Profile decomposition for solutions of the Navier-Stokes equations

Isabelle Gallagher

Bulletin de la Société Mathématique de France (2001)

  • Volume: 129, Issue: 2, page 285-316
  • ISSN: 0037-9484

Abstract

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We consider sequences of solutions of the Navier-Stokes equations in  ℝ 3 , associated with sequences of initial data bounded in  H ˙ 1 / 2 . We prove, in the spirit of the work of H.Bahouri and P.Gérard (in the case of the wave equation), that they can be decomposed into a sum of orthogonal profiles, bounded in  H ˙ 1 / 2 , up to a remainder term small in  L 3 ; the method is based on the proof of a similar result for the heat equation, followed by a perturbation–type argument. If  𝒜 is an “admissible” space (in particular  L 3 , B ˙ p , ∞ - 1 + 3 / p for  p < + ∞ or  ∇ B M O ), and if  ℬ N S 𝒜 is the largest ball in  𝒜 centered at zero such that the elements of  H ˙ 1 / 2 ∩ ℬ N S 𝒜 generate global solutions, then we obtain as a corollary ana prioriestimate for those solutions. We also prove that the mapping from data in  H ˙ 1 / 2 ∩ ℬ N S 𝒜 to the associate solution is Lipschitz.

How to cite

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Gallagher, Isabelle. "Profile decomposition for solutions of the Navier-Stokes equations." Bulletin de la Société Mathématique de France 129.2 (2001): 285-316. <http://eudml.org/doc/272321>.

@article{Gallagher2001,
abstract = {We consider sequences of solutions of the Navier-Stokes equations in $\{\mathbb \{R\}\}^3$, associated with sequences of initial data bounded in $\dot\{H\}^\{1/2\}$. We prove, in the spirit of the work of H.Bahouri and P.Gérard (in the case of the wave equation), that they can be decomposed into a sum of orthogonal profiles, bounded in $\dot\{H\}^\{1/2\}$, up to a remainder term small in $L^3$; the method is based on the proof of a similar result for the heat equation, followed by a perturbation–type argument. If $\{\mathcal \{A\}\}$ is an “admissible” space (in particular $L^3$, $\dot\{B\}^\{-1+3/p\}_\{p,\infty \}$ for $p &lt; +\infty $ or $\nabla \{B\hspace\{-0.55542pt\}M\hspace\{-0.55542pt\}O\} $), and if $\{\mathcal \{B\}\}_\{_\{\!\{N\hspace\{-1.111pt\}S\}\}\}^\{\{\mathcal \{A\}\}\}$ is the largest ball in $\{\mathcal \{A\}\}$ centered at zero such that the elements of $\dot\{H\}^\{1/2\}\cap \{\mathcal \{B\}\}_\{_\{\!\{N\hspace\{-1.111pt\}S\}\}\}^\{\{\mathcal \{A\}\}\}$ generate global solutions, then we obtain as a corollary ana prioriestimate for those solutions. We also prove that the mapping from data in $\dot\{H\}^\{1/2\}\cap \{\mathcal \{B\}\}_\{_\{\!\{N\hspace\{-1.111pt\}S\}\}\}^\{\{\mathcal \{A\}\}\}$ to the associate solution is Lipschitz.},
author = {Gallagher, Isabelle},
journal = {Bulletin de la Société Mathématique de France},
keywords = {Navier–Stokes; explosion; profiles; a priori estimate; admissible space},
language = {eng},
number = {2},
pages = {285-316},
publisher = {Société mathématique de France},
title = {Profile decomposition for solutions of the Navier-Stokes equations},
url = {http://eudml.org/doc/272321},
volume = {129},
year = {2001},
}

TY - JOUR
AU - Gallagher, Isabelle
TI - Profile decomposition for solutions of the Navier-Stokes equations
JO - Bulletin de la Société Mathématique de France
PY - 2001
PB - Société mathématique de France
VL - 129
IS - 2
SP - 285
EP - 316
AB - We consider sequences of solutions of the Navier-Stokes equations in ${\mathbb {R}}^3$, associated with sequences of initial data bounded in $\dot{H}^{1/2}$. We prove, in the spirit of the work of H.Bahouri and P.Gérard (in the case of the wave equation), that they can be decomposed into a sum of orthogonal profiles, bounded in $\dot{H}^{1/2}$, up to a remainder term small in $L^3$; the method is based on the proof of a similar result for the heat equation, followed by a perturbation–type argument. If ${\mathcal {A}}$ is an “admissible” space (in particular $L^3$, $\dot{B}^{-1+3/p}_{p,\infty }$ for $p &lt; +\infty $ or $\nabla {B\hspace{-0.55542pt}M\hspace{-0.55542pt}O} $), and if ${\mathcal {B}}_{_{\!{N\hspace{-1.111pt}S}}}^{{\mathcal {A}}}$ is the largest ball in ${\mathcal {A}}$ centered at zero such that the elements of $\dot{H}^{1/2}\cap {\mathcal {B}}_{_{\!{N\hspace{-1.111pt}S}}}^{{\mathcal {A}}}$ generate global solutions, then we obtain as a corollary ana prioriestimate for those solutions. We also prove that the mapping from data in $\dot{H}^{1/2}\cap {\mathcal {B}}_{_{\!{N\hspace{-1.111pt}S}}}^{{\mathcal {A}}}$ to the associate solution is Lipschitz.
LA - eng
KW - Navier–Stokes; explosion; profiles; a priori estimate; admissible space
UR - http://eudml.org/doc/272321
ER -

References

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