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Global weak solutions with higher regularity to the two-dimensional isentropic compressible Navier-Stokes and magnetohydrodynamic equations with far-field vacuum and unbounded density

Shuai Wang, Xin Zhong

Abstract

We establish the global existence of a class of weak solutions to the isentropic compressible Navier-Stokes and magnetohydrodynamic (MHD) equations on the whole plane under a suitably small initial energy. The solutions constructed here admit far-field vacuum and unbounded densities. Moreover, they possess an intermediate regularity regime between the finite-energy weak solutions of Lions-Feireisl and the framework of Hoff. This particularly extends our previous half-plane case with Dirichlet boundary conditions (arXiv:2601.11852) to the whole-plane MHD coupling, and we also generalize the works of Hoff (Comm. Pure Appl. Math. 55 (2002), pp. 1365-1407) and Suen and Hoff (Arch. Ration. Mech. Anal. 205 (2012), pp. 27-58) by allowing vacuum states and unbounded density. Our analysis lies in a new perspective that exploits the spatial integrability of the density and the resulting integrability of the pressure, together with the specific structure of the MHD system.

Global weak solutions with higher regularity to the two-dimensional isentropic compressible Navier-Stokes and magnetohydrodynamic equations with far-field vacuum and unbounded density

Abstract

We establish the global existence of a class of weak solutions to the isentropic compressible Navier-Stokes and magnetohydrodynamic (MHD) equations on the whole plane under a suitably small initial energy. The solutions constructed here admit far-field vacuum and unbounded densities. Moreover, they possess an intermediate regularity regime between the finite-energy weak solutions of Lions-Feireisl and the framework of Hoff. This particularly extends our previous half-plane case with Dirichlet boundary conditions (arXiv:2601.11852) to the whole-plane MHD coupling, and we also generalize the works of Hoff (Comm. Pure Appl. Math. 55 (2002), pp. 1365-1407) and Suen and Hoff (Arch. Ration. Mech. Anal. 205 (2012), pp. 27-58) by allowing vacuum states and unbounded density. Our analysis lies in a new perspective that exploits the spatial integrability of the density and the resulting integrability of the pressure, together with the specific structure of the MHD system.
Paper Structure (9 sections, 15 theorems, 152 equations)

This paper contains 9 sections, 15 theorems, 152 equations.

Table of Contents

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

Key Result

Theorem 1.1

Let the assumptions 1.6, 1.8, and 1.10 be satisfied. Then there is a positive constant $\varepsilon$ depending only on $\alpha, \hat{\rho}, M, a, \gamma, \mu, \lambda, \nu$, and $\eta_0$ such that if the problem a1--a3 admits a global weak solution $(\rho,\mathbf{u},\mathbf{B})$ in the sense of Definition d1.1 satisfying, for any $0<T<\infty$, with $\sigma=\sigma(t)\triangleq\min\{1,t\}$, and f

Theorems & Definitions (29)

  • Definition 1.1
  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • ...and 19 more