Well-posedness of the linearized plasma-vacuum interface problem in ideal incompressible MHD

Abstract

Abstract. We study the free boundary problem for the plasma-vacuum interface in ideal incompressible magnetohydrodynamics (MHD). In the vacuum region the magnetic field is described by the div-curl system of pre-Maxwell dynamics, while at the interface the total pressure is continuous and the magnetic field is tangent to the boundary. Under a suitable stability condition satisfied at each point of the plasma-vacuum interface, we prove the well-posedness of the linearized problem in Sobolev spaces.

Nonlinear Stability of the Current‐Vortex Sheet to the Incompressible MHD Equations

“…We believe that our method can be applied to solve the plasma-vacuum interface problem in ideal incompressible MHD and the other related free boundary problems (see [14], for example). We note that the well-posedness of the linearized plasma-vacuum interface problem has been proved by Morando, Trakhinin, and Trebeschi [20].…”

On a Priori Energy Estimates for Characteristic Boundary Value Problems

“…Instead, the linearized problem with general data F and g = 0 admits an a priori estimate similar to (10), with loss of one derivative in F and g, see [32]. For similar results in the case of the incompressible plasma -vacuum problem, see [25].…”

“…As a first example, we quote the multiplication by a matrix-valued function B ∈ C ∞ (0) (R n + ). It is clear that this makes an operator of order zero according to (38); indeed (25) gives for any vector-valued…”

“…, n and γ ≥ 1. Since the leading part of L only involves conormal derivatives, applying (25), (27), (28) then gives…”

Well-posedness of the Free Boundary Problem for Non-relativistic and Relativistic Compressible Euler Equations with a Vacuum Boundary Condition

“…Considering the pre-Maxwell equations for the magnetic field in vacuum, Secchi-Trakhinin proved in [36] the well-posedness of the nonlinear plasma-vacuum interface problem in ideal compressible MHD based on their linear well-posedness results in [35,41]. As for the incompressible case, we refer to Morando et al [28] for the well-posedness of the linearized problem, Hao [13] for nonlinear a priori estimates, and Sun et al [37] for nonlinear well-posedness. For the plasma-vacuum interface problem in RMHD, where the vacuum electric and magnetic fields satisfy Maxwell's equations, an a priori estimate for the linearized problem in the anisotropic Sobolev space H 1 * was provided by the first author in [42].…”