TY - JOUR

T1 - Well-Posedness of the Two-Dimensional Compressible Plasma-Vacuum Interface Problem

AU - Morando, Alessandro

AU - Secchi, Paolo

AU - Trakhinin, Yuri

AU - Trebeschi, Paola

AU - Yuan, Difan

N1 - Open access funding provided by Universit\u00E0 degli Studi di Brescia within the CRUI-CARE Agreement. The research of A. Morando, P. Secchi, P. Trebeschi was supported in part by the Italian MUR Project PRIN prot. 20204NT8W4. The research of Y. Trakhinin was supported by Mathematical Center in Akademgorodok under Agreement No. 075-15-2022-282 with the Ministry of Science and Higher Education of the Russian Federation. D. Yuan was supported by NSFC Grant No.12001045 and China Postdoctoral Science Foundation No. 2020M680428, No. 2021T140063. D. Yuan thanks the University of Brescia for its kind hospitality.

PY - 2024/6/4

Y1 - 2024/6/4

N2 - We consider the two-dimensional plasma-vacuum interface problem in ideal compressible magnetohydrodynamics (MHD). This is a hyperbolic-elliptic coupled system with a characteristic free boundary. In the plasma region the 2D planar flow is governed by the hyperbolic equations of ideal compressible MHD, while in the vacuum region the magnetic field obeys the elliptic system of pre-Maxwell dynamics. At the free interface moving with the velocity of plasma particles, the total pressure is continuous and the magnetic field on both sides is tangent to the boundary. The plasma-vacuum system is not isolated from the outside world, since it is driven by a given surface current which forces oscillations onto the system. We prove the local-in-time existence and uniqueness of solutions to this nonlinear free boundary problem, provided that at least one of the two magnetic fields, in the plasma or in the vacuum region, is non-zero at each point of the initial interface. The proof follows from the analysis of the linearized MHD equations in the plasma region and the elliptic system for the vacuum magnetic field, suitable tame estimates in Sobolev spaces for the full linearized problem, and a Nash–Moser iteration.

AB - We consider the two-dimensional plasma-vacuum interface problem in ideal compressible magnetohydrodynamics (MHD). This is a hyperbolic-elliptic coupled system with a characteristic free boundary. In the plasma region the 2D planar flow is governed by the hyperbolic equations of ideal compressible MHD, while in the vacuum region the magnetic field obeys the elliptic system of pre-Maxwell dynamics. At the free interface moving with the velocity of plasma particles, the total pressure is continuous and the magnetic field on both sides is tangent to the boundary. The plasma-vacuum system is not isolated from the outside world, since it is driven by a given surface current which forces oscillations onto the system. We prove the local-in-time existence and uniqueness of solutions to this nonlinear free boundary problem, provided that at least one of the two magnetic fields, in the plasma or in the vacuum region, is non-zero at each point of the initial interface. The proof follows from the analysis of the linearized MHD equations in the plasma region and the elliptic system for the vacuum magnetic field, suitable tame estimates in Sobolev spaces for the full linearized problem, and a Nash–Moser iteration.

UR - https://www.scopus.com/record/display.uri?eid=2-s2.0-85195403852&origin=inward&txGid=f6cfc23fa27c192fa03a14d1c9eb8977

UR - https://www.mendeley.com/catalogue/63973ceb-943d-39a9-ab2c-2609ca30e110/

U2 - 10.1007/s00205-024-02001-y

DO - 10.1007/s00205-024-02001-y

M3 - Article

VL - 248

JO - Archive for Rational Mechanics and Analysis

JF - Archive for Rational Mechanics and Analysis

SN - 0003-9527

IS - 4

M1 - 56

ER -