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# Stability by rescaled weak convergence for the Navier-Stokes equations

1 EDP
LAMA - Laboratoire d'Analyse et de Mathématiques Appliquées
Abstract : 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 $(u_{0, n})_{n\in \N}$ of initial data, bounded in some scaling invariant space, converges weakly to an initial data $u_0$ which generates a global regular solution, does $u_{0, n}$ generate a global regular solution ? A positive answer in general to this question would imply global regularity for any data, through the following examples~$u_{0,n} = n \vf_0(n\cdot)$ or~$u_{0,n} = \vf_0(\cdot-x_n)$ 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.
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https://hal.archives-ouvertes.fr/hal-00868384
Contributor : Isabelle Gallagher <>
Submitted on : Tuesday, October 1, 2013 - 12:20:14 PM
Last modification on : Wednesday, December 9, 2020 - 3:45:17 AM
Long-term archiving on: : Thursday, January 2, 2014 - 6:25:40 AM

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### Identifiers

• HAL Id : hal-00868384, version 1
• ARXIV : 1310.0256

### Citation

Hajer Bahouri, Jean-Yves Chemin, Isabelle Gallagher. Stability by rescaled weak convergence for the Navier-Stokes equations. 2013. ⟨hal-00868384⟩

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