Global existence and large time asymptotic behavior of strong solutions to the Cauchy problem of 2D density-dependent magnetohydrodynamic equations with vacuum

Abstract

This paper concerns the Cauchy problem of the nonhomogeneous incompressible Magnetohydrodynamic equations on the whole two-dimensional space with vacuum as far-field density. We establish the global existence and uniqueness of strong solutions to the 2D Cauchy problem provided that the initial density and the initial magnetic field decay not too slow at infinity. In particular, the initial data can be arbitrarily large and the initial density can contain vacuum states and even have compact support. Furthermore, we also obtain the large time decay rates of the gradients of velocity, magnetic field and pressure.