Well-posedness of Free Boundary Problem in Non-relativistic and Relativistic Ideal Compressible Magnetohydrodynamics
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Well-posedness of Free Boundary Problem in Non-relativistic and Relativistic Ideal Compressible Magnetohydrodynamics Yuri Trakhinin
& Tao Wang
Communicated by N. Masmoudi
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.
Contents 1. Introduction . . . . . . . . . . . . . . . . . . 2. Nonlinear Problems and Main Theorems . . . 3. Well-posedness of the Linearized Problem . . 3.1. Main Result for the Linearized Problem . 3.2. Properties of Anisotropic Sobolev Spaces 3.3. Well-posedness in L 2 . . . . . . . . . . . 3.4. Higher-order Energy Estimates . . . . . . 3.5. Proof of Theorem 3.1 . . . . . . . . . . . 4. Nash–Moser Iteration . . . . . . . . . . . . . 4.1. Approximate Solution . . . . . . . . . . .
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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
Y. Trakhinin, Y. Trakhinin 4.2. Iteration Scheme . . . . . . . . . . . . . . . . . . . . 4.3. Inductive Hypothesis . . . . . . . . . . . . . . . . . . 4.4. Estimates of the Error Terms . . . . . . . . . . . . . . 4.5. Proof of Theorem 2.1 . . . . . . . . . . . . . . . . . . 5. Relativistic Case . . . . . . . . . . . . . . . . . . . . . . . 6. Appendix A: Conventional Notation in the Vector Calculus 7. Appendix B: Symmetrization for RMHD . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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1. Introduction This paper concerns the well-posedness of the free boundary problem for a plasma–vacu
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