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Zbl 0982.35081
Chen, Zhi Min; Xin, Zhouping
Homogeneity criterion for the Navier-Stokes equations in the whole spaces. (English)
[J] J. Math. Fluid Mech. 3, No.2, 152-182 (2001). ISSN 1422-6928; ISSN 1422-6952/e

The Cauchy problem for the nonstationary Navier-Stokes system is considered in $\bbfR^n\times (0,\infty)$, $n\ge 2$, $$\align &{\partial v\over\partial t}-\Delta v+ v\cdot\nabla v+\nabla p= 0,\quad\nabla\cdot v= 0\quad\text{in }x\in\bbfR^n,\quad t>0,\\ & v(x,0)= a(x).\endalign$$ Here, $v(x, t)$ is the vector of velocity of the liquid, $p(x, t)$ is the pressure.\par It is proved that the problem has a unique small regular solution in the homogeneous Besov space $\dot B^{-1+ n/p}_{p,\infty}(\bbfR^n)$ and in a homogeneous space $\widehat M_n(\bbfR^n)$ 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 $C([0,\infty); L_n(\bbfR^n))$ is shown, too.
[Il'ya Sh.Mogilevskij (Tver)]

MSC 2000:: *35Q30 Stokes and Navier-Stokes equations
76D03 Existence, uniqueness, and regularity theory
76D05 Navier-Stokes equations (fluid dynamics)

Keywords: nonstationary Navier-Stokes equations; existence; uniqueness; Cauchy problem; homogeneous Besov space; small forward self-similar solutions

Citations: Zbl 0681.35072

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