Global Existence and Large Time Behavior of Strong Solutions for 3D Nonhomogeneous Heat-Conducting Magnetohydrodynamic Equations

Abstract

We are concerned with an initial boundary value problem of nonhomogeneous heat-conducting magnetohydrodynamic equations in a bounded simply connected smooth domain $\Omega \subseteq {\mathbb {R}}^3$, with Navier-slip boundary conditions for the velocity and magnetic field and Neumann boundary condition for the temperature. We prove the global existence of a unique strong solution provided that $\left( \Vert \sqrt{\rho _0}{\mathbf {u}}_0\Vert _{L^2}^2 +\Vert {\mathbf {b}}_0\Vert _{L^2}^2\right)$ $\left( \Vert {{\,\mathrm{curl}\,}}{\mathbf {u}}_0\Vert _{L^2}^2 +\Vert {{\,\mathrm{curl}\,}}{\mathbf {b}}_0\Vert _{L^2}^2\right)$ is suitably small. Moreover, we also obtain large time decay rates of the solution.