Non-contraction of intermediate admissible discontinuities for 3-D planar isentropic magnetohydrodynamics

We investigate non-contraction of large perturbations around intermediate entropic shock waves and contact discontinuities for the three-dimensional planar compressible isentropic magnetohydrodynamics (MHD). To do that, we take advantage of criteria developed by Kang and Vasseur in [6], and non-contraction property is measured by pseudo distance based on relative entropy.


Introduction
This article is devoted to the study of non-contraction property of certain intermediate entropic shock waves and contact discontinuities for the three-dimensional planar compressible inviscid isentropic MHD, which takes the form in Lagrangian coordinates: (1.1) Here v denotes specific volume, and (u 1 , u 2 , u 3 ) and (β, B 2 , B 3 ) represent the three-dimensional fluid velocity and magnetic field, respectively. Those only depend on a single direction e 1 measured by x, and no dynamics with respect to other variables. Notice that β is constant due to the divergence-free condition of magnetic field of full MHD. As an ideal isentropic polytropic gas, the pressure p is assumed to satisfy (1.2) p(v) = v −γ , γ > 1.
The system (1.1) has a convex entropy η as in terms of the conservative variables U := (v, q 2 , q 3 , u 1 , u 2 , u 3 ) where q i := vB i for i = 2, 3.
In particular, η is strictly convex because we consider non-vacuum states for MHD.
Using the entropy η, we define its relative entropy function by It is well-known that the relative entropy η(·|·) is positive-definite, but looses the symmetry unless η(u) = |u| 2 . Nevertheless the relative entropy is comparable to the square of L 2 distance for any bounded solutions. (See for example [6,7,11]) Recently in [11], Vasseur has shown contraction for large perturbations around extremal shocks (1-shock and n-shock) of the hyperbolic system of conservation laws satisfying physical conditions, which is satisfied by Euler systems of gas dynamics. In order to measure the distance between any bounded entropic solution and extremal shock, he used a spatially inhomogeneous pseudo-distance as follows: for a given weight a > 0, the spatially inhomogeneous pseudo-distance d a is defined by where S(t, x) denotes a given extremal shock (u l , u r , σ), i.e., Based on this pseudo-distance, it has been shown in [11] that there exists suitable weight a > 0 such that the extremal shock is contractive up to suitable Lipschitz shift α(t) in the sense that for all bounded entropic solution u ∈ BV loc ((0, ∞) × R) n , x + α(t)), S(t, x))dx is non-increasing in time. On the other hand for intermediate admissible discontinuities, the authors in [6] developed criteria to identify whether the intermediate entropic shocks and contact discontinuities are contractive or not in the pseudo-distance as above. Applying the criteria into the two-dimensional planar isentropic MHD, it turns out in [6] that there is no weight for the contraction of certain intermediate shock waves.
In this article, we use the criteria developed in [6] to show non-contraction of certain intermediate shocks and contact discontinuities. More precisely, we prove that there is no weight a > 0 such that the intermediate discontinuities satisfy contraction in that sense (1.3) with weight a.
Concerning studies on stability of shock waves to the viscous model of (1.1), we refer to [1,2,4,5,10], in which it turns out that the viscous shock waves (including intermediate waves) are Evans stable under small perturbation. This implies Lopatinski stability because Lopatinski stability condition is necessary for stability of viscous profile (See [12]). Thus small BV -perturbations of the inviscid MHD entropic shock waves are stable thanks to Lopatinski stability (See [3,8,9]). Notice that this is not in contradiction with our result on non-contraction, because our framework is based on large perturbation around the entropic discontinuity, thus our results says that some large perturbation can increase as time goes on. In Remark 2.1, we explain more precisely a meaning of non-contraction for entropic discontinuities in our framework.
The rest of the paper is organized as follows. In Section 2, we present a criterion developed in [6] and six characteristic fields of the 6 × 6 system (1.1), which are used to show noncontraction for intermediate entropic shock waves of (1.1) in Section 3, and for contact discontinuities in Section 4.

