Large, global solutions to the Navier-Stokes equations, slowly varying in one direction
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- by Jean-Yves Chemin and Isabelle Gallagher 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
- M. Cannone, Y. Meyer, and F. Planchon, Solutions auto-similaires des équations de Navier-Stokes, Séminaire sur les Équations aux Dérivées Partielles, 1993–1994, École Polytech., Palaiseau, 1994, pp. Exp. No. VIII, 12 (French). MR 1300903, DOI 10.1108/09533239410052824
- J.-Y. Chemin, Remarques sur l’existence globale pour le système de Navier-Stokes incompressible, SIAM J. Math. Anal. 23 (1992), no. 1, 20–28 (French, with English summary). MR 1145160, DOI 10.1137/0523002
- Jean-Yves Chemin and Isabelle Gallagher, On the global wellposedness of the 3-D Navier-Stokes equations with large initial data, Ann. Sci. École Norm. Sup. (4) 39 (2006), no. 4, 679–698 (English, with English and French summaries). MR 2290141, DOI 10.1016/j.ansens.2006.07.002
- Jean-Yves Chemin and Isabelle Gallagher, Wellposedness and stability results for the Navier-Stokes equations in $\textbf {R}^3$, Ann. Inst. H. Poincaré C Anal. Non Linéaire 26 (2009), no. 2, 599–624. MR 2504045, DOI 10.1016/j.anihpc.2007.05.008
- Hiroshi Fujita and Tosio Kato, On the Navier-Stokes initial value problem. I, Arch. Rational Mech. Anal. 16 (1964), 269–315. MR 166499, DOI 10.1007/BF00276188
- Isabelle Gallagher, The tridimensional Navier-Stokes equations with almost bidimensional data: stability, uniqueness, and life span, Internat. Math. Res. Notices 18 (1997), 919–935. MR 1481611, DOI 10.1155/S1073792897000597
- I. Gallagher, D. Iftimie, and F. Planchon, Asymptotics and stability for global solutions to the Navier-Stokes equations, Ann. Inst. Fourier (Grenoble) 53 (2003), no. 5, 1387–1424 (English, with English and French summaries). MR 2032938
- Yoshikazu Giga and Tetsuro Miyakawa, Solutions in $L_r$ of the Navier-Stokes initial value problem, Arch. Rational Mech. Anal. 89 (1985), no. 3, 267–281. MR 786550, DOI 10.1007/BF00276875
- Dragoş Iftimie, The 3D Navier-Stokes equations seen as a perturbation of the 2D Navier-Stokes equations, Bull. Soc. Math. France 127 (1999), no. 4, 473–517 (English, with English and French summaries). MR 1765551
- Dragoş Iftimie, Geneviève Raugel, and George R. Sell, Navier-Stokes equations in thin 3D domains with Navier boundary conditions, Indiana Univ. Math. J. 56 (2007), no. 3, 1083–1156. MR 2333468, DOI 10.1512/iumj.2007.56.2834
- Herbert Koch and Daniel Tataru, Well-posedness for the Navier-Stokes equations, Adv. Math. 157 (2001), no. 1, 22–35. MR 1808843, DOI 10.1006/aima.2000.1937
- O. A. Ladyzhenskaya, The mathematical theory of viscous incompressible flow, Mathematics and its Applications, Vol. 2, Gordon and Breach Science Publishers, New York-London-Paris, 1969. Second English edition, revised and enlarged; Translated from the Russian by Richard A. Silverman and John Chu. MR 0254401
- A. Mahalov, E. S. Titi, and S. Leibovich, Invariant helical subspaces for the Navier-Stokes equations, Arch. Rational Mech. Anal. 112 (1990), no. 3, 193–222. MR 1076072, DOI 10.1007/BF00381234
- J. Leray, Essai sur le mouvement d’un liquide visqueux emplissant l’espace, Acta Matematica, 63, 1933, pages 193–248.
- A. S. Makhalov and V. P. Nikolaenko, Global solvability of three-dimensional Navier-Stokes equations with uniformly high initial vorticity, Uspekhi Mat. Nauk 58 (2003), no. 2(350), 79–110 (Russian, with Russian summary); English transl., Russian Math. Surveys 58 (2003), no. 2, 287–318. MR 1992565, DOI 10.1070/RM2003v058n02ABEH000611
- G. Ponce, R. Racke, T. C. Sideris, and E. S. Titi, Global stability of large solutions to the $3$D Navier-Stokes equations, Comm. Math. Phys. 159 (1994), no. 2, 329–341. MR 1256992
- Geneviève Raugel and George R. Sell, Navier-Stokes equations on thin $3$D domains. I. Global attractors and global regularity of solutions, J. Amer. Math. Soc. 6 (1993), no. 3, 503–568. MR 1179539, DOI 10.1090/S0894-0347-1993-1179539-4
- M. R. Ukhovskii and V. I. Iudovich, Axially symmetric flows of ideal and viscous fluids filling the whole space, J. Appl. Math. Mech. 32 (1968), 52–61. MR 0239293, DOI 10.1016/0021-8928(68)90147-0
Additional Information
- Jean-Yves Chemin
- Affiliation: Laboratoire J.-L. Lions UMR 7598, Université Paris VI, 175 rue du Chevaleret, 75013 Paris, France
- Email: chemin@ann.jussieu.fr
- Isabelle Gallagher
- Affiliation: Institut de Mathématiques de Jussieu UMR 7586, Université Paris VII, 175 rue du Chevaleret, 75013 Paris, France
- MR Author ID: 617258
- Email: Isabelle.Gallagher@math.jussieu.fr
- Received by editor(s): October 29, 2007
- Published electronically: January 20, 2010
- © Copyright 2010 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 362 (2010), 2859-2873
- MSC (2000): Primary 35Q30, 76D05, 76D03
- DOI: https://doi.org/10.1090/S0002-9947-10-04744-6
- MathSciNet review: 2592939