Decay rates for the Viscous Incompressible MHD with and without Surface Tension

Abstract

In this paper, we consider a layer of a viscous incompressible electrically conducting fluid interacting with the magnetic filed in a horizontally periodic setting. The upper boundary bounded by a free boundary and below bounded by a flat rigid interface. We prove the global well-posedness of the problem for both the case with and without surface tension. Moreover, we show that the global solution decays to the equilibrium exponentially in the case with surface tension, however the global solution decays to the equilibrium at an almost exponential rate in the case without surface tension.