Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On physically grounded boundary conditions for the compressible MHD system

Jan Brezina, Eduard Feireisl

TL;DR

This work addresses the challenge of defining physically meaningful boundary conditions for the compressible MHD system on bounded domains with heterogeneous media by employing a penalization strategy for transport coefficients and studying singular limits. The authors establish weak solutions for the primitive system, derive a robust energy framework, and prove convergence to limit problems that realize boundary types such as PEC, PMC, and isolator conditions as transmission limits. The main contributions are the rigorous justification of various boundary regimes within a unified asymptotic framework and the resulting weak formulations, which provide a solid foundation for numerical simulations on geometrically complex domains. Overall, the results offer a principled approach to boundary modeling in compressible MHD and pave the way for reliable computational experiments in challenging geometries.

Abstract

We consider a general compressible MHD system, where the magnetic field propagates in a heterogeneous medium. Using suitable penalization in terms of the transport coefficients we perform several singular limits. As a result we obtain: 1. A rigorous justification of physically grounded boundary conditions for the compressible MHD system on a bounded domain. 2. Existence of weak solutions for arbitrary finite energy initial data in the situation the Maxwell induction equation holds also outside the fluid domain. 3. A suitable theoretical platform for numerical experiments on domains with geometrically complicated boundaries.

On physically grounded boundary conditions for the compressible MHD system

TL;DR

This work addresses the challenge of defining physically meaningful boundary conditions for the compressible MHD system on bounded domains with heterogeneous media by employing a penalization strategy for transport coefficients and studying singular limits. The authors establish weak solutions for the primitive system, derive a robust energy framework, and prove convergence to limit problems that realize boundary types such as PEC, PMC, and isolator conditions as transmission limits. The main contributions are the rigorous justification of various boundary regimes within a unified asymptotic framework and the resulting weak formulations, which provide a solid foundation for numerical simulations on geometrically complex domains. Overall, the results offer a principled approach to boundary modeling in compressible MHD and pave the way for reliable computational experiments in challenging geometries.

Abstract

We consider a general compressible MHD system, where the magnetic field propagates in a heterogeneous medium. Using suitable penalization in terms of the transport coefficients we perform several singular limits. As a result we obtain: 1. A rigorous justification of physically grounded boundary conditions for the compressible MHD system on a bounded domain. 2. Existence of weak solutions for arbitrary finite energy initial data in the situation the Maxwell induction equation holds also outside the fluid domain. 3. A suitable theoretical platform for numerical experiments on domains with geometrically complicated boundaries.
Paper Structure (21 sections, 2 theorems, 75 equations, 1 figure)

This paper contains 21 sections, 2 theorems, 75 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Problem with smooth transport coefficients
  3. Weak formulation of the primitive system
  4. Energy balance
  5. Existence and regularity of the weak solutions
  6. Penalization of transport coefficients
  7. Singular limit in the fluid equations
  8. Singular limit in the equation of continuity
  9. Singular limit in the momentum equation
  10. Main results
  11. Weak form of the limit induction equation
  12. Perfect electric isolator
  13. Perfect magnetic conductor (PMC)
  14. Perfect electric conductor (PEC)
  15. Isolator type boundary conditions
  16. ...and 6 more sections

Key Result

Proposition 3.1

Under the scaling Z3, Z4, Z8, we have passing to a suitable subsequence as the case may be. The limit satisfies the equation of continuity for any $\varphi \in C^1(\mathbb{T}^d)$, and the momentum equation for any $0 \leq \tau \leq T$ and any $\boldsymbol{\varphi} \in C^1_c(\Omega_F; R^d)$.

Figures (1)

  • Figure :

Theorems & Definitions (5)

  • Definition 2.1: Weak solutions to the primitive system
  • Remark 2.2
  • Definition 2.3: Energy weak solution
  • Proposition 3.1
  • Theorem 4.1: Singular limits