Regularity criterion for weak solution to the 3D micropolar fluid equations (Q642764)

Summary: Regularity criterion for the 3D micropolar fluid equations is investigated. We prove that, for some \(T>0\), if \(\int^T_0 \|v_{x_3}\|^\rho_{L^\rho}\,dt<\infty\), where \(3/\rho+2/\rho\leq 1\) and \(\rho\geq 3\), then the solution \((v,w)\) can be extended smoothly beyond \(t=T\). The derivative \(v_{x_3}\) can be substituted with any directional derivative of \(v\).