A Global Existence Result for the Anisotropic Rotating Magnetohydrodynamical Systems

Abstract

In this article, we study an anisotropic rotating system arising in magnetohydrodynamics (MHD) in the whole space R 3 , in the case where there are no diffusivity in the vertical direction and a vanishing diffusivity in the horizontal direction (when the rotation goes to infinity). We first prove the local existence and uniqueness of a strong solution and then, using Strichartz-type estimates, we prove that this solution exists globally in time for large initial data, when the rotation is fast enough.1991 Mathematics Subject Classification. 76D03; 76D05; 76U05, 76W05.

On the fast rotation asymptotics of a non-homogeneous incompressible MHD system

“…The case of rotating MHD equations has also received some attention in the past years. Once again, most of the available results concern the case of homogeneous flows: for instance, we mention papers [10,30], concerning the stability of boundary layers in homogeneous rotating MHD, and [27], about the stabilising effect the rotation has on solution lifespan. See also references therein, as well as chapter 10 of [6], for further references.…”

“…This justifies the use of an electrostatic approximation in the Maxwell equations, which are simplified by omitting the time derivative of the electric field. Obviously, this is not a wild assumption since we intend to work on planetary or stellar fluids subject to the body's rotation (see also [10], [27]). Thus, Ampère's circuital law reads…”

“…where we have used also equation (27) in passing from the first to the second line. By virtue of the uniform bounds we have already obtained on σ , j , η , j and f , j , we see that…”