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Homogeneity criterion for the Navier-Stokes equations in the whole spaces. (English) Zbl 0982.35081

The Cauchy problem for the nonstationary Navier-Stokes system is considered in n ×(0,), n2,

v t-Δv+v·v+p=0,·v=0inx n ,t>0,v(x,0)=a(x)·

Here, v(x,t) is the vector of velocity of the liquid, p(x,t) is the pressure.

It is proved that the problem has a unique small regular solution in the homogeneous Besov space B ˙ p, -1+n/p ( n ) and in a homogeneous space M ^ n ( n ) which contains the Morrey-type space of measures appeared in Y. Giga and T. Miyakawa [Commun. Partial Differ. Equations 14, 577-618 (1989; Zbl 0681.35072)]. These results imply the existence of small forward self-similar solutions to the Navier-Stokes equations. The uniqueness of solution in C([0,);L n ( n )) is shown, too.

MSC:
35Q30Stokes and Navier-Stokes equations
76D03Existence, uniqueness, and regularity theory
76D05Navier-Stokes equations (fluid dynamics)