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Martingale solutions for the compressible MHD systems with stochastic external forces

Huaqiao Wang

Abstract

In this paper we consider the three-dimensional compressible MHD system with stochastic external forces in a bounded domain. We obtain the existence of martingale solution which is a weak solution for the fluid variables, the Brownian motion on a probability space. The construction of the solution is based on the Galerkin approximation method, stopping time, the compactness method and Jakubowski Skorokhod theorem, etc.

Martingale solutions for the compressible MHD systems with stochastic external forces

Abstract

In this paper we consider the three-dimensional compressible MHD system with stochastic external forces in a bounded domain. We obtain the existence of martingale solution which is a weak solution for the fluid variables, the Brownian motion on a probability space. The construction of the solution is based on the Galerkin approximation method, stopping time, the compactness method and Jakubowski Skorokhod theorem, etc.

Paper Structure

This paper contains 24 sections, 23 theorems, 295 equations.

Table of Contents

  1. Introduction
  2. Definition of solutions and assumptions
  3. Main results
  4. Preliminaries
  5. The Faedo-Galerkin approximation and a priori estimates
  6. Approximation scheme and a priori estimates
  7. Tightness Property
  8. Application of Jakubowski-Skorokhod Theorem
  9. Passage to the limit for $n\to\infty$
  10. The Vanishing Viscosity Limit
  11. On the equation $\mathrm{div} v=f$
  12. Uniform (on viscosity) estimates of density
  13. Tightness property
  14. The vanishing viscosity limit
  15. The effective viscous flux
  16. ...and 9 more sections

Key Result

Theorem 1.2

Suppose that the initial data $(\rho_0, m_0,B_0)$ satisfies G-initialdata and assumption holds. Let $\gamma>3/2$. If $D\subset\mathbb{R}^3$ is a bounded domain of class $C^{2+\alpha}$ with $\alpha>0$, then there exists a martingale solution $\left((\Omega,\mathscr{F},\textrm{P}),\mathscr{F}_t,\beta_ to the problem MHD-boundary for any given $T>0$ and for all $1\le p<\infty$.

Theorems & Definitions (36)

  • Definition 1.1
  • Theorem 1.2
  • Lemma 2.1
  • Proposition 3.1
  • proof
  • Remark 3.2
  • Lemma 3.3
  • Proposition 3.4
  • proof
  • proof : Proof of Lemma \ref{['p-timelemma']}
  • ...and 26 more