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Global weak solutions and incompressible limit of two-dimensional isentropic compressible magnetohydrodynamic equations with ripped density and large initial data

Shuai Wang, Guochun Wu, Xin Zhong

Abstract

We establish the global existence of weak solutions of the isentropic compressible magnetohydrodynamic equations with ripped density in the whole plane provided the bulk viscosity coefficient is properly large. Moreover, we show that such solutions converge globally in time to a weak solution of the inhomogeneous incompressible magnetohydrodynamic equations when the bulk viscosity coefficient tends to infinity. In particular, the initial energy can be arbitrarily large and vacuum states are allowed in interior regions. Our analysis depends on the effective viscous flux and a Desjardins-type logarithmic interpolation inequality as well as structure of the system under consideration. To the best of our knowledge, this paper provides the first incompressible limit of the isentropic compressible magnetohydrodynamic equations for the large bulk viscosity.

Global weak solutions and incompressible limit of two-dimensional isentropic compressible magnetohydrodynamic equations with ripped density and large initial data

Abstract

We establish the global existence of weak solutions of the isentropic compressible magnetohydrodynamic equations with ripped density in the whole plane provided the bulk viscosity coefficient is properly large. Moreover, we show that such solutions converge globally in time to a weak solution of the inhomogeneous incompressible magnetohydrodynamic equations when the bulk viscosity coefficient tends to infinity. In particular, the initial energy can be arbitrarily large and vacuum states are allowed in interior regions. Our analysis depends on the effective viscous flux and a Desjardins-type logarithmic interpolation inequality as well as structure of the system under consideration. To the best of our knowledge, this paper provides the first incompressible limit of the isentropic compressible magnetohydrodynamic equations for the large bulk viscosity.

Paper Structure

This paper contains 10 sections, 13 theorems, 143 equations.

Table of Contents

  1. Introduction
  2. Background and motivation
  3. Main results
  4. Strategy of the proof
  5. Preliminaries
  6. Auxiliary inequalities and facts
  7. Uniform estimates for $F$, $\boldsymbol\omega$, and $\nabla\mathbf{u}$
  8. A priori estimates
  9. Proof of Theorem \ref{['t1.1']}
  10. Proof of Theorem \ref{['t1.2']}

Key Result

Theorem 1.1

Let c1 and c2 be satisfied, there exists a positive number $D$ depending only on $\tilde{\rho}$, $\hat{\rho}$, $a$, $\gamma$, $\nu$, and $\mu$ such that if then the Cauchy problem a1--a3 admits a global weak solution $(\rho,\mathbf{u},\mathbf{B})$ in the sense of Definition d1.1 satisfying where $\sigma\triangleq\min\{1,t\}$.

Theorems & Definitions (26)

  • Definition 1.1
  • Theorem 1.1
  • Theorem 1.2
  • Definition 1.2
  • Remark 1.1
  • Remark 1.2
  • Remark 1.3
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • ...and 16 more