Global Well-posedness of Incompressible MHD System with Density-dependent Viscosity in $\mathbb{R}^2$

Abstract

We study the Cauchy problem for the two-dimensional inhomogeneous incompressible Magnetohydrodynamic equations with density-dependent viscosity and vacuum. For initial data with positive mass, finite second moment of density, and zero total momentum, we establish the existence and uniqueness of global strong solutions when the viscosity is sufficiently close to one in $L^\infty$. The smallness threshold may depend on the initial data, while no separate smallness condition is imposed on the initial velocity or magnetic field. We obtain uniform-in-time bounds for the density second moment and its $H^1\cap W^{1,q}$ norm, where $q>2$. Moreover, for every $η\in(0,1)$, $\|\sqrtρu(\cdot,t)\|_{L^2}+\|b(\cdot,t)\|_{L^2}$ decays at the rate $t^{-η/2}$, and the $H^1$ norms of $\nabla u$, $P$, and $\nabla b$ decay at the rate $t^{-(1+η)/2}$. The proof combines a velocity decomposition based on Hardy-space estimates with magnetic-field integrability and time-weighted energy estimates.