Local existence results in Sobolev spaces for generalized magnetohydrodynamics equations

Abstract

We consider generalized magnetohydrodynamics (MHD) equations in with fractional diffusions and for the velocity and magnetic field , respectively, where are nonnegative constants and is the fractional Laplacian of order . Using the smoothing effect of for any , we establish local existence results for the generalized MHD equations when the initial data belong to with possibly different orders and . Here, denotes the standard ‐based Sobolev space on of order . To prove our existence results, we also derive new product and commutator estimates in fractional Sobolev spaces which are of independent interest in themselves.