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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. GĂ©rard (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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