Global well-posedness to the 3D nonhomogeneous magnetohydrodynamic equations with density-dependent viscosity and large initial velocity

Abstract

We investigate the global well-posedness of strong solutions to the three-dimensional (3D) nonhomogeneous magnetohydrodynamic equations with density-dependent viscosity and vacuum. Combined energy method with the structure of the system under consideration, we obtain global existence and uniqueness of strong solutions provided that $\Vert \rho _0\Vert _{L^1}+\Vert {\mathbf {b}}_0\Vert _{L^2}$ is suitably small. In particular, the initial velocity can be arbitrarily large. Moreover, we also derive exponential decay rates of the solution. As a direct application, we show global strong solutions for the 3D nonhomogeneous Navier–Stokes equations with density-dependent viscosity as long as the initial mass is properly small. This work improves Liu’s result (Z Angew Math Phys 70:Paper No. 107, 2019), where the author requires the smallness condition on $\Vert \rho _0\Vert _{L^\infty }+\Vert {\mathbf {b}}_0\Vert _{L^3}$. Moreover, we also extend the local strong solutions obtained by Song (Z Angew Math Phys 69:Paper No. 23, 2018) to be a global one.