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Regularity criterion via the pressure on weak solutions to the 3D Navier-Stokes equations. (English) Zbl 1126.35047

The authors consider a Leray-Hopf weak solution (u,p) of the Navier-Stokes equations in 3 ×(0,T). They prove that this solution is regular provided that the initial velocity u 0 belongs to L 2 ( 3 )L q ( 3 ) for some q>3 and that the pressure satisfies

0 T p(t) B ˙ , 0 dt<·

The main tool in obtaining this result is an a-priori-estimate of the form

sup 0tT u(t) L s C(u 0 L s +(CT) 1 s +e) exp(C 0 T p(t) B ˙ , 0 dt) ,3<s4,

which is derived with the help of the Paley-Littlewood decomposition.

35Q30Stokes and Navier-Stokes equations
76D03Existence, uniqueness, and regularity theory
35B65Smoothness and regularity of solutions of PDE