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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)].

##### MSC:
 35Q30 Navier-Stokes equations 76D05 Navier-Stokes equations for incompressible viscous fluids 35D05 Existence of generalized solutions of PDE (MSC2000)
##### Keywords:
smallness conditions; global solution
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##### References:
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