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Incompressible Limit of Compressible Ideal MHD Flows inside a Perfectly Conducting Wall

Jiawei Wang, Junyan Zhang

TL;DR

The paper establishes the incompressible limit of compressible ideal MHD flows in a domain with a perfectly conducting wall, addressing severe regularity obstacles when the magnetic field is tangent to the boundary. It introduces space-time anisotropic Sobolev spaces and a Mach-number weighted energy framework that exploits a hidden structure in the Lorentz force to trade a normal derivative for two tangential derivatives, yielding uniform-in-$\varepsilon$ estimates. The authors prove local well-posedness with uniform energy bounds and rigorously justify the limit to incompressible MHD with a transported entropy, under well-prepared data, and extend the approach to the 2D setting. The results bridge compressible and incompressible MHD in wall-bound settings and point toward applications to free-boundary problems in ideal MHD.

Abstract

We prove the incompressible limit of compressible ideal magnetohydrodynamic(MHD) flows in a reference domain where the magnetic field is tangential to the boundary. Unlike the case of transversal magnetic fields, the linearized problem of our case is not well-posed in standard Sobolev space $H^m~(m\geq 2)$, while the incompressible problem is still well-posed in $H^m$. The key observation to overcome the difficulty is a hidden structure contributed by Lorentz force in the vorticity analysis, which reveals that one should trade one normal derivative for two tangential derivatives together with a gain of Mach number weight $\varepsilon^2$. Thus, the energy functional should be defined by using suitable anisotropic Sobolev spaces. The weights of Mach number should be carefully chosen according to the number of tangential derivatives, such that the energy estimates are uniform in Mach number. Besides, part of the proof is similar to the study of compressible water waves, so our result opens the possibility to study the incompressible limit of free-boundary problems in ideal MHD.

Incompressible Limit of Compressible Ideal MHD Flows inside a Perfectly Conducting Wall

TL;DR

The paper establishes the incompressible limit of compressible ideal MHD flows in a domain with a perfectly conducting wall, addressing severe regularity obstacles when the magnetic field is tangent to the boundary. It introduces space-time anisotropic Sobolev spaces and a Mach-number weighted energy framework that exploits a hidden structure in the Lorentz force to trade a normal derivative for two tangential derivatives, yielding uniform-in- estimates. The authors prove local well-posedness with uniform energy bounds and rigorously justify the limit to incompressible MHD with a transported entropy, under well-prepared data, and extend the approach to the 2D setting. The results bridge compressible and incompressible MHD in wall-bound settings and point toward applications to free-boundary problems in ideal MHD.

Abstract

We prove the incompressible limit of compressible ideal magnetohydrodynamic(MHD) flows in a reference domain where the magnetic field is tangential to the boundary. Unlike the case of transversal magnetic fields, the linearized problem of our case is not well-posed in standard Sobolev space , while the incompressible problem is still well-posed in . The key observation to overcome the difficulty is a hidden structure contributed by Lorentz force in the vorticity analysis, which reveals that one should trade one normal derivative for two tangential derivatives together with a gain of Mach number weight . Thus, the energy functional should be defined by using suitable anisotropic Sobolev spaces. The weights of Mach number should be carefully chosen according to the number of tangential derivatives, such that the energy estimates are uniform in Mach number. Besides, part of the proof is similar to the study of compressible water waves, so our result opens the possibility to study the incompressible limit of free-boundary problems in ideal MHD.
Paper Structure (29 sections, 12 theorems, 185 equations)

This paper contains 29 sections, 12 theorems, 185 equations.

Table of Contents

  1. Introduction
  2. The equation of state and sound speed
  3. An overview of previous results
  4. The main theorems
  5. Organization of the paper
  6. Difficulties and strategies
  7. Choice of function spaces
  8. Key observation: hidden structure of Lorentz force in the vorticity analysis
  9. Reduction of pressure and design of energy functional
  10. Further applications of our method
  11. Uniform energy estimates
  12. $L^2$ estimate
  13. Reduction of normal derivatives
  14. Reduction of pressure
  15. Div-Curl analysis
  16. ...and 14 more sections

Key Result

Theorem 1.1

Let $\varepsilon\in(0,1)$ be fixed. Let $(u_0, B_0, \rho_0,S_0)\in H^8(\Omega)\times H^8(\Omega)\times H^8(\Omega)\times H^8(\Omega)$ be the initial data of CMHD3 satisfying the compatibility conditions comp cond up to 7-th order and for some $M>0$ independent of $\varepsilon$. Then there exists $T>0$ depending only on $M$, such that CMHD3 admits a unique solution $(p(t),u(t),B(t),S(t))$ that ver

Theorems & Definitions (29)

  • Remark 1.1
  • Theorem 1.1: Local well-posedness and uniform-in-$\varepsilon$ estimates
  • Remark 1.2: Correction of $E_4(t)$
  • Remark 1.3: "Prepared" initial data
  • Remark 1.4: Weights of Mach number of $p$
  • Remark 1.5: Relations with anisotropic Sobolev space
  • Remark 1.6: Choice of regularity
  • Remark 1.7: Compatibility conditions and regularity of initial data
  • Theorem 1.2: Incompressible limit
  • Remark 1.8: The space for the convergence of initial data
  • ...and 19 more