Chaotic behavior of solutions of the Navier-Stokes system in $\Bbb R^N$

Abstract

In this paper, we study the relationship between the long time behavior of a solution $u(t,x)$ of the Navier-Stokes system with no external force in $\mathbb R^N $ and the asymptotic behavior as $|x|\to \infty $ of its initial value $u_0$. In particular, for initial values $u_0$ small in a certain sense, we show that if the sequence of dilations $\lambda _n u_0(\lambda _n\cdot)$ converges weakly to $z(\cdot)$ as $\lambda _n\to \infty $, then the rescaled solution $\sqrt{t} u(t,\cdot\sqrt t)$ converges uniformly on $\mathbb R^N $ to ${{\boldsymbol{\mathcal S}}}(1)z$ along the subsequence $t_n=\lambda _n^2$, where ${{\boldsymbol{\mathcal S}}}(t)$ is the Navier-Stokes flow. If $N=2$ or $3$, we show there exists an initial value $U_0$ such that the set of all possible $z$ attainable in this fashion is a closed ball $B$ of an infinite-dimensional space. The resulting ``universal" solution is therefore asymptotically close along appropriate subsequences to all solutions with initial values in $B$. Moreover, for a fixed $t_0>0$, ${{\boldsymbol{\mathcal S}}}(t_0)$ followed by an appropriate dilation generates a chaotic discrete dynamical system.