and noting that Bi± (V±,0 , Φ±,0 ) = 0, we obtain Bi± (V±,n+1 , Φ±,n+1 ) =

n ∑ (

B±′

1 i,(V±,j+ 2

j=0

,Sθj

) ± ˙ ±,j , δΦ±,j ) + ef (δ V i,j

(5.23)

) ˙ +,j , δ V ˙ −,j ) + eg (δ V 2,j ,

(5.24)

Φ±,j )

for i = 1, 3. Similarly, one has B2 (V

+,n+1

,V

−,n+1

)=

n ∑ ( j=0

B′

1

1

2,(V+,j+ 2 ,V−,j+ 2 )

f ± ± where the errors eg 2,j can be defined as that of ei,j in (5.20)–(5.22) with Bi being replaced by B2 . Observe that, if the limit of (V±,n , Φ±,n ) exists which is expected to be a solution, then the left-hand sides of equations (5.12) and (5.23)–(5.24) should tend to fa± and zero respectively when n → ∞. Thus, with respect to the well-posed boundary condition form of the linear problem (4.8), ˙ ±,n of the approximate solutions to be the solutions to the following we define the increments δ V problem:  ′ ˙ ±,n = f ± , δV L  1 n  +Sθn Φ±,n ) +V±,n+ 2 ,Ψ± e,(U±  a a   ˙ +,n , δ V ˙ −,n ) = h+ − h− + λ+ (U± + V±,n+ 21 )h+ − λ− (U± + V±,n+ 21 )h− B1 (δ V a a 1,n 1,n 3,n 3,n     B ′ ˙ +,n , δ V ˙ −,n ) = gñ (δ V on bΩT , +,n+ 1 −,n+ 1 2,(V

2

,V

2

on bΩT ,

)

(5.25) where ˙ +,n , δ V ˙ −,n ) B1 (δ V

( +,n ) ±,n+ 12 +,n +,n ˙ = λ+ (U± ) δV − (Ψ+ )x2 δ V˙ 6+,n − (Ψ+ )x3 δ V˙ 7+,n a +V a + Sθ n Φ a + Sθ n Φ 5 ( ) +,n +,n − δ V˙ 2+,n − (Ψ+ )x2 δ V˙ 3+,n − (Ψ+ )x3 δ V˙ 4+,n a + Sθn Φ a + Sθn Φ ( ) −,n −,n ±,n+ 21 ) δ V˙ 5−,n − (Ψ− )x2 δ V˙ 6−,n − (Ψ− )x3 δ V˙ 7−,n −λ− (U± a + Sθ n Φ a + Sθn Φ a +V ( ) −,n −,n + δ V˙ 2−,n − (Ψ− )x2 δ V˙ 3−,n − (Ψ− )x3 δ V˙ 4−,n a + Sθn Φ a + Sθ n Φ


17

± with λ± (·) being defined in (4.7), fn± , gñ , h± 1,n , and h3,n are defined by

 n ( n−1 ) ( n−1 ) n ∑ ± ∑ ∑ ∑   fj + Sθn e±,j = Sθn fa± , g˜j + Sθn eg 2,j = 0,  j=0 j=0 j=0 j=0 ( n−1 ) ( n−1 ) n n ∑ ∑ g ∑ ∑ g  ± ± ±   h1,j + Sθn e1,j = 0, h3,j + Sθn e± 3,j = 0, j=0

j=0

j=0

(5.26)

j=0

± by induction on n, with f0± = Sθ0 fa± and g˜0 = h± 1,0 = h3,0 = 0. ± To construct δΦ±,n from B1 (V± , Φ± ) defined in (5.18), we clearly have ±′ ± ± ± ± ± B1,(V + (Ua,3 + V3± )∂x2 Θ± + (Ua,4 + V4± )∂x3 Θ± ± ,Φ± ) (W , Θ ) =∂t Θ ± ± ± ± ± − W2± + (Ψ± a + Φ )x2 W3 + (Ψa + Φ )x3 W4 .

