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 $B_{2,\infty }^{\sigma }({\mathbb{R}}^d)$ 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 ${\mathring{B}}_{2,1}^{1+\frac{d}{2}-2\alpha }({\mathbb{R}}^d)$ and $b_0$ in ${\mathring{B}}_{2,1}^{1+\frac{d}{2}-\alpha }({\mathbb{R}}^d)$ guarantees the existence and uniqueness. These regularity requirements appear to be optimal.

arXiv comment35 pages

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