Unique weak solutions of the magnetohydrodynamic equations with fractional dissipation

This paper examines the existence and uniqueness of weak solutions to the d-dimensional magnetohydrodynamic (MHD) equations with fractional dissipation (-Delta)(alpha)upsilon and fractional magnetic diffusion (-Delta)(beta)b. The aim is at the uniqueness of weak solutions in the weakest possible inhomogeneous Besov spaces. We establish the local existence and uniqueness in the functional setting u is an element of L-infinity(0T;B-2,1(d/2-2 alpha+1)(R-d)) and b is an element of L-infinity (0,T;B-2,1(D/2) R-d))when alpha > 1/2, beta >= 0 and alpha + beta >= 1. The case when alpha = 1 with nu > 0 and eta = 0 has previously been studied in [7, 19]. However, their approaches can not be directly extended to the fractional case when alpha < 1 due to the breakdown of a bilinear estimate. By decomposing the bilinear term into different frequencies, we are able to obtain a suitable upper bound on the bilinear term for alpha < 1, which allows us to close the estimates in the aforementioned Besov spaces.