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Well-posedness of Free Boundary Problem in Non-relativistic and Relativistic Ideal Compressible Magnetohydrodynamics

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Abstract

We consider the free boundary problem for non-relativistic and relativistic ideal compressible magnetohydrodynamics in two and three spatial dimensions with the total pressure vanishing on the plasma–vacuum interface. We establish the local-in-time existence and uniqueness of solutions to this nonlinear characteristic hyperbolic problem under the Rayleigh–Taylor sign condition on the total pressure. The proof is based on certain tame estimates in anisotropic Sobolev spaces for the linearized problem and a modification of the Nash–Moser iteration scheme. Our result is uniform in the speed of light and appears to be the first well-posedness result for the free boundary problem in ideal compressible magnetohydrodynamics with zero total pressure on the moving boundary.

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Acknowledgements

The authors would like to thank the anonymous referees for helpful comments and suggestions that helped to improve the quality of the paper.

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Authors and Affiliations

  1. Sobolev Institute of Mathematics, Koptyug av. 4, Novosibirsk, Russia, 630090

    Yuri Trakhinin

  2. Novosibirsk State University, Pirogova str. 1, Novosibirsk, Russia, 630090

    Yuri Trakhinin

  3. School of Mathematics and Statistics, Wuhan University, Wuhan, 430072, China

    Tao Wang

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The research of Yuri Trakhinin was partially supported by RFBR (Russian Foundation for Basic Research) under Grant 19-01-00261-a and by Mathematical Center in Akademgorodok under Agreement No. 075-15-2019-1675 with the Ministry of Science and Higher Education of the Russian Federation. The research of Tao Wang was partially supported by the National Natural Science Foundation of China under Grants 11971359 and 11731008

Appendices

Appendix A: Conventional Notation in the Vector Calculus

For readers’ convenience, we collect the conventional notation in the vector calculus. The spatial dimension is denoted by \(d=2,3\). We abbreviate the partial differentials as

$$\begin{aligned} \partial _t:=\frac{\partial }{\partial t},\quad \partial _i:=\frac{\partial }{\partial x_i} \quad \text {for }i=1,\ldots ,d. \end{aligned}$$

We denote the gradient by \(\nabla :=(\partial _1,\ldots ,\partial _d)^{{\mathsf {T}}}\). For any \(d\times d\) matrix \(F=(F_{ij})\), vectors \(u=(u_1,\ldots ,u_d)^{{\mathsf {T}}}\) and \(v=(v_1,\ldots ,v_d)^{{\mathsf {T}}}\), and scalar a, the symbol \(u\otimes v\) denotes the \(d\times d\) matrix with (i, j)-entry \(u_i v_j\), and

$$\begin{aligned}&\nabla \cdot F:= (\partial _j F_{1j},\ldots ,\partial _j F_{dj})^{{\mathsf {T}}}, \quad \nabla \cdot u:= \partial _i u_{i},\\&u\times v:= \left\{ \begin{aligned}&u_1 v_2 -u_2 v_1\quad&\text {if }d=2,\\&(u_2 v_3 -u_3 v_2,u_3 v_1 -u_1 v_3,u_1 v_2 -u_2 v_1)^{{\mathsf {T}}}\quad&\text {if }d=3, \end{aligned} \right. \\&\nabla \times v:= \left\{ \begin{aligned}&\partial _1 v_2 -\partial _2 v_1\quad&\text {if }d=2,\\&(\partial _2 v_3 -\partial _3 v_2,\partial _3 v_1 -\partial _1 v_3,\partial _1 v_2 -\partial _2 v_1)^{{\mathsf {T}}}\quad&\text {if }d=3, \end{aligned} \right. \\&u\times a = -a\times u:=(u_2 a, -u_1 a)^{{\mathsf {T}}}, \quad \nabla \times a :=(\partial _2a , -\partial _1 a )^{{\mathsf {T}}} \quad \text {if }d=2. \end{aligned}$$

The notation above was employed by Kawashima [17, p. 144] to write down the electromagnetic fluid system in two spatial dimensions. The compressible MHD equations (1.1) with \(d=2\) follow from the assumption that all the quantities in (1.1) are independent of \(x_3\) and the components \(v_3\) and \(H_3\) are identically zero.

