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 $(-\Delta)^\alpha u$ 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 $B^\sigma_{2,\infty}(\mathbb R^d)$ with $\sigma> 1+\frac{d}{2}-\alpha$ leads to a unique local solution. For the case $\alpha\ge 1$, $u_0$ in the homogeneous Besov space $\mathring B^{1+\frac{d}{2}-2\alpha}_{2,1}(\mathbb R^d)$ and $b_0$ in $ \mathring B^{1+\frac{d}{2}-\alpha}_{2,1}(\mathbb R^d)$ guarantees the existence and uniqueness. These regularity requirements appear to be optimal.