Global regularity for solutions to 2D generalized MHD equations with multiple exponential upper bound uniformly in time

Abstract

This paper studies strong solutions to 2D generalized MHD equations with velocity dissipation given by Λ2α and magnetic diffusion given by reducing about double logarithmic diffusion from standard Laplacian diffusion that is given by a Fourier multiplier with its symbol given by|ξ|2/(log⁡log⁡(ek+1+|ξ|)log⁡log⁡log⁡(ek+1+|ξ|)⋯log⁡log⁡⋯log︸k(ek+1+|ξ|)). We prove that there exists a unique global solution in Sobolev spaces having at least a (k+1)-multiple exponential upper bound uniformly in times when α>14 which implies that it is difficult to obtain the global regular solution even for reducing logarithm-type diffusion for magnetic field if the dissipation is small for velocity field. Furthermore, this implies that the well-known global regularity problem on the 2D resistive MHD equations is a critical open problem.