Global well-posedness and exponential decay of 2D nonhomogeneous Navier-Stokes and magnetohydrodynamic equations with density-dependent viscosity and vacuum

TL;DR

This paper proves the global existence, uniqueness, and exponential decay of strong solutions for 2D nonhomogeneous magnetohydrodynamic and Navier-Stokes equations with density-dependent viscosity, allowing vacuum and without compatibility conditions.

Contribution

It establishes the first global well-posedness and decay results for these equations with vacuum and density-dependent viscosity in 2D bounded domains.

Findings

Global existence of strong solutions under smallness condition on initial data

Exponential decay rates of solutions over time

No compatibility condition needed despite vacuum presence

Abstract

We establish global well-posedness of strong solutions for the nonhomogeneous magnetohydrodynamic equations with density-dependent viscosity and initial density allowing vanish in two-dimensional (2D) bounded domains. Applying delicate energy estimates and Desjardins' interpolation inequality, we derive the global existence of a unique strong solution provided that $\|\nabla\mu(\rho_0)\|_{L^q}$ is suitably small. Moreover, we also obtain exponential decay rates of the solution. In particular, there is no need to impose some compatibility condition on the initial data despite the presence of vacuum. As a direct application, it is shown that the similar result also holds for the nonhomogeneous Navier-Stokes equations with density-dependent viscosity.