Well-posedness of initial boundary value problems for 2D compressible MHD equations in domains with corners

TL;DR

This paper establishes the well-posedness of initial boundary value problems for 2D compressible ideal MHD equations in domains with corners, introducing new anisotropic Sobolev spaces to handle geometric complexities.

Contribution

The paper introduces a novel class of anisotropic Sobolev spaces to address corner geometries, enabling well-posedness analysis for 2D compressible MHD equations with complex boundaries.

Findings

Well-posedness established for linear and nonlinear MHD problems.

Existence of weak solutions proved via duality in high-order tangential spaces.

Normal derivative estimates achieved through structure and Helmholtz decomposition.

Abstract

In this paper, the well-posedness is studied for the initial boundary value problem of the two-dimensional compressible ideal magnetohydrodynamic (MHD) equations in bounded perfectly conducting domains with corners. The presence of corners yields intrinsic analytic obstacles: the lack of smooth tangential vectors to the boundary prevents the use of classical anisotropic Sobolev spaces, and due to the coupling of normal derivatives near corners, one can not follow the usual way to estimate the normal derivatives of solutions from the equations. To overcome these difficulties, a new class of anisotropic Sobolev spaces $H^m_*(\Omega)$ is introduced to treat corner geometries. Within this framework, the well-posedness theory is obtained for both linear and nonlinear problems of the compressible ideal MHD equations with the impermeable and perfectly conducting boundary conditions. The associated linearized problem is studied in several steps: first one deduces the existence of weak solutions by using a duality argument in high order tangential spaces, then verifies that it is indeed a strong solution by several smoothing procedures preserving traces to get a weak-strong uniqueness result, afterwards the estimates of normal derivatives are obtained by combining the structure of MHD equations with the Helmholtz-type decomposition for both of velocity and magnetic fields.