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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 ${ℝ}^{n}×\left(0,\infty \right)$, $n\ge 2$,

$\begin{array}{cc}& \frac{\partial v}{\partial t}-{\Delta }v+v·\nabla v+\nabla p=0,\phantom{\rule{1.em}{0ex}}\nabla ·v=0\phantom{\rule{1.em}{0ex}}\text{in}\phantom{\rule{4.pt}{0ex}}x\in {ℝ}^{n},\phantom{\rule{1.em}{0ex}}t>0,\hfill \\ & v\left(x,0\right)=a\left(x\right)·\hfill \end{array}$

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

It is proved that the problem has a unique small regular solution in the homogeneous Besov space ${\stackrel{˙}{B}}_{p,\infty }^{-1+n/p}\left({ℝ}^{n}\right)$ and in a homogeneous space ${\stackrel{^}{M}}_{n}\left({ℝ}^{n}\right)$ 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 $C\left(\left[0,\infty \right);{L}_{n}\left({ℝ}^{n}\right)\right)$ is shown, too.

##### MSC:
 35Q30 Stokes and Navier-Stokes equations 76D03 Existence, uniqueness, and regularity theory 76D05 Navier-Stokes equations (fluid dynamics)