2025/04/06 by Hairong Liu, Tao Luo, Liu, Hairong +3 · 1 citation
Mathematics · #Analysis of PDEs (math.AP) #FOS: Mathematics #Gas Dynamics and Kinetic Theory #Navier-Stokes equation solutions #Nonlinear Partial Differential Equations
paper · pdf · doi:10.48550/arxiv.2504.04376
openalex publication_date 2025/04/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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.