Existence theorem and regularity criteria for the generalized MHD equations

In this paper, we study the generalized MHD equations with fractional dissipative terms (−Δ)αu(−Δ)αu and (−Δ)βb(−Δ)βb respectively for the velocity field and magnetic field. We consider the special case α=βα=β of generalized MHD equations in R3R3, prove the existence of local in time solution and set up certain kind of regularity criteria for weak solutions which improve some known regularity conditions. The tools we mainly use are Littlewood–Paley theory and Bony’s paraproduct decomposition.