Appendix B: Symmetrization for RMHD

Let us deduce the symmetric system (5.10) from (5.6)–(5.8). First, the last equation for S in (5.10) is exactly (5.8). In view of (5.2) and (5.9), we have \(\varGamma =(1+\epsilon ^2|w|^2)^{1/2}\) and \(v=\varGamma ^{-1}w\), so that

$$\begin{aligned} \partial _{\alpha } \varGamma =\epsilon ^2 v_i\partial _{\alpha } w_i,\ \ \partial _{\alpha } v=\varGamma ^{-1}\partial _{\alpha } w-\epsilon ^2 \varGamma ^{-1} v v_i \partial _{\alpha } w_i \quad \text {for }\alpha =0,\ldots , d. \end{aligned}$$
(5.10)

It follows from the identities (5.8), (5.6a), and (B.1) that

$$\begin{aligned} \varGamma (\partial _t+v_i\partial _i) p=-\rho a^2(\epsilon ^2 v_i \partial _t w_i +\partial _i w_i ), \end{aligned}$$

which immediately gives the first equation for p in (5.10). Using (5.6d) and (5.7), we have \((\partial _t+v\cdot \nabla ) H-(H\cdot \nabla )v+H\nabla \cdot v=0\), which, together with (B.1), yields

$$\begin{aligned} (\partial _t+v\cdot \nabla ) H +{\mathcal {M}}_i \partial _i w=0, \end{aligned}$$
(5.11)

with \( {\mathcal {M}}_i:=\varGamma ^{-1} \{ H\otimes \varvec{e}_i -H_i I_d -\epsilon ^2 (v_i H- H_i v )\otimes v \}. \) Thanks to (5.6b) and (B.1), we infer from (5.6c) that

$$\begin{aligned} \left( \rho h\varGamma + {\epsilon ^2{\varGamma ^{-1}} |H|^2} \right) (I_d -\epsilon ^2 v\otimes v)(\partial _t +v\cdot \nabla ) w\qquad \qquad \qquad&\\ + \epsilon ^2 v\partial _t p+\nabla p+\sum _{i=1}^{4}{\mathcal {T}}_i=0,&\end{aligned}$$

where

$$\begin{aligned} {\mathcal {T}}_1:=\;&\frac{1}{2} v\partial _t|b|^2 -\epsilon ^2\partial _t ((v\cdot H)H ),\\ {\mathcal {T}}_2:=\;&-\nabla \cdot (\varGamma ^{-2}H\otimes H),\quad {\mathcal {T}}_3:=\frac{1}{2}\epsilon ^{-2}\nabla |b|^2,\\ {\mathcal {T}}_4:= \;&-\epsilon ^2 \nabla \cdot ( (v\cdot H)(H\otimes v+v\otimes H) ) +\epsilon ^2 v\nabla \cdot ( (v\cdot H) H) \\ =\;&-\epsilon ^2 (v\cdot H)( (v\cdot \nabla )H+(H\cdot \nabla )v+H\nabla \cdot v ) -\epsilon ^2 H (v\cdot \nabla ) (v\cdot H) . \end{aligned}$$

By virtue of (5.5), (B.1), and (5.7), we deduce

$$\begin{aligned}&{\mathcal {T}}_1= \epsilon ^2 \varGamma ^{-1} \big \{\epsilon ^2 (v\cdot H) (H\otimes v+v\otimes H)- |b|^2 v\otimes v-H\otimes H\big \}\partial _t w +{\mathcal {T}}_{1a},\\&{\mathcal {T}}_2= -\varGamma ^{-2} (H\cdot \nabla )H_i +2\epsilon ^2 \varGamma ^{-3} H_i H\otimes v \partial _i w, \\&{\mathcal {T}}_3= (\varGamma ^{-2} H_i+ \epsilon ^2 (v\cdot H) v_i )\nabla H_i +\varGamma ^{-1}\varvec{e}_i\otimes (\epsilon ^2(v\cdot H) H-|b|^2 v )\partial _i w, \end{aligned}$$

with \({\mathcal {T}}_{1a}:=\epsilon ^2 (v\varGamma ^{-2} H_i +\epsilon ^2 (v\cdot H) v v_i -H v_i)\partial _t H_i-\epsilon ^2 (v\cdot H)\partial _t H.\) Then we utilize (B.1) and (B.2) for calculating the terms \({\mathcal {T}}_4\) and \({\mathcal {T}}_{1a}\) respectively to derive the equations for w in (5.10). Noticing that \({\mathcal {M}}_0{\mathcal {M}}_i={\mathcal {N}}_i\), we obtain the equations for H in (5.10) from the left-multiplication of (B.2) by \({\mathcal {M}}_0\).

