ENERGY CONSERVATION AND REGULARITY FOR THE 3D MAGNETO-HYDRODYNAMICS EQUATIONS

This paper studies the energy conservation and regularity problems for the 3D magneto-hydrodynamics (MHD) equations. We first establish some uniform bounds on some invariant quantities in terms of suitable weak solution (u; b) is an element of L-2,L-infinity (0, T; BMO(Omega)). As the applications, first, we show that as the solution (u; b) approaches a finite blowup time T, the BMO norm must blow up at a rate p c/root T-t with some absolute constant c > 0. Then, a regularity criteria for suitable weak solutions is proved which allows the vertical part of the velocity and magnetic to be large under the norm of L-2,L-infinity ([-1, 0]; BMO(R-3)). Finally, we prove that any suitable weak solution of the MHD equations in L(2,infinity)1(0, T; BMO (Omega)) satisfies the local energy equality for any bounded Lipschitz domain Omega subset of R-3. As a corollary, we prove that any suitable weak solution of MHD equations in L-2,L-infinity (0; T; BMOloc(R-3)) satisfies the energy equality.