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Regularity criteria in terms of pressure for the 3-D Navier-Stokes equations in a generic domain. (English) Zbl 1054.35062
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 $$(0,T)$$.

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