From (5.23) with i = 1, we first define δΦ+,n by the following problem: { +′ ˙ +,n , δΦ+,n ) = h+ B (δ V in ΩT , 1,n +,n+ 1 +,n 1,(V

2

,Sθn Φ

)

δΦ+,n |t≤0 = 0,

(5.27)

(5.28)

where h+ 1,n is given in (5.26). Denote by ± +,n +,n ˙ +,n − (Ψ+ hg )x2 δ V˙ 6+,n − (Ψ+ )x3 δ V˙ 7+,n a + Sθ n Φ a + Sθn Φ 3,n =δ V5 ) ( ±,n+ 12 ± Ua,6 + V6 +,n +,n , − ((δΦ )x2 , (δΦ )x3 ) ±,n+ 12 ± Ua,7 + V7

(5.29)

and − + + ± ±,n+ 12 g − ± ±,n+ 21 g ˙ +,n , δ V ˙ −,n ). hg ) h+ ) h− 1,n = h1,n + λ (Ua + V 3,n − λ (Ua + V 3,n − B1 (δ V

Then we determine δΦ−,n by solving the following problem:  − B −′ ˙ −,n , δΦ−,n ) = hg (δ V 1 1,n 1,(V−,n+ 2 ,Sθn Φ−,n ) δΦ−,n | = 0.

in ΩT ,

(5.30)

t≤0

By employing Proposition 4.3 for problems (5.28) and (5.30), we obtain δΦ+,n = δΦ−,n

on {x1 = 0}.

In order to keep the boundary {x1 = 0} being uniformly characteristic of constant multiplicity 1 at each iteration step (5.25), we define the modified state V±,n+ 2 by requiring  ±,n+ 12 ± ±  (∂t + Ua,3 ∂x2 + Ua,4 ∂x3 )(Sθn Φ±,n ) − V2      ±,n+ 12 ±,n+ 12  ±,n ±,n  +(Ψ± ) x2 V 3 + (Ψ± )x3 V4 = 0, a + Sθn Φ a + Sθ n Φ (5.31)  ±,n+ 12 ±,n+ 12 ±,n+ 21   V5 − (Ψ± + Sθn Φ±,n )x2 V6 − (Ψ± + Sθn Φ±,n )x3 V7 a a      ± ± −(Sθn Φ±,n )x2 Ua,6 − (Sθn Φ±,n )x3 Ua,7 =0 on {x1 = 0}, which leads to define ±,n+ 12

Vj

= Sθn Vj±,n

for j ̸= 2, 5,

(5.32)

18


 ±,n+ 21 ±,n ±,n  V2 = (Ψ± )x2 Sθn V3±,n + (Ψ± )x3 Sθn V4±,n  a + Sθn Φ a + Sθ n Φ    ± ± ±,n  +(∂t + Ua,3 ∂x2 + Ua,4 ∂x3 )(Sθn Φ ),

and

(5.33) ±,n+ 21  ±,n ±,n ± ±,n ± ±,n  V = (Ψ + S Φ ) S V + (Ψ + S Φ ) S V  θ x θ θ x θ a a n 2 n n 3 n 5 6 7    ± ± +Ua,6 (Sθn Φ±,n )x2 + Ua,7 (Sθn Φ±,n )x3 . ˙ ±,n from (5.25) and then δΦ±,n The steps for determining (δV±,n , δΦ±,n ) are to solve first δ V ±,n from (5.28) and (5.30), and to obtain δV finally from (5.11). 5.3. Convergence of the Iteration Scheme. Fix any s0 ≥ 9, α ≥ s0 + 6, and s1 ∈ [α + 7, 2α + 4 − s0 ]. Let the zero-th order approximate solutions for the initial data (U± 0 , ψ0 ) constructed in §5.1 satisfy ˙ ± ∥s +3,T + ∥Ψ ˙ ± ∥s +3,T + ∥f ± ∥s −4,T ≤ ε, ∥U ∥f ± ∥α+3,T /ε is small, (5.34) a

