Global strong solutions of three-dimensional heat conducting incompressible magnetohydrodynamic equations with vacuum

Abstract

This paper is devoted to studying global existence and large time behavior of strong solutions of the three-dimensional heat conducting incompressible magnetohydrodynamic flows with initial vacuum and a vacuum far field. Under the scaling invariant quantity (1+‖ρ0‖∞)2(‖ρ0u0‖22+‖b0‖22)(‖∇u0‖22+‖∇b0‖22) is suitably small, the global well-posedness of strong solution to the Cauchy problem is proved. Here, the explicit dependence on the initial norms is given and the smallness depends only on the known parameters in the system. This implies that the global well-posedness result is also valid for some large initial data. Furthermore, the algebraic decay rates of the global solution are obtained.