Wellposedness and stability results for the Navier-Stokes equations in ${\mathbf R}^{3}$

Details Wellposedness and stability results for the Navier-Stokes equations in ${\mathbf R}^{3}$ Wellposedness and stability results for the Navier-Stokes equations in ${\mathbf R}^{3}$

2006-11-02

arxiv.org/abs/math/0611044v2

Mathematics

Analysis of PDEs

28 pages misprints corrected

Scientific paper

In a previous work, we presented a class of initial data to the three dimensional, periodic, incompressible Navier-Stokes equations, generating a global smooth solution although the norm of the initial data may be chosen arbitrarily large. The aim of this article is twofold. First, we adapt the construction to the case of the whole space: we prove that if a certain nonlinear function of the initial data is small enough, in a Koch-Tataru type space, then there is a global solution to the Navier-Stokes equations. We provide an example of initial data satisfying that nonlinear smallness condition, but whose norm is arbitrarily large in $ C^{-1}$. Then we prove a stability result on the nonlinear smallness assumption. More precisely we show that the new smallness assumption also holds for linear superpositions of translated and dilated iterates of the initial data, in the spirit of a construction by the authors and H. Bahouri, thus generating a large number of different examples.

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