a

1

a

1

1

a

˙± for some small constant ε > 0, with Ψ a = χ(±x1 )ψa (t, x2 , x3 ). The key estimates for proving the convergence of the iteration scheme are as follows: Proposition 5.1. For the solution sequence (δV±n , δΦ±,n ) given by (5.25) and (5.28)–(5.30), we have  ∥δV±,n ∥s,T + ∥δΦ±,n ∥s,T ≤ εθns−α−2 ∆n for s ∈ [s0 , s1 ],    ∥L(V±,n , Φ±,n )V±,n − f ± ∥ s−α−3 for s ∈ [s0 , s1 − 4], a s,T ≤ 2εθn (5.35) ± ±,n ±,n s−α−3  ∥B (V , Φ )∥ ≤ 2εθ for s ∈ [s0 , s1 − 4], s,T  n 1   |B2 (V+,n , V−,n )|s−1,T ≤ εθns−α−3 for s ∈ [s0 , s1 − 2] for any n ≥ 0, where ∆n = θn+1 − θn . This proposition is obtained by induction on n ≥ 0. Suppose that estimates (5.35) hold for all g g ±,2 0 ≤ n ≤ m − 1. From the definition of (e±,1 i,n , ei,n ) (i = 1, 3) given in (5.21), we conclude  g ±,1 ±,1 ∥eg 2 s+s0 −2α−4 ∆n , 1,n ∥s,T + ∥e3,n ∥s,T ≤ Cε θn (5.36) g ±,2 ±,2 2 s+s0 −2α−2 ∥eg ∥ ≤ Cε θ ∆ ∥ + ∥ e s,T n s,T n 3,n 1,n for all 0 ≤ n ≤ m − 1 and s ∈ [s0 , s1 − 1]. From (5.33), we have ±,n+ 21

V2

± ± ±,n − V2±,n =(∂t + Ua,3 ∂x2 + Ua,4 ∂x3 )(Sθn − I)Φ±,n + (Ψ± )x2 (Sθn − I)V3±,n a + Sθ n Φ ±,n + (Ψ± )x3 (Sθn − I)V4±,n + ((Sθn − I)Φ±,n )x2 V3±,n a + Sθn Φ

+ ((Sθn − I)Φ±,n )x3 V4±,n + B1± (V±,n , Φ±,n ), which implies the estimate: ±,n+ 1

2 ∥V2 − V2±,n ∥s,T ≤ Cεθns−α for all s0 ≤ s ≤ s1 − 4 and 0 ≤ n ≤ m − 1. Similarly, one has

±,n+ 1

2 − V5±,n ∥s,T ≤ Cεθns−α ∥V5 holding for all s0 ≤ s ≤ s1 − 4 and 0 ≤ n ≤ m − 1. Therefore, we obtain

|Bi± (V±,n+ 2 , Sθn ϕn )|s,T 1

≤ |Bi± (V±,n+ 2 , Sθn ϕn ) − Bi± (V ±,n , ϕn )|s,T + |Bi± (V ±,n , ϕn )|s,T 1

≤ Cεθns+1−α


19

for s0 − 1 ≤ s ≤ s1 − 5, 0 ≤ n ≤ m − 1, and i = 1, 3, by using an estimate of B3± (V±,n , Φ±,n ) similar to (5.35) derived from problem (5.25) (cf. [36]). Thus, from (5.22), we deduce g g ±,4 2 s+s0 −2α ∆n ∥e±,4 1,n ∥s,T + ∥e3,n ∥s,T ≤ Cε θn

(5.37)

for all 0 ≤ n ≤ m − 1 and s ∈ [s0 , s1 − 6]. Combining (5.36) with (5.37), it follows that g ± ± 2 s+s0 −2α ∥eg ∆n 1,n ∥s,T + ∥e3,n ∥s,T ≤ Cε θn

(5.38)

for all 0 ≤ n ≤ m − 1 and s ∈ [s0 , s1 − 6]. From (5.26), one immediately deduces ± ± ^ ^ h± i,m = (Sθm−1 − Sθm )Ei,m−1 + Sθm ei,m−1

∑m−2 g ± ± with E^ i,m−1 = n=0 ei,n for i = 1, 3, which implies ± 2 s+s0 −2α ∆m |h± 1,m |s,T + |h3,m |s,T ≤ Cε θm

