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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 \(\mathbb R^n\times (0,\infty)\), \(n\geq 2\), \[ \begin{aligned} &{\partial v\over\partial t}-\Delta v+ v\cdot\nabla v+\nabla p= 0,\quad\nabla\cdot v= 0\quad\text{in }x\in\mathbb R^n,\quad t>0,\\ & v(x,0)= a(x).\end{aligned} \] 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 \(\dot B^{-1+ n/p}_{p,\infty}(\mathbb R^n)\) and in a homogeneous space \(\widehat M_n(\mathbb R^n)\) 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 \(C([0,\infty); L_n(\mathbb R^n))\) is shown, too.

MSC:
35Q30 Navier-Stokes equations
76D03 Existence, uniqueness, and regularity theory for incompressible viscous fluids
76D05 Navier-Stokes equations for incompressible viscous fluids
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