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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 $\Bbb R^n\times (0,\infty)$, $n\ge 2$, $$\align &{\partial v\over\partial t}-\Delta v+ v\cdot\nabla v+\nabla p= 0,\quad\nabla\cdot v= 0\quad\text{in }x\in\Bbb R^n,\quad t>0,\\ & v(x,0)= a(x).\endalign$$ 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 $\dot B^{-1+ n/p}_{p,\infty}(\Bbb R^n)$ and in a homogeneous space $\widehat M_n(\Bbb R^n)$ which contains the Morrey-type space of measures appeared in {\it Y. Giga} and {\it 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 $C([0,\infty); L_n(\Bbb R^n))$ is shown, too.

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