Global solutions to the two-dimensional compressible MHD equations with density-dependent bulk viscosity

Abstract

We study the two-dimensional periodic resistive compressible magnetohydrodynamic equations with constant shear viscosity and magnetic diffusivity, bulk viscosity $\rho^\beta$, and pressure $\rho^\gamma$. For every $\beta>1$ and $\gamma>1$, we construct global finite-energy renormalized weak solutions from strictly positive bounded $W^{1,q}$ densities, $q>2$, and arbitrary $H^1$ velocity and magnetic data. The construction combines approximation-independent finite density moments, generalized effective-flux identities, and a joint logarithmic renormalization of the density and its entropy defect. An augmented relative energy containing the $\beta$-power density potential yields weak--strong uniqueness against positive $H^2$ strong solutions for the same full pressure range. We also establish global strong existence for a large-background family when $\gamma>\max\{5-2\beta,\beta+1\}$. Writing $m=\int\rho_0$, the density perturbation is bounded in the scaled norm $m^{(\gamma-3)/2}\|\rho_0-m\|_{H^2}$, while the fixed $H^2$ velocity and magnetic norms need not be small. For sufficiently large $m$, frequency-dependent acoustic dissipation, staged vorticity weights, and a separate initial-layer flux estimate close a simultaneous bootstrap. The admissible pressure exponents have infimum $7/3$, with the endpoint excluded. For $\beta>2$ the relevant damping rate is the minimum of the viscous and pressure-relaxation rates; keeping both rates extends the argument without an upper restriction on $\beta$. The strong theorem has no upper restriction on $\gamma$ under this scaled density hypothesis; for fixed additive density perturbations it applies in the stated lower range up to $\gamma=3$.