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Blow-up rate estimates for weak solutions of the Navier-Stokes equations. (English) Zbl 1009.76015
Summary: We investigate the interior regularity of Leray weak solutions u of Navier-Stokes equations in a domain Ω n with n3. It is shown that u is regular in a neighbourhood of a point (x 0 ,t 0 )Ω×(0,T) if there exist constants 0θ<1 and small ε>0 such that lim k esssup Q 1/k (x 0 ,t 0 ) |t-t 0 | θ/2 |x-x 0 | 1-θ |u(x,t)|<ε with Q 1/k (x 0 ,t 0 )={x n ;|x-x 0 |<1/k}×(t 0 -1/k 2 ,t 0 +1/k 2 ). If (x 0 ,t 0 ) is an irregular point of u, there exists a sequence of non-zero measure sets E k i Q 1/k i (x 0 ,t 0 ) for i=1,2,, such that the blow-up rate estimate |u(x,t)|ε|t-t 0 | -θ/2 |x-x 0 | -1+θ , (x,t)E k i holds.
76D03Existence, uniqueness, and regularity theory
35Q30Stokes and Navier-Stokes equations
76D05Navier-Stokes equations (fluid dynamics)