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ISSN 1088-6850(e) ISSN 0002-9947(p)

Large, global solutions to the Navier-Stokes equations, slowly varying in one direction

Author(s): Jean-Yves Chemin; Isabelle Gallagher
Journal: Trans. Amer. Math. Soc. 362 (2010), 2859-2873.
MSC (2000): Primary 35Q30, 76D05, 76D03
Posted: January 20, 2010
MathSciNet review: 2592939
Retrieve article in: PDF

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Abstract: In two earlier papers by the authors, classes of initial data for the three dimensional, incompressible Navier-Stokes equations were presented, generating a global smooth solution although the norm of the initial data may be chosen arbitrarily large. The aim of this article is to provide new examples of arbitrarily large initial data giving rise to global solutions, in the whole space. Contrary to the previous examples, the initial data has no particular oscillatory properties, but varies slowly in one direction. The proof uses the special structure of the nonlinear term of the equation.

References:

1.: M. Cannone, Y. Meyer and F. Planchon, Solutions autosimilaires des équations de Navier-Stokes, Séminaire ``Équations aux Dérivées Partielles'' de l'École polytechnique, Exposé VIII, 1993-1994. MR 1300903 (95k:35157)
2.: J.-Y. Chemin, Remarques sur l'existence pour le système de Navier-Stokes incompressible, SIAM Journal on Mathematical Analysis, 23, 1992, pages 20-28. MR 1145160 (93a:35118)
3.: J.-Y. Chemin and I. Gallagher, On the global wellposedness of the 3-D Navier-Stokes equations with large initial data, Annales de l'École Normale Supérieure, 39, 2006, pages 679-698. MR 2290141
4.: J.-Y. Chemin and I. Gallagher, Wellposedness and stability results for the Navier-Stokes equations in ${\bf R}^{3}$ , Annales de l'Institut Henri Poincaré, Analyse Non Linéaire, 26, 2009, pages 599-624. MR 2504045
5.: H. Fujita and T. Kato, On the Navier-Stokes initial value problem I, Archive for Rational Mechanics and Analysis, 16, 1964, pages 269-315. MR 0166499 (29:3774)
6.: I. Gallagher, The tridimensional Navier-Stokes equations with almost bidimensional data: stability, uniqueness and life span, International Mathematical Research Notices, 18, 1997, pages 919-935. MR 1481611 (98m:76049)
7.: I. Gallagher, D. Iftimie and F. Planchon, Asymptotics and stability for global solutions to the Navier-Stokes equations, Annales de l'Institut Fourier, 53, 2003, no. 5, pages 1387-1424. MR 2032938 (2005b:35219)
8.: Y. Giga and T. Miyakawa, Solutions in of the Navier-Stokes initial value problem, Archive for Rational Mechanics and Analysis, 89, 1985, no. 3, pages 267-281. MR 786550 (86m:35138)
9.: D. Iftimie, The 3D Navier-Stokes equations seen as a perturbation of the 2D Navier-Stokes equations, Bulletin de la Société Mathématique de France, 127, 1999, pages 473-517. MR 1765551 (2001e:35139)
10.: D. Iftimie, G. Raugel and G.R. Sell, Navier-Stokes equations in thin 3D domains with Navier boundary conditions, Indiana University Mathematics Journal, 56, 2007, pages 1083-1156. MR 2333468
11.: H. Koch and D. Tataru, Well-posedness for the Navier-Stokes equations, Advances in Mathematics, 157, 2001, pages 22-35. MR 1808843 (2001m:35257)
12.: O. Ladyzhenskaya, The mathematical theory of viscous incompressible flow. Second English edition, revised and enlarged. Mathematics and its Applications, Vol. 2. Gordon and Breach Science Publishers, New York-London-Paris, 1969, xviii+224 pp. MR 0254401 (40:7610)
13.: S. Leibovich, A. Mahalov and E. Titi, Invariant helical subspaces for the Navier-Stokes equations. Arch. Rational Mech. Anal., 112, 1990, no. 3, pages 193-222. MR 1076072 (91h:35252)
14.: J. Leray, Essai sur le mouvement d'un liquide visqueux emplissant l'espace, Acta Matematica, 63, 1933, pages 193-248.
15.: A. Mahalov and B. Nicolaenko, Global solvability of three-dimensional Navier-Stokes equations with uniformly high initial vorticity, (Russian. Russian summary) Uspekhi Mat. Nauk, 58, 2003, pages 79-110; translation in Russian Math. Surveys, 58, 2003, pages 287-318. MR 1992565 (2004h:35177)
16.: G. Ponce, R. Racke, T. Sideris and E. Titi, Global stability of large solutions to the D Navier-Stokes equations, Comm. Math. Phys., 159, 1994, no. 2, pages 329-341. MR 1256992 (95a:35115)
17.: G. Raugel and G.R. Sell, Navier-Stokes equations on thin D domains. I. Global attractors and global regularity of solutions, Journal of the American Mathematical Society, 6, 1993, pages 503-568. MR 1179539 (93j:35134)
18.: M. Ukhovskii and V. Iudovich, Axially symmetric flows of ideal and viscous fluids filling the whole space, Prikl. Mat. Meh., 32, 1968, pages 59-69 (Russian); translated as J. Appl. Math. Mech., 32, 1968, pages 52-61. MR 0239293 (39:650)

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