(5.39)

for all s ≥ s0 . ± On the other hand, for fm and g˜m given in (5.25), as in [15], we have ± s−α−3 ∥fm ∥s,T + |˜ gm |s+1,T ≤ C(εδ + ε2 )θm ∆m

for all s ≥ s0 , with δ = ∥fa± ∥α+2,T /ε being small. Applying Corollary 4.2 for problem (5.25) with n = m and using the above estimate and (5.39), we find ˙ ±,m ∥s,T ≤ C(εδ + ε2 )θs−α−3 ∆m ∥δ V (5.40) m for all s ≥ s0 , by noting α ≥ s0 + 5. By applying a classical estimate for problem (5.28) of δΦ+,m and using (5.40), it follows that s−α−3 ∥δΦ+,m ∥s,T ≤ CT (εδ + ε2 )θm ∆m

(5.41)

± 2 s−α−2 ∥hg ∆m 3,m ∥s,T ≤ C(εδ + ε )θm

(5.42)

for all s ≥ s0 . Thus, from (5.29), we have

for all s ≥ s0 . − For the function hg 1,m given in (5.30), it is easy to have − − + − + − + ± ±,m+ 21 − ± ±,m+ 12 hg )(hg )(hg 1,m = h1,m + λ (Ua + V 3,m − h3,m ) − λ (Ua + V 3,m − h3,m ),

which implies − 2 s−α−2 ∥hg ∆m 1,m ∥s,T ≤ C(εδ + ε )θm

(5.43)

for all s ≥ s0 . Applying the classical estimate again for problem (5.30) and using (5.43), we have s−α−2 ∥δΦ−,m ∥s,T ≤ CT (εδ + ε2 )θm ∆m

(5.44)

for all s ≥ s0 . Thus, from (5.40), (5.41), and (5.44), we have s−α−2 ∥δV±,m ∥s,T ≤ C(εδ + ε2 )θm ∆m

for all s ≥ s0 .

(5.45)

20


From (5.41) and (5.44)–(5.45), we immediately obtain the estimates of ∥δV±,m ∥s,T and ∥δΦ±,m ∥s,T given in (5.35) by choosing δ = ∥fa± ∥α+2,T /ε small. The remaining estimates of (5.35) can be verified directly, and the details can be found in [15]. Convergence of the Iteration Scheme: From the first result of (5.35), we have ∑ ∥(δV±,n , δΦ±,n )∥α,T < ∞,

(5.46)

n≥0

which implies that there exist (V± , Φ± ) ∈ B α (ΩT ) such that (V±,n , Φ±,n ) −→ (V± , Φ± )

in B α (ΩT ).

(5.47) ±

±

From the other results given in (5.35), we obtain that the limit functions (V , Φ ) satisfy  ± ± ± ±  in ΩT , L(V , Φ )V = fa ± ± ± (5.48) B1 (V , Φ ) = 0 in ΩT ,   + − B2 (V , V ) = 0 on bΩT . On the other hand, from the second result given in (5.48), we obtain that the constraint B3± (V± , Φ± ) = 0 also holds on bΩT if it is true at {t = 0}, by using Lemma 3.1. Note that δΦ+,n = δΦ−,n for all n immediately imply Φ+ = Φ− on {x1 = 0} as well. Thus the second result given in (5.48) leads to one of the Rankine-Hugoniot condition: + − vN = vN

