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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.

MSC:

35Q30 Navier-Stokes equations
76D05 Navier-Stokes equations for incompressible viscous fluids
35B35 Stability in context of PDEs
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References:

[1] Auscher, P.; Dubois, S.; Tchamitchian, P., On the stability of global solutions to navier – stokes equations in the space, Journal de mathématiaques pures et appliquées, 83, 673-697, (2004) · Zbl 1107.35096
[2] Bahouri, H.; Chemin, J.-Y.; Gallagher, I., Refined Hardy inequalities, Annali di scuola normale di Pisa, classe di scienze, serie V, 5, 375-391, (2006) · Zbl 1121.43006
[3] Bahouri, H.; Gérard, P., High frequency approximation of solutions to critical nonlinear wave equations, American journal of mathematics, 121, 131-175, (1999) · Zbl 0919.35089
[4] M. Cannone, Y. Meyer, F. Planchon, Solutions autosimilaires des équations de Navier-Stokes, Séminaire “Équations aux Dérivées Partielles” de l’École polytechnique, Exposé VIII, 1993-1994
[5] Chemin, J.-Y., Fluides parfaits incompressibles, Astérisque, vol. 230, (1998), Oxford University Press, English translation: J.-Y. Chemin, Perfect Incompressible Fluids
[6] Chemin, J.-Y., Théorèmes d’unicité pour le système de navier – stokes tridimensionnel, Journal d’analyse mathématique, 77, 27-50, (1999) · Zbl 0938.35125
[7] J.-Y. Chemin, Localization in Fourier space and Navier-Stokes system, in: Phase Space Analysis of Partial Differential Equations, Proceedings 2004, CRM series, Pubbl. Cent. Ric. Mat. Ennio Giorgi, Scuola Normale di Pisa, pp. 53-136
[8] Chemin, J.-Y.; Gallagher, I., On the global wellposedness of the 3-D navier – stokes equations with large initial data, Annales scientifiques de l’école normale supérieure de Paris, 39, 4, 679-698, (2006) · Zbl 1124.35052
[9] Foias, C.; Saut, J.-C., Asymptotic behaviour, as \(t \rightarrow \infty\), of solutions of the navier – stokes equations and nonlinear spectral manifolds, Indiana mathematical journal, 33, 3, 459-477, (1984) · Zbl 0565.35087
[10] Fujita, H.; Kato, T., On the navier – stokes initial value problem I, Archive for rational mechanics and analysis, 16, 269-315, (1964) · Zbl 0126.42301
[11] Gallagher, I., The tridimensional navier – stokes equations with almost bidimensional data: stability, uniqueness and life span, International mathematical research notices, 18, 919-935, (1997) · Zbl 0893.35098
[12] Gallagher, I., Profile decomposition for the navier – stokes equations, Bulletin de la société mathématique de France, 129, 285-316, (2001) · Zbl 0987.35120
[13] Gallagher, I.; Iftimie, D.; Planchon, F., Asymptotics and stability for global solutions to the navier – stokes equations, Annales de l’institut Fourier, 53, 5, 1387-1424, (2003) · Zbl 1038.35054
[14] Gérard, P., Description du défaut de compacité de l’injection de Sobolev, ESAIM contrôle optimal et calcul des variations, 3, 213-233, (1998)
[15] Giga, Y.; Miyakawa, T., Solutions in \(L^r\) of the navier – stokes initial value problem, Archive for rational mechanics and analysis, 89, 3, 267-281, (1985) · Zbl 0587.35078
[16] Iftimie, D., The 3D navier – stokes equations seen as a perturbation of the 2D navier – stokes equations, Bulletin de la société mathématique de France, 127, 473-517, (1999) · Zbl 0946.35059
[17] Kato, T., Strong \(L^p\) solutions of the navier – stokes equations in \(\mathbf{R}^m\) with applications to weak solutions, Mathematische zeitschrift, 187, 471-480, (1984) · Zbl 0545.35073
[18] Koch, H.; Tataru, D., Well-posedness for the navier – stokes equations, Advances in mathematics, 157, 22-35, (2001) · Zbl 0972.35084
[19] Ladyzhenskaya, O., The mathematical theory of viscous incompressible flow, Mathematics and its applications, vol. 2, (1969), Gordon and Breach, Science Publishers New York-London-Paris, xviii+224 pp · Zbl 0184.52603
[20] Leibovich, S.; Mahalov, A.; Titi, E., Invariant helical subspaces for the navier – stokes equations, Archive for rational mechanics and analysis, 112, 3, 193-222, (1990) · Zbl 0708.76044
[21] Lemarié-Rieusset, P.-G., Recent developments in the navier – stokes problem, Research notes in mathematics, vol. 431, (2002), Chapman & Hall/CRC · Zbl 1034.35093
[22] Leray, J., Essai sur le mouvement d’un liquide visqueux emplissant l’espace, Acta matematica, 63, 193-248, (1933) · JFM 59.0763.02
[23] Ponce, G.; Racke, R.; Sideris, T.; Titi, E., Global stability of large solutions to the 3D navier – stokes equations, Communitacions in mathematical physics, 159, (1994) · Zbl 0795.35082
[24] Ukhovskii, M.; Iudovich, V., Axially symmetric flows of ideal and viscous fluids filling the whole space, Prikladnaya matematika i mekhanika, Journal of applied mathematics and mechanics, 32, 52-61, (1968), (in Russian); translated as
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