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Zbl 1054.35062
Zhou, Yong
Regularity criteria in terms of pressure for the 3-D Navier-Stokes equations in a generic domain. (English)
[J] Math. Ann. 328, No. 1-2, 173-192 (2004). ISSN 0025-5831; ISSN 1432-1807/e

The initial boundary value problem is considered in $\Omega\times(0,T)$ $$\align &\frac{\partial v}{\partial t}-\Delta v+ (v\cdot\nabla )v +\nabla p=0,\quad \nabla\cdot v=0\ \ \text{in\ } \Omega\times(0,T)\\ &v=0 \quad \text{on\ }\partial\Omega\times(0,T)\\ &v(x,0)=0\quad \text{in\ } \Omega.\endalign$$ Here $\Omega\subset \bbfR^3$ is the half-space $\bbfR^3_+$or a bounded domain with smooth boundary, or an exterior domain with smooth boundary. \par It is proved that if $v(x,t)$ is a Leray-Hopf weak solution of the problem and $p(x,t)$ or $\nabla p(x,t)$ satisfies to certain conditions of integrability then $v(x,t)$ is a smooth solution in $(0,T)$.
[Il'ya Sh. Mogilevskij (Tver)]

MSC 2000:: *35Q30 Stokes and Navier-Stokes equations
35B65 Smoothness of solutions of PDE
76D03 Existence, uniqueness, and regularity theory

Keywords: Navier-Stokes equations; regularity criterion; integrability of pressure

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Zentralblatt MATH Berlin [Germany]