Preliminaries
2.1. Criteria on non-contraction of intermediate entropic discontinuities. In this section, we present a criterion in [6] for non-contraction of admissible discontinuities of hyperbolic system of conservation laws: For the system (2.4) satisfying Liu and Lax entropy conditions, the authors in [6] have developed sufficient conditions to identify non-contraction of intermediate entropic discontinuities as follows.
Theorem 2.1. For a fixed 1 < i < n, let (u l , u r , σ l,r ) be a given i-th entropic discontinuity satisfying Liu and Lax entropy conditions. Assume that there are 1 ≤ j < i < k ≤ n such that j-and k-characteristic fields are genuinely nonlinear. Then the following statements holds.
• (1) For 0 < a < 1, we assume that there is a C 1 j-th rarefaction curve R j u l (s) with j < i such that R j u l (0) = u l and the backward curve R j,− u l (s) of R j u l (s), i.e., λ j (R j,− u l (s)) < λ j (u l ), intersects with the (n − 1)-dimensional surface Σ a . Then, (u l , u r , σ l,r ) does not satisfy contraction in the sense (1.3) with weight a.
• (2) For a > 1, we assume that there is a C 1 k-th rarefaction curve R k ur (s) with k > i such that R k ur (0) = u r and the forward curve R k,+ ur (s) of R k ur (s), i.e., λ k (R k,+ ur (s)) > λ k (u r ), intersects with the (n − 1)-dimensional surface Σ a . Then, (u l , u r , σ l,r ) does not satisfy contraction in the sense (1.3) with weight a.
• (3) For a = 1, we assume that one of the assumptions of (1) and (2) is satisfied.
Then, (u l , u r , σ l,r ) does not satisfy contraction in the sense (1.3) with weight a. , we see the definition of non-contraction in our framework as follows. We say that an admissible discontinuity (u l , u r , σ) does not satisfy contraction in the pseudo distance (1.3) with weight a if for any Lipschitz curve α(t) with α(0) = 0, there are some entropy solutionū and small constant T 0 > 0 such that for all 0 < t < T 0 , In fact, the authors in [6] constructed a specific (local) smooth solutionū satisfying (2.5), which evolves from a smooth initial datā whereū is the point appeared in Theorem 2.1, which is the intersection point of suitable rarefaction wave and surface Σ a , thusū depends on the weight a, thus on the pseudo distance.

2.2.
Characteristic fields for the 6 × 6 system (1.1). We here present six characteristic fields of the system (1.1). For simplicity of computation, we use non-conservative variable W := (v, B 2 , B 3 , u 1 , u 2 , u 3 ) and rewrite (1.1) as a quasilinear form: where the 6 × 6 matrix A is given by where c := −p ′ (v) denotes the sound speed. By a straightforward computation, we have the characteristic polynomial of A as i.e., Then since If we consider the case of |B| = 0, then we have Here we assume that β = 0. Thus we have six eigenvalues By a straightforward computation, we have the corresponding eigenvectors Then we can check that for each i = 1, 3, 4, 6, (λ i , r i ) are genuinely nonlinear whereas (λ 2 , r 2 ) and (λ 4 , r 4 ) are linearly degenerate as follows. Indeed since and for each i = 2, 3, we have dλ 1 · r 1 > 0. Using the similar argument, we have dλ i · r i > 0 for i = 3, 4, 6. On the other hand, it is easy to get dλ i · r i = 0 for i = 2, 5.

non-contraction of shock waves
In this Section, we show that there is no weight a > 0 such that certain entropic shock waves satisfy contraction in the sense (1.3) with weight a. Let (U l , U r , σ 3 ) be any 3-shock waves satisfying the Rankine-Hugoniot condition: This is true at least for weak shock. But, this relation can be justified for shock waves of arbitrary amplitude. We give this justification in the Appendix for the reader's convenience.
On the other hand, since it follows from (3.8) that for each i = 2, 3, Moreover, since (2.7) and Lax condition yield (3.10) Notice that we do not see the sign of v l − β 2 σ −2 3 , thus of B i,r , because we have not found the explicit formulation of the speed σ 3 . Based on the observation above, we here consider the specific condition that the 3-shock wave satisfies one of the following cases : For each i = 2, 3, (B i,l , B i,r ) satisfies one of B i,l > B i,r ≥ 0 or B i,l < B i,r ≤ 0 or B i,l = B i,r = 0. (3.11) Remark 3.1. In the above assumption (3.11), when B 2,l = 0 and B 3,l = 0 simultaneously, it follows from (2.6) that corresponding to the eigenvalue − β √ v l is spanned by independent two eigenvectors: This implies that dλ 1 (U ) · r 1 (U ) = 0 except for U = U l as an umbilical point. Likewise, if For a given 4-shock wave (Ũ l ,Ũ r , σ 4 ), using the same arguments as above, we have [ṽ] > 0,