on Γ

given in (3.8). Therefore, we conclude ¯± ∈ Theorem 5.2. Let α ≥ 15 and s1 ∈ [α + 7, 2α − 5]. Let ψ0 ∈ H 2s1 +3 (R2 ) and U± 0 − U 2(s1 +2) 3 B (R+ ) satisfy the compatibility conditions of problem (3.18) up to order s1 + 2, and let conditions (3.7)–(3.8) and (5.34) be satisfied. Then there exists a solution V± ∈ B α (ΩT ), ϕ ∈ H α−1 (bΩT ) to problem (5.4). Then Theorem 3.2 in Section 3 directly follows from Theorem 5.2. 6. Concluding Remarks and Open Problems Characteristic discontinuities (compressible vortex sheets and entropy waves), along with shock and rarefaction waves, occur ubiquitously in nature and are fundamental waves in the entropy solutions to hyperbolic systems of conservation laws in several space variables. The stability problems for characteristic discontinuities are fundamental, especially in shock reflection-diffraction and various wave interactions. Their mathematical rigorous treatments are truly challenging. What we have known is still very limited. Most of problems involving characteristic discontinuities are longstanding and still open. In particular, the following problems have not well understood, which deserve our attention: 1. As discussed in §2.2, another kind of characteristic discontinuities for the two-dimensional full Euler equations in gas dynamics is entropy waves. Similarly, they occur in the higher dimensional situations. It would be interesting to analyze entropy waves to explore new phenomena and features of these waves in two-dimensions and even higher dimensions. 2. In §3–§5, we have established the stability of current-vortex sheets when the jump of the tangential velocity is dominated by the jump of the non-paralleled tangential magnetic fields in the sense that λ± determined in (3.21) satisfy condition (3.20); also see Remark 3.1. The next


21

concern is the stability/instability issue of current-vortex sheets in three-dimensional MHD when the jump of the tangential velocity is not dominated by the jump of the tangential magnetic fields, especially when the magnetic fields are parallel to each other on both sides of the front. 3. From the Rankine-Hugoniot conditions in (3.6) with mN = 0 on Γ, besides the case (3.7)–(3.8) for the current-vortex sheets, there is another kind of characteristic discontinuities on which + − + − ψt = vN = vN , HN = HN ̸= 0

on Γ,

which implies [vτ ] = [Hτ ] = 0, that is, [H] = [v] = 0,

[p] = 0,

but [S] ̸= 0 equivalently [ρ] ̸= 0. Such a wave is called a current-entropy wave. It is important to understand the stability/instability of current-entropy waves in three-dimensional MHD. 4. There are other different characteristic/noncharacteristic discontinuities in MHD; see BlokhinTrakhinin [8], Trakhinin [37], and the references cited therein. It would be interesting to study these discontinuities and related problems in MHD and explore their new phenomena/features. 5. For the Euler equations in gas dynamics, it has been shown in Chen-Zhang-Zhu [14] and Chen-Kukreja [13] that two-dimensional steady-state vortex sheets are always stable under the two-dimensional steady perturbations of the incoming supersonic fluid flow. For shock reflectiondiffraction problems, the solutions are self-similar, and most of Mach reflection-diffraction configurations involve a vorticity wave formed by a vortex sheet. It is important to understand the compressible vortex sheets for the Euler equations in the self-similar coordinates. In particular, when a vortex sheet forms a vorticity wave, it is useful to understand which spaces of functions the solutions of the vorticity waves belong. 6. Another important direction is to analyze various interaction between shock fronts with vortex sheets in multidimensional compressible fluid flows. It would be interesting to explore possible nonlinear approaches to see whether the corresponding estimates of solutions have no derivative loss with respect to initial data for the problems addressed; also see Coutand-Shkoller [17]. It is clear that the solution to these problems involving characteristic discontinuities requires further new mathematical ideas, techniques, and approaches, which will be also useful for solving other longstanding problems in nonlinear partial differential equations, especially various boundary value problems, free boundary problems, among others, in hyperbolic conservation laws. Acknowledgements: The research of Gui-Qiang Chen was supported in part by the National Science Foundation under Grants NSF grants DMS-0935967, DMS-0807551, the UK EPSRC Science and Innovation award to the Oxford Centre for Nonlinear PDE (EP/E035027/1), the NSFC under a joint project Grant 10728101, and the Royal Society–Wolfson Research Merit Award (UK). The research of Ya-Guang Wang was supported in part by the National Science Foundation of China under Grants 10971134 and 11031001. The second author would like to express his gratitude to the Department of Mathematics of Northwestern University (USA) for the hospitality, where this work was initiated when he visited there during the Spring Quarter 2005. Both authors would like to thank Y. Trakhinin for the helpful discussion on this problem.

22


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