Regularity criteria in terms of pressure for the 3-D Navier-Stokes equations in a generic domain. (English) Zbl 1054.35062

Zhou, Yong

Math. Ann. 328, No. 1-2, 173-192 (2004).

The initial boundary value problem is considered in \(\Omega\times(0,T)\) \[ \begin{aligned} &\frac{\partial v}{\partial t}-\Delta v+ (v\cdot\nabla )v +\nabla p=0,\quad \nabla\cdot v=0\;\;\text{in\;} \Omega\times(0,T)\\ &v=0 \quad \text{on\;}\partial\Omega\times(0,T)\\ &v(x,0)=0\quad \text{in\;} \Omega.\end{aligned} \] Here \(\Omega\subset \mathbb{R}^3\) is the half-space \(\mathbb{R}^3_+\)or a bounded domain with smooth boundary, or an exterior domain with smooth boundary.
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 \((0,T)\).

Reviewer: Il’ya Sh. Mogilevskij (Tver)

MSC:

35Q30	Navier-Stokes equations
35B65	Smoothness and regularity of solutions to PDEs
76D03	Existence, uniqueness, and regularity theory for incompressible viscous fluids

Keywords:

Navier-Stokes equations; regularity criterion; integrability of pressure