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Global Strong Solutions to the incompressible Magnetohydrodynamic Equations with Density-Dependent Viscosity and Vacuum in 3D Exterior Domains

Bing Yuan, Rong Zhang, Peng Zhou

TL;DR

This work proves global existence and decay for the 3D nonhomogeneous incompressible MHD equations with density-dependent viscosity in exterior domains, allowing vacuum in the density. A key hypothesis is $B_0\in L^p$ with $1\le p<12/7$, which yields enhanced decay and enables global strong solutions when the initial velocity and magnetic-field gradients are small. The authors develop a comprehensive energy-based framework, including time-weighted a priori estimates for $u$, $B$, and their time derivatives, and establish precise large-time behavior in the exterior setting with slip and perfect-conductor boundary conditions. The results extend previous Cauchy-domain findings to exterior domains with vacuum, and also cover the constant-viscosity special case, highlighting optimal decay rates for the magnetic field.

Abstract

The nonhomogeneous incompressible Magnetohydrodynamic Equations with density-dependent viscosity is studied in three-dimensional (3D) exterior domains with slip boundary conditions. The key is the constraint of an additional initial value condition $B_0\in L^p (1\leqslant p<12/7)$, which increase decay-in-time rates of the solutions, thus we obtain the global existence of strong solutions provided the gradient of the initial velocity and initial magnetic field is suitably small. In particular, the initial density is allowed to contain vacuum states and large oscillations. Moreover, the large-time behavior of the solution is also shown.

Global Strong Solutions to the incompressible Magnetohydrodynamic Equations with Density-Dependent Viscosity and Vacuum in 3D Exterior Domains

TL;DR

This work proves global existence and decay for the 3D nonhomogeneous incompressible MHD equations with density-dependent viscosity in exterior domains, allowing vacuum in the density. A key hypothesis is with , which yields enhanced decay and enables global strong solutions when the initial velocity and magnetic-field gradients are small. The authors develop a comprehensive energy-based framework, including time-weighted a priori estimates for , , and their time derivatives, and establish precise large-time behavior in the exterior setting with slip and perfect-conductor boundary conditions. The results extend previous Cauchy-domain findings to exterior domains with vacuum, and also cover the constant-viscosity special case, highlighting optimal decay rates for the magnetic field.

Abstract

The nonhomogeneous incompressible Magnetohydrodynamic Equations with density-dependent viscosity is studied in three-dimensional (3D) exterior domains with slip boundary conditions. The key is the constraint of an additional initial value condition , which increase decay-in-time rates of the solutions, thus we obtain the global existence of strong solutions provided the gradient of the initial velocity and initial magnetic field is suitably small. In particular, the initial density is allowed to contain vacuum states and large oscillations. Moreover, the large-time behavior of the solution is also shown.
Paper Structure (4 sections, 18 theorems, 162 equations)

This paper contains 4 sections, 18 theorems, 162 equations.

Table of Contents

  1. Introduction
  2. Auxiliary lemma
  3. A priori estimates
  4. Proof of Theorem 1.1-1.2

Key Result

Theorem 1.1

Let $\Omega$ be the exterior of a simply connected bounded domain $D$ in $\mathbb{R}^3$ and its boundary $\partial\Omega$ is smooth. For constants $\bar{\rho}>0, q \in(3, \infty)$, $p\in[1,12/7)$, assume that the initial data $\left(\rho_0, m_0, B_0\right)$ satisfy Then for there exists some small positive constant $\varepsilon_0$ depending only on $p, q, \bar{\rho}, \underline{\mu}, \bar{\mu},

Theorems & Definitions (31)

  • Theorem 1.1
  • Theorem 1.2
  • Remark 1.1
  • Remark 1.2
  • Remark 1.3
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Lemma 2.5
  • ...and 21 more