Local Existence of Strong Solutions to the Generalized MHD Equations

Abstract

This paper devotes to consider the local existence of the strong solutions to the generalized MHD system with fractional dissipative terms $Λ^{2α}u$ for the velocity field and $Λ^{2α}b$ for the magnetic field, respectively. We construct the approximate solutions by the Fourier truncation method, and use energy method to obtain the local existence of strong solutions in $H^s (\mathbb{R}^n)$ $(s > max \{\frac{n}{2} + 1 − 2α, 0\})$ for any $α ≥ 0.$