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

TL;DR

This paper proves the existence and uniqueness of weak solutions for the d-dimensional micropolar equations with fractional dissipation, establishing optimal regularity conditions in Besov spaces for different dissipation parameters.

Contribution

It extends the theory of micropolar equations by including fractional dissipation and determines optimal Besov space conditions for solution uniqueness.

Findings

Unique weak solutions exist for specified fractional dissipation parameters.

Optimal Besov space regularity indices are identified for solution uniqueness.

2D micropolar equations with standard Laplacian dissipation have unique solutions in B^0_{2,1}.

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.