三维微极Rayleigh-Bénard对流方程在Triebel-Lizorkin空间中的正则性准则

Regularity Criteria for the Three-Dimensional Micropolar Rayleigh-Bénard Convection Equations in Triebel-Lizorkin Spaces

摘要: 本文考虑三维微极Rayleigh-Bénard对流方程，设$\left(u,\omega ,\theta \right)\left(t,x\right)$ 是方程在$\left(0,T\right)$ 上的Leray-Hopf弱解。证明当压力满足$q>\frac{12}{5}$ ，${\displaystyle {\int }_0^T{\Vert \pi \Vert }_{{\dot{F}}_{q,\frac{10q}{5q+6}\left({ℝ}^3\right)}^0}^{\frac{10q}{5q-6}}}d\tau <\infty$ ，或者压力的梯度满足$\frac{12}{11}<q<4$ ，${\displaystyle {\int }_0^T{\Vert \nabla \pi \Vert }_{{\dot{F}}_{q,\frac{8q}{12-3q}\left({ℝ}^3\right)}^0}^{\frac{8q}{11q-12}}}d\tau <\infty$ 时，解$(u,\omega ,\theta )$ 可以延拓到$t=T$ 。

Abstract: This paper considers the three-dimensional micropolar Rayleigh-Bénard convection equations. We establish regularity criteria for weak solutions, based on the following conditions concerning the pressure: $q>\frac{12}{5}$ , ${\displaystyle {\int }_0^T{\Vert \pi \Vert }_{{\dot{F}}_{q,\frac{10q}{5q+6}\left({ℝ}^3\right)}^0}^{\frac{10q}{5q-6}}}d\tau <\infty$ , or$\frac{12}{11}<q<4$ , ${\displaystyle {\int }_0^T{\Vert \nabla \pi \Vert }_{{\dot{F}}_{q,\frac{8q}{12-3q}\left({ℝ}^3\right)}^0}^{\frac{8q}{11q-12}}}d\tau <\infty$ then the solution can extended beyond $t=T$ .