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The Cauchy problem for Navier-Stokes equations. (English) Zbl 0880.35091
The 3D Cauchy problem for the Navier-Stokes system is considered \[ \frac{\partial u}{\partial t}-\nu\Delta u+(u\cdot\nabla )u +\nabla p=f,\qquad \text{div} u=0\quad \text{in}\quad \mathbb{R}^3\times (0,T) \]
\[ u(x,0)=a(x),\quad x\in \mathbb{R}^3, \quad u(x,t)\to 0\quad \text{as}\quad |x|\to\infty \] It is proved that the strong generalized solution (from \(H^1\)) of the problem exists and is unique for sufficiently small \(T\). The author proves that if the data of the problem \(a\) and \(f\) are sufficiently small then the solution is global, i.e. \(T=+\infty\). The smallness conditions in the present paper are weaker than the similar conditions in H. Beirão da Veiga [Indiana Univ. Math. J. 36, 149-166 (1987; Zbl 0601.35093)].

35Q30 Navier-Stokes equations
76D05 Navier-Stokes equations for incompressible viscous fluids
35D05 Existence of generalized solutions of PDE (MSC2000)
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