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In this paper, we consider a layer of a viscous incompressible electricallyconducting fluid interacting with the magnetic filed in a horizontally periodicsetting. The upper boundary bounded by a free boundary and below bounded by aflat rigid interface. We prove the global well-posedness of the problem forboth the case with and without surface tension. Moreover, we show that theglobal solution decays to the equilibrium exponentially in the case withsurface tension, however the global solution decays to the equilibrium at analmost exponential rate in the case without surface tension.

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