Next we show that the matrix \(B_0(V)\) is positive definite in the non-vacuum region \(\{\rho _*<\rho <\rho ^* \}\). For any \(\varvec{u}\in {\mathbb {R}}^d\setminus \{ 0\}\), we get \(\varvec{u}^{{\mathsf {T}}}{\mathcal {M}}_0\varvec{u}\ge \varGamma ^{-1} |\varvec{u}|^2\), that is, \({\mathcal {M}}_0\ge \varGamma ^{-1}I_d\). Since

$$\begin{aligned} \begin{pmatrix} \dfrac{\varGamma }{\rho a^2}&{} \epsilon ^2 v^{{\mathsf {T}}} \\ \epsilon ^2 v &{}{\mathcal {A}}_0 \end{pmatrix} = \varvec{P}^{{\mathsf {T}}}\begin{pmatrix} \dfrac{\varGamma }{\rho a^2}&{} 0 \\ 0 &{}{\mathcal {A}}_0-\epsilon ^4\dfrac{\rho a^2}{\varGamma } v v^{{\mathsf {T}}} \end{pmatrix}\varvec{P} \ \ \text {with } \varvec{P}:=\begin{pmatrix} 1 &{} \dfrac{\rho a^2}{\varGamma }\epsilon ^2 v^{{\mathsf {T}}} \\ 0 &{}I_d \end{pmatrix}, \end{aligned}$$

it suffices to show that the matrix \({\mathcal {A}}_0-\epsilon ^4 \rho a^2 \varGamma ^{-1} v v^{{\mathsf {T}}}\) is positive definite. For any \(\varvec{u}\in {\mathbb {R}}^d\setminus \{ 0\}\), we have

$$\begin{aligned} \varvec{u}^{{\mathsf {T}}}\big ({\mathcal {A}}_0-\epsilon ^4 \rho a^2 \varGamma ^{-1} v v^{{\mathsf {T}}}\big )\varvec{u} ={\mathcal {T}}_5+{\mathcal {T}}_6, \end{aligned}$$

where \({\mathcal {T}}_5:=\rho h \varGamma |\varvec{u}|^2 -(\epsilon ^2\rho h \varGamma +\epsilon ^4 \rho a^2 \varGamma ^{-1}) (v\cdot \varvec{u})^2\) and

owing to (5.2). By virtue of (2.19) and (5.2), we infer

$$\begin{aligned} {\mathcal {T}}_5\ge \;&\rho h \varGamma |\varvec{u}|^2 -(\epsilon ^2\rho h \varGamma +\epsilon ^4 \rho a^2 \varGamma ^{-1}) |v|^2|\varvec{u}|^2\\ =\;&\rho h \varGamma ^{-1} |\varvec{u}|^2 \left( 1 -c_\mathrm{s}^2 \epsilon ^4 |v|^2\right) \ge \rho h \varGamma ^{-1} |\varvec{u}|^2 (1-\epsilon ^2 |v|^2) =\rho h \varGamma ^{-3} |\varvec{u}|^2. \end{aligned}$$

Therefore, we obtain that \({\mathcal {A}}_0-\epsilon ^4 \rho a^2 \varGamma ^{-1} v v^{{\mathsf {T}}} \) and \(B_0(V)\) are positive definite.

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Trakhinin, Y., Wang, T. Well-posedness of Free Boundary Problem in Non-relativistic and Relativistic Ideal Compressible Magnetohydrodynamics. Arch Rational Mech Anal 239, 1131–1176 (2021). https://doi.org/10.1007/s00205-020-01592-6

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