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Stability of strong solutions to the full compressible magnetohydrodynamic system with non-conservative boundary conditions

Hana Mizerová

Abstract

We define a dissipative measure-valued (DMV) solution to the system of equations governing the motion of a general compressible, viscous, electrically and heat conducting fluid driven by non-conservative boundary conditions. We show the stability of strong solutions to the full compressible magnetohydrodynamic system in a large class of these DMV solutions. In other words, we prove a DMV-strong uniqueness principle: a DMV solution coincides with the strong solution emanating from the same initial data as long as the latter exists.

Stability of strong solutions to the full compressible magnetohydrodynamic system with non-conservative boundary conditions

Abstract

We define a dissipative measure-valued (DMV) solution to the system of equations governing the motion of a general compressible, viscous, electrically and heat conducting fluid driven by non-conservative boundary conditions. We show the stability of strong solutions to the full compressible magnetohydrodynamic system in a large class of these DMV solutions. In other words, we prove a DMV-strong uniqueness principle: a DMV solution coincides with the strong solution emanating from the same initial data as long as the latter exists.

Paper Structure

This paper contains 14 sections, 5 theorems, 95 equations.

Table of Contents

  1. Introduction
  2. Full compressible magnetohydrodynamic model
  3. Dissipative measure-valued solutions
  4. Young measure and initial condition
  5. Definition of a dissipative measure-valued solution
  6. Relative energy inequality
  7. Suitable form of relative energy inequality
  8. DMV-strong uniqueness principle
  9. Conditional DMV-strong uniqueness
  10. Assuming bounded measure-valued solution
  11. Assuming constant coefficients
  12. Assuming perfect gas law
  13. Unconditional DMV-strong uniqueness
  14. Equation of state

Key Result

Lemma 4.1

Let the coefficients $\mu,$$\eta,$$\kappa$ and $\zeta$ be continuously differentiable functions of $(\varrho,\vartheta)\in(0,\infty)^2.$ Let the thermodynamic functions $p$, $e$ and $s$ be continuously differentiable functions of $(\varrho,\vartheta)\in(0,\infty)^2$ satisfying Gibb's equation gibbs

Theorems & Definitions (12)

  • Definition 3.1: Dissipative measure-valued solution
  • Remark 3.2
  • Lemma 4.1: Relative energy inequality for MHD system
  • Theorem 5.1: Conditional result: bounded DMV
  • proof
  • Theorem 5.2: Conditional result: constant coefficients
  • proof
  • Theorem 5.3: Conditional result: perfect gas law
  • proof
  • Theorem 5.4: Unconditional result
  • ...and 2 more