On the stability of global solutions to the three-dimensional Navier-Stokes equations

Bahouri, Hajer; Chemin, Jean-Yves; Gallagher, Isabelle

Journal de l’École polytechnique — Mathématiques, Volume 5 (2018), pp. 843-911.

We prove a weak stability result for the three-dimensional homogeneous incompressible Navier-Stokes system. More precisely, we investigate the following problem: if a sequence ${\left(u_{0,n}\right)}_{n\in \mathbb{N}}$ of initial data, bounded in some scaling invariant space, converges weakly to an initial data $u_0$ which generates a global smooth solution, does $u_{0,n}$ generate a global smooth solution? A positive answer in general to this question would imply global regularity for any data, through the following examples $u_{0,n}=n{\varphi }_0(n·)$ or $u_{0,n}={\varphi }_0(·-x_n)$ with $|x_n|\to \infty$. We therefore introduce a new concept of weak convergence (rescaled weak convergence) under which we are able to give a positive answer. The proof relies on profile decompositions in anisotropic spaces and their propagation by the Navier-Stokes equations.

On démontre un résultat de stabilité faible pour les équations de Navier-Stokes tridimensionnelles, incompressibles et homogènes. Plus précisément on étudie le problème suivant : si une suite de données initiales ${\left(u_{0,n}\right)}_{n\in \mathbb{N}}$, bornée dans un espace invariant d’échelle, converge faiblement vers une donnée $u_0$ qui engendre une solution globale régulière, est-ce que $u_{0,n}$ engendre une solution globale régulière ? Une réponse affirmative à cette question en général aurait pour conséquence la régularité globale pour toute donnée initiale, via les exemples $u_{0,n}=n{\varphi }_0(n·)$ ou $u_{0,n}={\varphi }_0(·-x_n)$ avec $|x_n|\to \infty$. On introduit donc un nouveau concept de convergence faible (convergence faible remise à l’échelle) sous lequel on peut donner une réponse affirmative. La démonstration repose sur des décompositions en profils dans des espaces de régularité anisotrope, et leur propagation par les équations de Navier-Stokes.

Classification: 35Q30, 42B37
Keywords: Navier-Stokes equations, anisotropy, Besov spaces, profile decomposition
Mot clés : Équations de Navier-Stokes, anisotropie, espaces de Besov, décompositions en profils

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