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( − Δ ) α u and fractional magnetic diffusion( − Δ ) β 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 ∈ L ∞ ( 0 , T ; B 2 , 1 d / 2 − 2 α + 1( R d ) )and b ∈ L ∞ ( 0 , T ; B 2 , 1 d / 2R d ) )when α > 1 / 2 , β ≥ 0 and α + β ≥ 1 . The case when α = 1 with ν > 0 and η = 0 has previously been studied in [7, 19]. However, their approaches can not be directly extended to the fractional case when α < 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 α < 1 , which allows us to close the estimates in the aforementioned Besov spaces.