Strong solutions to the 3-D compressible MHD equations with density-dependent viscosities in exterior domains with far-field vacuum

TL;DR

The paper proves local existence and uniqueness of strong solutions to the 3D compressible MHD equations with density-dependent viscosities (specifically $\mu(\rho)=\alpha\rho$, $\lambda(\rho)=\beta\rho$) in exterior domains with far-field vacuum, under Navier-slip boundary conditions for the velocity and perfect conductivity for the magnetic field. It introduces a reformulation using $\phi=\rho^{(\gamma-1)/2}$, $\psi=\nabla\rho/\rho$, and $J=H/\rho$ to manage degeneracy near vacuum and derives uniform in $\sigma$ a priori estimates for a linearized problem with vacuum. A fixed-point/iteration scheme built on these linearized results yields a local strong solution to the reformulated system, and a limiting argument $\sigma\to0$ recovers a local strong solution to the original problem, with vacuum confined to the far field. The work extends known results from Cauchy and bounded-domain settings to 3D exterior domains with vacuum, and establishes uniqueness and regularity within the constructed framework. This advances the mathematical understanding of degenerate MHD systems and provides the foundation for potential global and stability analyses in future work.

Abstract

This paper investigates the existence and uniqueness of local strong solutions to the three-dimensional compressible magnetohydrodynamic (MHD) equations with density-dependent viscosities in an exterior domain. The system models the dynamics of electrically conducting fluids, such as plasmas, and incorporates the effects of magnetic fields on fluid motion. We focus on the case where the viscosity coefficients are proportional to the fluid density, and the far-field density approaches vacuum. By introducing a reformulation of the problem using new variables to handle the degeneracy near vacuum, we establish the local well-posedness of strong solutions for arbitrarily large initial data, even in the presence of far-field vacuum. Our analysis employs energy estimates, elliptic regularity theory, and a careful treatment of the Navier-slip boundary conditions for the velocity and perfect conductivity conditions for the magnetic field. To the best of our knowledge, such results are not available even for the Cauchy problem to the 3-D compressible MHD equations with degenerate viscosities.

Theorems & Definitions (28)

• Definition 1.1
• Theorem 1.2
• Remark 1.3
• Remark 1.4
• Remark 1.5
• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• Lemma 2.4
• Lemma 2.5