On the regularity criterion of weak solution for the 3D viscous Magneto-hydrodynamics equations

Here u, b describe the flow velocity vector and the magnetic field vector respectively, p is a scalar pressure, ν > 0 is the kinematic viscosity, η > 0 is the magnetic diffusivity, while u0 and b0 are the given initial velocity and initial magnetic field with ∇ · u0 = ∇ · b0 = 0. If ν = η = 0, (1.1) is called the ideal MHD equations. As same as the 3D Navier-Stokes equations, the regularity of weak solution for the 3D MHD equations remains open[17]. For the 3D Navier-Stokes equations, the Serrin-type criterion states that a Leray-Hopf weak solution u is regular provided the following condition holds[1, 9, 11, 16, 19]: