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

35 pages