Wellposedness and stability results for the Navier-Stokes equations in \(\mathbb R^3\). (English) Zbl 1165.35038

Summary: In [J.-Y. Chemin and I. Gallagher, Ann. Sci. Éc. Norm. Supér. (4) 39, No. 4, 679–698 (2006; Zbl 1124.35052)] a class of initial data to the three dimensional, periodic, incompressible Navier-Stokes equations was presented, 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 of [loc. cit] 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 [H. Koch and D. Tataru, Adv. Math. 157, No. 1, 22–35 (2001; Zbl 0972.35084)] 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 in [H. Bahouri, J.-Y. Chemin and I. Gallagher, Ann. Sc. Norm. Super. Pisa, Cl. Sci. (5) 5, No. 3, 375–391 (2006; Zbl 1121.43006)], thus generating a large number of different examples.


35Q30 Navier-Stokes equations
76D05 Navier-Stokes equations for incompressible viscous fluids
35B35 Stability in context of PDEs
Full Text: DOI arXiv EuDML


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