Non-uniform decay of MHD equations with and without magnetic diffusion
  We consider the long time behavior of solutions to the magnetohydrodynamics equations in two and three spatial dimensions. It is shown that in the absence of magnetic diffusion, if strong bounded solutions were to exist their energy cannot present any asymptotic oscillatory behavior, the diffusivity of the velocity is enough to prevent such oscillations. When magnetic diffusion is present and the data is only in L^2, it is shown that the solutions decay to zero without a rate, and this nonuniform decay is optimal. 
Ruben Agapito, Maria Schonbek