Profile decomposition for solutions of the Navier-Stokes equations

摘要

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. Gerard (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.