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

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.

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