Unique weak solutions of the non-resistive magnetohydrodynamic equations with fractional dissipation

Abstract

This paper examines the uniqueness of weak solutions to the d-dimensional magnetohydrodynamic (MHD) equations with the fractional dissipation $(-Δ)^αu$ and without the magnetic diffusion. Important progress has been made on the standard Laplacian dissipation case $α=1$. This paper discovers that there are new phenomena with the case $α<1$. The approach for $α=1$ can not be directly extended to $α<1$. We establish that, for $α<1$, any initial data $(u_0, b_0)$ in the inhomogeneous Besov space $B^σ_{2,\infty}(\mathbb R^d)$ with $σ> 1+\frac{d}{2}-α$ leads to a unique local solution. For the case $α\ge 1$, $u_0$ in the homogeneous Besov space $\mathring B^{1+\frac{d}{2}-2α}_{2,1}(\mathbb R^d)$ and $b_0$ in $ \mathring B^{1+\frac{d}{2}-α}_{2,1}(\mathbb R^d)$ guarantees the existence and uniqueness. These regularity requirements appear to be optimal.

35 pages