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Asymptotic behavior of global solutions to the Navier-Stokes equations in $$\mathbb{R}^3$$. (English) Zbl 0910.35096
The author considers the Cauchy problem to the Navier-Stokes equations $\frac{\partial u}{\partial t}-\Delta u+(u\cdot\nabla)u + \nabla p=0,\qquad \text{div }u=0,\quad x\in \mathbb{R}^3,\;t>0,\tag{1}$
$u(x,0)=u_0(x),\qquad x\in \mathbb{R}^3.$ It is proved that if the norm of $$u_0$$ in Besov space is sufficiently small then the problem has a global unique solution. Another result is related to the asymptotic behavior of the above solution. Let $$u(x,t)$$ be a solution of the problem (1) and $$v(x,t)= \lim_{\lambda \to\infty}\lambda u(\lambda x,\lambda^2 t)$$ (it corresponds to replacing $$x$$ by $$x/\sqrt{t}$$ and tending $$t\to\infty$$). If $$v(x,t)$$ is the solution of the Navier-Stokes system with initial data $$v_0(x)=\lim_{\lambda \to\infty}\lambda u(\lambda x,0)$$, then $$v(x,t)$$ is a self-similar solution. The following result is proved: If $$\sqrt{t} u(\sqrt{t}x,t)\to V(x)$$ in $$L^p$$ as $$t\to\infty$$ for $$p\in(3,+\infty)$$ then $$v(x,t)= (1/\sqrt{t})V(x/\sqrt{t})$$ is a self-similar solution of (1).

##### MSC:
 35Q30 Navier-Stokes equations 35B40 Asymptotic behavior of solutions to PDEs
##### Keywords:
Cauchy problem; self-similar solution
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