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 , where and is a domain of , . Homogeneous Dirichlet boundary conditions are imposed on the lateral boundary . Here, the author restricts the study to the cases where is the whole 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 is a strong solution of the Navier-Stokes system and that the gradient of the pressure belongs to , with , and and , the author proves that is a smooth solution of the Navier-Stokes system. The proof of the regularity result is obtained using some properties of the spaces , through Sobolev embeddings, and interpolation tools.