A global existence result for the anisotropic magnetohydrodynamical systems

Abstract

We study an anisotropic system arising in magnetohydrodynamics (MHD) in the whole space R^3 , in the case where there are no diffusivity in the vertical direction and only a small diffusivity in the horizontal direction (of size $\epsilon$$\alpha$ with 0 \textless{} $\alpha$ $\le$ $\alpha$0, for some $\alpha$0 \textgreater{} 0). We prove the local existence and uniqueness of a strong solution and then, using Strichartz-type estimates, we prove that this solution globally exists in time for large initial data, when the rotation is fast enough.