Profile decomposition for solutions of the Navier-Stokes equations

Abstract

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 ∇BMO), and if ℬ NS 𝒜 is the largest ball in 𝒜 centered at zero such that the elements of H ˙ 1/2 ∩ℬ NS 𝒜 generate global solutions, then we obtain as a corollary an a priori estimate for those solutions. We also prove that the mapping from data in H ˙ 1/2 ∩ℬ NS 𝒜 to the associate solution is Lipschitz.

Locations

• Bulletin de la Société mathématique de France - View - PDF