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Regularity criterion via two components of vorticity on weak solutions to the Navier-Stokes equations in 3 . (English) Zbl 1091.35064

The authors give conditions on vorticity guaranteeing the smoothness of weak solutions to the Navier-Stokes system in 3 . Since the pioneering work of Beirao da Veiga, this procedure has been used several times in the work of Kozono, Ogawa, Taniuchi. In a paper of Chae and Choe the conditions of Serrin type are imposed only on two components of vorticity in L q (L r ), while a result of Kozono and Yatsu requires conditions on two components of vorticity in L q (BMO) and r=. In this paper, an analogous result is proved in homogeneous Besov spaces B ˙ r,σ 0 .

The main result reads as follows: Let T>0. Suppose u(t,x) be a weak Leray-Hopf solution to the Navier-Stokes system on 3 ×(0,T) with solenoidal initial value u 0 H 1 ( 3 ). Set w=curlu=[w 1 ,w 2 ,w 3 ],w ˜=[w 1 ,w 2 ,0] and assume that

0 T w ˜ B ˙ r,σ 0 q dt<·

Then u is regular provided 2 q+3 r=2; 3 2<r, σ2r 3.

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