## Profile decomposition for solutions of the Navier-Stokes equations.(English)Zbl 0987.35120

Summary: 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 $$p < +\infty$$ or $$\nabla \text{BMO}$$), and if $${\mathcal B}_{NS}^{{\mathcal A}}$$ is the largest ball in $${\mathcal A}$$ centered at zero such that the elements of $$\dot H^{1/2}\cap {\mathcal B}_{NS}^{{\mathcal A}}$$ 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 $$\dot H^{1/2}\cap {\mathcal B}_{NS}^{{\mathcal A}}$$ to the associate solution is Lipschitz.

### MSC:

 35Q30 Navier-Stokes equations 76D05 Navier-Stokes equations for incompressible viscous fluids 35B45 A priori estimates in context of PDEs
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