Profile decompositions and applications to Navier-Stokes
Koch, Gabriel S.
Journées équations aux dérivées partielles, (2010), p. 1-13 / Harvested from Numdam
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In this expository note, we collect some recent results concerning the applications of methods from dispersive and hyperbolic equations to the study of regularity criteria for the Navier-Stokes equations. In particular, these methods have recently been used to give an alternative approach to the L 3,∞ Navier-Stokes regularity criterion of Escauriaza, Seregin and Šverák. The key tools are profile decompositions for bounded sequences of functions in critical spaces.

Publié le : 2010-01-01
DOI : https://doi.org/10.5802/jedp.69 
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     author = {Koch, Gabriel S.},
     title = {Profile decompositions and applications to Navier-Stokes},
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     year = {2010},
     pages = {1-13},
     doi = {10.5802/jedp.69},
     language = {en},
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Koch, Gabriel S. Profile decompositions and applications to Navier-Stokes. Journées équations aux dérivées partielles,  (2010), pp. 1-13. doi : 10.5802/jedp.69. http://gdmltest.u-ga.fr/item/JEDP_2010____A12_0/

[1] Hajer Bahouri, Albert Cohen, and Gabriel Koch. A general construction method for profile decompositions. in progress.

[2] Hajer Bahouri and Patrick Gérard. High frequency approximation of solutions to critical nonlinear wave equations. Amer. J. Math., 121(1):131–175, 1999. | MR 1705001 | Zbl 0919.35089

[3] H. Brezis and J.-M. Coron. Convergence of solutions of H-systems or how to blow bubbles. Arch. Rational Mech. Anal., 89(1):21–56, 1985. | MR 784102 | Zbl 0584.49024

[4] L. Caffarelli, R. Kohn, and L. Nirenberg. Partial regularity of suitable weak solutions of the Navier-Stokes equations. Comm. Pure Appl. Math., 35(6):771–831, 1982. | MR 673830 | Zbl 0509.35067

[5] L. Escauriaza, G. A. Seregin, and V. Šverák. L 3,∞ -solutions of Navier-Stokes equations and backward uniqueness. Uspekhi Mat. Nauk, 58(2(350)):3–44, 2003. | MR 1992563 | Zbl 1064.35134

[6] Isabelle Gallagher. Profile decomposition for solutions of the Navier-Stokes equations. Bull. Soc. Math. France, 129(2):285–316, 2001. | Numdam | MR 1871299 | Zbl 0987.35120

[7] Isabelle Gallagher, Dragoş Iftimie, and Fabrice Planchon. Non-explosion en temps grand et stabilité de solutions globales des équations de Navier-Stokes. C. R. Math. Acad. Sci. Paris, 334(4):289–292, 2002. | MR 1891005 | Zbl 0997.35051

[8] Isabelle Gallagher, Dragoş Iftimie, and Fabrice Planchon. Asymptotics and stability for global solutions to the Navier-Stokes equations. Ann. Inst. Fourier (Grenoble), 53(5):1387–1424, 2003. | Numdam | MR 2032938 | Zbl 1038.35054

[9] Isabelle Gallagher, Gabriel Koch, and Fabrice Planchon. A profile decomposition approach to the L t ∞ (L x 3 ) Navier-Stokes regularity criterion. arXiv:1012.0145.

[10] Patrick Gérard. Description du défaut de compacité de l’injection de Sobolev. ESAIM Control Optim. Calc. Var., 3:213–233 (electronic), 1998. | Numdam | MR 1632171 | Zbl 0907.46027

[11] Stéphane Jaffard. Analysis of the lack of compactness in the critical Sobolev embeddings. J. Funct. Anal., 161(2):384–396, 1999. | MR 1674639 | Zbl 0922.46030

[12] Carlos Kenig and Gabriel Koch. An alternative approach to the Navier-Stokes equations in critical spaces. to appear in Annales de l’Institut Henri Poincaré (C) Analyse Non Linéaire (preprint: arXiv:0908.3349).

[13] Carlos E. Kenig and Frank Merle. Global well-posedness, scattering and blow-up for the energy-critical, focusing, non-linear Schrödinger equation in the radial case. Invent. Math., 166(3):645–675, 2006. | MR 2257393 | Zbl 1115.35125

[14] Carlos E. Kenig and Frank Merle. Global well-posedness, scattering and blow-up for the energy-critical focusing non-linear wave equation. Acta Math., 201(2):147–212, 2008. | MR 2461508 | Zbl 1183.35202

[15] Carlos E. Kenig and Frank Merle. Scattering for H 1/2 bounded solutions to the cubic, defocusing NLS in 3 dimensions. Trans. Amer. Math. Soc., 362(4):1937–1962, 2010. | MR 2574882 | Zbl 1188.35180

[16] Sahbi Keraani. On the blow up phenomenon of the critical nonlinear Schrödinger equation. J. Funct. Anal., 235(1):171–192, 2006. | MR 2216444 | Zbl 1099.35132

[17] Gabriel Koch. Profile decompositions for critical Lebesgue and Besov space embeddings. arXiv:1006.3064.

[18] O. A. Ladyženskaja. Uniqueness and smoothness of generalized solutions of Navier-Stokes equations. Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), 5:169–185, 1967. | MR 236541 | Zbl 0194.12805

[19] J. Leray. Sur le mouvement d’un liquide visqueux emplissant l’espace. Acta Math., 63:193–248, 1934. | MR 1555394

[20] P.-L. Lions. The concentration-compactness principle in the calculus of variations. The limit case. I. Rev. Mat. Iberoamericana, 1(1):145–201, 1985. | MR 834360 | Zbl 0704.49005

[21] J. Nečas, M. Růžička, and V. Šverák. On Leray’s self-similar solutions of the Navier-Stokes equations. Acta Math., 176(2):283–294, 1996. | Zbl 0884.35115

[22] Giovanni Prodi. Un teorema di unicità per le equazioni di Navier-Stokes. Ann. Mat. Pura Appl. (4), 48:173–182, 1959. | MR 126088 | Zbl 0148.08202

[23] James Serrin. The initial value problem for the Navier-Stokes equations. In Nonlinear Problems (Proc. Sympos., Madison, Wis, pages 69–98. Univ. of Wisconsin Press, Madison, Wis., 1963. | MR 150444 | Zbl 0115.08502

[24] Sergio Solimini. A note on compactness-type properties with respect to Lorentz norms of bounded subsets of a Sobolev space. Ann. Inst. H. Poincaré Anal. Non Linéaire, 12(3):319–337, 1995. | Numdam | MR 1340267 | Zbl 0837.46025

[25] V. Šverák and W. Rusin. Minimal initial data for potential Navier-Stokes singularities. arXiv:0911.0500, 2009.