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Morrey space techniques applied to the interior regularity problem of the Navier-Stokes equations. (English) Zbl 1022.35036

A new criterion of the interior regularity to the weak solutions of the evolutionary Navier-Stokes equations is formulated in the framework of Morrey spaces. Namely, the following theorem is proved: Let 7 2<q5, (t 0 ,x 0 )(0,T)×Ω, 0<r 0 <dist(Ω,x 0 ) and 0<t 0 -r 0 2 with Ω the boundary of Ω. Suppose that u is a weak solution of equation

u/t-Δu+·(uu)+π=0in(0,T)×Ω
divu=0in(0,T)×Ω·

Then u is regular in (t 0 -r 1 2 ,t 0 ]×B r 1 (x 0 ) for r 1 =r 0 /2 provided that

supr 1-5/q t-r 2 t |x-y|<r |u(s,y)| q dyds 1/q <ϵ

where the constant ϵ is sufficiently small and the supremum is taken over all (t-r 2 ,t]×B r (x)(t 0 -r 0 2 ,t 0 ]×B r 0 (x 0 )·

MSC:
35Q30Stokes and Navier-Stokes equations
76D03Existence, uniqueness, and regularity theory