English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

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

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