Unique weak solutions of the d-dimensional micropolar equation with fractional dissipation

Abstract

This article examines the existence and uniqueness of weak solutions to the d-dimensional micropolar equations ($d=2$ or $d=3$) with general fractional dissipation $(-\Delta)^{\alpha}u$ and $(-\Delta)^{\beta}w$. The micropolar equations with standard Laplacian dissipation model fluids with microstructure. The generalization to include fractional dissipation allows simultaneous study of a family of equations and is relevant in some physical circumstances. We establish that, when $\alpha\ge \frac12$ and $\beta\ge \frac12$, any initial data $(u_0, w_0)$ in the critical Besov space $u_0\in B^{1+\frac{d}{2}-2\alpha}_{2,1}(\mathbb R^d)$ and $w_0\in B^{1+\frac{d}{2}-2\beta}_{2,1}(\mathbb R^d)$ yields a unique weak solution. For $\alpha\ge 1$ and $\beta=0$, any initial data $u_0\in B^{1+\frac{d}{2}-2\alpha}_{2,1}(\mathbb R^d)$ and $w_0\in B^{\frac{d}{2}}_{2,1}(\mathbb R^d)$ also leads to a unique weak solution as well. The regularity indices in these Besov spaces appear to be optimal and can not be lowered in order to achieve the uniqueness. Especially, the 2D micropolar equations with the standard Laplacian dissipation, namely $\alpha=\beta=1$ have a unique weak solution for $(u_0, w_0)\in B^0_{2,1}$. The proof involves the construction of successive approximation sequences and extensive {\it a priori} estimates in Besov space settings.