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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).

35Q30 Navier-Stokes equations
35B40 Asymptotic behavior of solutions to PDEs
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