ON THE STABILITY OF GLOBAL SOLUTIONS TO THE THREE DIMENSIONAL NAVIER-STOKES EQUATIONS

Hajer Bahouri (1) , Jean-Yves Chemin (2) , Isabelle Gallagher (3, 4)

1 LAMA - Laboratoire d'Analyse et de Mathématiques Appliquées
2 LJLL - Laboratoire Jacques-Louis Lions
3 DMA - Département de Mathématiques et Applications - ENS Paris
4 UPD7 - Université Paris Diderot - Paris 7

We prove a weak stability result for the three-dimensional homogeneous incom-pressible Navier-Stokes system. More precisely, we investigate the following problem : if a sequence (u_{0,n})_{ n∈\in N} of initial data, bounded in some scaling invariant space, converges weakly to an initial data u0 which generates a global smooth solution, does u0,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} = nϕ0(n·) or u_{0,n}) = ϕ0(· − x_n) with |x_n| → ∞. 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.

Domaines: Equations aux dérivées partielles [math.AP]

Hajer Bahouri, Jean-Yves Chemin, Isabelle Gallagher. ON THE STABILITY OF GLOBAL SOLUTIONS TO THE THREE DIMENSIONAL NAVIER-STOKES EQUATIONS. Journal de l'École polytechnique — Mathématiques, 2018, ⟨10.5802/jep.84⟩. ⟨hal-01710490v2⟩