Advances in Differential Equations

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

Thierry Cazenave, Flávio Dickstein, and Fred B. Weissler

Full-text: Access denied (no subscription detected) We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

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.

Article information

Source
Adv. Differential Equations Volume 10, Number 4 (2005), 361-398.

Dates
First available in Project Euclid: 18 December 2012

Permanent link to this document
https://projecteuclid.org/euclid.ade/1355867870

Mathematical Reviews number (MathSciNet)
MR2122695

Subjects
Primary: 35Q30: Navier-Stokes equations [See also 76D05, 76D07, 76N10]
Secondary: 35B40: Asymptotic behavior of solutions 37D45: Strange attractors, chaotic dynamics 37L99: None of the above, but in this section 37N10: Dynamical systems in fluid mechanics, oceanography and meteorology [See mainly 76-XX, especially 76D05, 76F20, 86A05, 86A10] 76D03: Existence, uniqueness, and regularity theory [See also 35Q30]

Citation

Cazenave, Thierry; Dickstein, Flávio; Weissler, Fred B. Chaotic behavior of solutions of the Navier-Stokes system in $\Bbb R^N$. Adv. Differential Equations 10 (2005), no. 4, 361--398. https://projecteuclid.org/euclid.ade/1355867870.


Export citation