UNIQUE WEAK SOLUTIONS OF THE NON-RESISTIVE MAGNETOHYDRODYNAMIC EQUATIONS WITH FRACTIONAL DISSIPATION

This paper examines the uniqueness of weak solutions to the d-dimensional magnetohydrodynamic (MHD) equations with the fractional dissipation (-Delta)(alpha)u and without the magnetic diffusion. Important progress has been made on the standard Laplacian dissipation case alpha = 1. This paper discovers that there are new phenomena with the case alpha < 1. The approach for alpha = 1 can not be directly extended to alpha < 1. We establish that, for alpha < 1, any initial data (u(0),b(0)) in the inhomogeneous Besov space B-2,infinity(sigma)(R-d) with sigma > 1 + d/2 - alpha leads to a unique local solution. For the case alpha >= 1, u(0) in the homogeneous Besov space (B) over circle (1+d/2-2 alpha)(2,1) (R-d) and b(0) in c (B) over circle (1+d/2-2 alpha)(2,1) (R-d) guarantees the existence and uniqueness. These regularity requirements appear to be optimal.