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On a Serrin-type regularity criterion for the Navier-Stokes equations in terms of the pressure. (English) Zbl 1131.35060
The author proves a regularity result for a weak solution (in the sense of Leray-Schauder) of the Navier-Stokes system posed in the space-time region $ \Omega \times (0,T)$, where $T>0$ and $\Omega $ is a domain of $\Bbb R^{n}$, $ n\geq 3$. Homogeneous Dirichlet boundary conditions are imposed on the lateral boundary $\partial \Omega \times (0,T)$. Here, the author restricts the study to the cases where $\Omega $ is the whole $\Bbb R^{n}$ or in a spatially periodic situation. He extends a previous result by {\it L. C. Berselli} and {\it G. P. Galdi} [Proc. Am. Math. Soc. 130, No. 12, 3585--3595 (2002; Zbl 1075.35031)]. Assuming that $u\in L^{\infty }([0,T];L^{2}(\Bbb R^{n}))\cap L^{2}([0,T];H_{0}^{1}(\Bbb R^{n}))$ is a strong solution of the Navier-Stokes system and that the gradient of the pressure belongs to $L_{x,t}^{r,s}$, with $n/r+2/s\leq 3$, and $n/3<r<+\infty $ and $ 2/3<s<+\infty $, the author proves that $u$ is a smooth solution of the Navier-Stokes system. The proof of the regularity result is obtained using some properties of the spaces $L_{x,t}^{r,s}$, through Sobolev embeddings, and interpolation tools.

35Q30Stokes and Navier-Stokes equations
35B65Smoothness and regularity of solutions of PDE
76D03Existence, uniqueness, and regularity theory
76D05Navier-Stokes equations (fluid dynamics)
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