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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 ${\Omega }×\left(0,T\right)$, where $T>0$ and ${\Omega }$ is a domain of ${ℝ}^{n}$, $n\ge 3$. Homogeneous Dirichlet boundary conditions are imposed on the lateral boundary $\partial {\Omega }×\left(0,T\right)$. Here, the author restricts the study to the cases where ${\Omega }$ is the whole ${ℝ}^{n}$ or in a spatially periodic situation. He extends a previous result by L. C. Berselli and G. P. Galdi [Proc. Am. Math. Soc. 130, No. 12, 3585–3595 (2002; Zbl 1075.35031)].

Assuming that $u\in {L}^{\infty }\left(\left[0,T\right];{L}^{2}\left({ℝ}^{n}\right)\right)\cap {L}^{2}\left(\left[0,T\right];{H}_{0}^{1}\left({ℝ}^{n}\right)\right)$ is a strong solution of the Navier-Stokes system and that the gradient of the pressure belongs to ${L}_{x,t}^{r,s}$, with $n/r+2/s\le 3$, and $n/3 and $2/3, 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.

##### MSC:
 35Q30 Stokes and Navier-Stokes equations 35B65 Smoothness and regularity of solutions of PDE 76D03 Existence, uniqueness, and regularity theory 76D05 Navier-Stokes equations (fluid dynamics)