Since (2.7) and Lax condition yield
Thus we consider the analogous condition that the 4-shock wave satisfies one of the following cases : For each i = 2, 3, (B i,l ,B i,r ) satisfies one of We are now ready to show that for any a > 0, there is no a-contraction of such intermediate shocks as follows.
Since dv ·r 1 = v α + (α + − β 2 v ) > 0, v is strictly monotone along the integral curve of the vector field r 1 , which means that the 1-rarefaction wave can be parameterized by v. Moreover since dλ 1 · r 1 > 0, −r 1 is the tangent vector field of the backward 1-rarefaction wave R 1,− U l , which implies that v decreases along In order to claim that R 1,− U l intersects with Σ a for any a < 1, we use the fact that for a < 1, η(U |U l ) ≤ aη(U |U r ) is equivalent to which is rewritten as for some constants c 1 , c 2 . This implies that ⇐⇒ v > c * and |q 2 | + |q 3 | + |u 1 | + |u 2 | + |u 3 | ≤ c * for some constants c * , c * > 0, since ∞ 0 p(s)ds = +∞, and all positive terms on u i and q i are quadratic in the left-hand side, whereas linear in the right-hand side of (3.14). Therefore there exists 0 < v * ≪ c * such that which implies that R 1,− U l intersects with Σ a for a < 1, because R 1,− U l is a continuous curve issuing from U l ∈ {U | η(U |U l ) < aη(U |U r )}.
On the other hand, we claim that the forward 6-rarefaction wave R 6,+ Ur issuing from U r intersects with the surface Σ a for any a ≥ 1. Since dλ 6 · r 6 > 0 and dv · r 6 15) r 6 is the tangent vector of the forward 6-rarefaction wave R 6,+ Ur , and the parameter v + decreases along R 6,+ Ur . Moreover R 6,+ Ur is well-defined for all v + ∈ (0, v r ]. For any fixed a ≥ 1, we consider a continuous functional F a defined by We first show that the functional F 1 (when a = 1) satisfies we use (3.15) to compute (3.17) Using (3.8), we have Since (1.2) and (2.7) yields it follows from (3.9) that I 1 → −∞ as v + → 0+.
Hence for all a > 0, the 3-shock wave (U l , U r , σ 2 ) does not satisfies contraction in the sense (1.3) with weight a thanks to Theorem 2.1.

non-contraction of contact discontinuities
We here show that there is no weight a > 0 such that certain contact discontinuities satisfy contraction in the sense (1.3) with weight a. Let (U l , U r ) be a given 2-contact discontinuity (or 5-contact discontinuity) of the system (1.1)-(1.2). Since i-contact discontinuity (U l , U r ) is a integral curve of the vector field r i for each i = 2, 5, we have  Proof. We follow the same arguments as the proof of Theorem 3.1. First of all, we can see that the backward 1-rarefaction wave R 1,− U l issuing from U l intersects with Σ a for a < 1, and the forward 6-rarefaction wave R 6,+ Ur issuing from U r intersects with Σ a for a > 1. In order to show that one of R 1,− U l and R 6,+ Ur intersects with the Σ 1 , we consider a functional F (U ) := η(U |U l ) − η(U |U r ).
• Case (B) (For each i = 2, 3, (B i,l , B i,r ) satisfies one of B i,l > B i,r > 0 or B i,l < B i,r < 0) Using the same arguments as previous case, we see that R 6,+ Ur intersects with Σ 1 under those constraints.