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

Wen Guo, Ya-Guang Wang

TL;DR

This work addresses local well-posedness for the initial-boundary value problem of the two-dimensional compressible ideal MHD equations in domains with corners, where corner singularities obstruct standard energy methods. The authors develop corner-adapted anisotropic Sobolev spaces $H^m_*(\Omega)$ and a linearized framework around a smooth background, obtaining rigorous tangential and normal derivative estimates via a Div-Curl–Helmholtz approach and Lax-Phillips duality for weak solutions. They upgrade weak solutions to strong ones through smoothing while preserving traces, establish weak-strong uniqueness, and prove local well-posedness for the nonlinear problem via a Picard iteration, under impermeable and perfectly conducting boundary conditions and corner-angle restrictions $\omega_n<\pi/[\frac{m}{2}]$ (for $m\ge 7$ linear, $m\ge 8$ nonlinear). The results provide a rigorous foundation for hyperbolic IBVPs with corner geometries and coupled velocity–magnetic fields, with implications for plasmas in polygonal domains and related confinement configurations.

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 $H^m_*(Ω)$ 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.

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

TL;DR

This work addresses local well-posedness for the initial-boundary value problem of the two-dimensional compressible ideal MHD equations in domains with corners, where corner singularities obstruct standard energy methods. The authors develop corner-adapted anisotropic Sobolev spaces and a linearized framework around a smooth background, obtaining rigorous tangential and normal derivative estimates via a Div-Curl–Helmholtz approach and Lax-Phillips duality for weak solutions. They upgrade weak solutions to strong ones through smoothing while preserving traces, establish weak-strong uniqueness, and prove local well-posedness for the nonlinear problem via a Picard iteration, under impermeable and perfectly conducting boundary conditions and corner-angle restrictions (for linear, nonlinear). The results provide a rigorous foundation for hyperbolic IBVPs with corner geometries and coupled velocity–magnetic fields, with implications for plasmas in polygonal domains and related confinement configurations.

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 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.

Paper Structure

This paper contains 21 sections, 49 theorems, 340 equations.

Table of Contents

  1. Introduction
  2. A priori estimates for the linearized MHD equations
  3. Preliminaries
  4. Properties of the anisotropic Sobolev spaces $H^m_*$
  5. Formulation of linearized problems
  6. A priori estimates of solutions to the linearized problem
  7. Estimates of tangential derivatives
  8. Estimates of normal derivatives
  9. Linearized problem with a smooth background state
  10. The associated boundary value problem
  11. Well-posedness in tangential spaces
  12. Normal regularity
  13. Compatibility lifting
  14. Proof of Theorem \ref{['thm-LIBVP']}
  15. Linearized problem with a non-smooth background state
  16. ...and 6 more sections

Key Result

Theorem 1.1

For any fixed integer $m\ge 8$, assume that the angle at the corner $q_n$ satisfies for each $1\le n \le N$ with the notation $[\cdot]$ denoting the integer part of a number, and the initial data given in icd, $(u_0, b_0, p_0, s_0) \in H^{m}(\Omega)$ satisfies the compatibility conditions of the problem MHD-bcd up to order $m-1$ on $\partial\Omega\backslash\{q_1,\ldots, q_N\}$. The

Theorems & Definitions (94)

  • Definition 1.1: Tangential vector fields $w^1$ and $w^2$
  • Definition 1.2
  • Remark 1.1
  • Theorem 1.1
  • Remark 1.2: Regularity of initial data
  • Remark 1.3: Angle assumptions
  • Lemma 2.1
  • Remark 2.1
  • Lemma 2.2
  • proof
  • ...and 84 more