Well-posedness for density-dependent incompressible fluids with non-Lipschitz velocity
[Existence de solutions fortes pour Navier-Stokes non-homogène avec vitesse non-Lipschitz]
Annales de l'Institut Fourier, Tome 62 (2012) no. 5, pp. 1717-1763.

Ce papier est dédié à l’étude de Cauchy pour le système de Navier-Stokes non homogène dans N avec N2. Nous adressons la question du caractère bien posé pour des données initiales grandes et petites ayant une régularité critique dans des espaces de Besov aussi proches que possible de ceux utilisés par Cannone, Meyer et Planchon pour Navier Stokes incompressible (où u 0 B p,r N p-1 avec 1p<+,1r+). Cela améliore l’analyse classique où la vitesse initiale u 0 est supposée appartenir à B p,1 N p-1 de telle manière que la vitesse u reste Lipschitz. Notre résultat utilise de nouvelles estimées pour l’équation de transport introduites par Bahouri, Chemin et Danchin lorsque la vitesse u n’est pas nécessairement Lipschitz mais seulement log Lipschitz. De plus, cela donne une première réponse de résultat au problème des solutions autosimilaires.

This paper is dedicated to the study of the initial value problem for density dependent incompressible viscous fluids in N with N2. We address the question of well-posedness for large and small initial data having critical Besov regularity in functional spaces as close as possible to the ones imposed in the incompressible Navier Stokes system by Cannone, Meyer and Planchon (where u 0 B p,r N p-1 with 1p<+,1r+). This improves the classical analysis where u 0 is considered belonging in B p,1 N p-1 such that the velocity u remains Lipschitz. Our result relies on a new a priori estimate for transport equation introduce by Bahouri, Chemin and Danchin when the velocity u is not necessary Lipschitz but only log Lipschitz. Furthermore it gives a first kind of answer to the problem of self-similar solution.

DOI : 10.5802/aif.2734
Classification : 76D03, 76D05, 35S50
Keywords: Navier-Stokes equations Cauchy problem, Littlewood-Paley theory, losing estimates for the transport equation
Mot clés : équations de Navier-Stokes, problème de Cauchy, Littlewood-Paley théorie, estimées avec perte pour l’équation de transport
Haspot, Boris 1

1 Université Paris Dauphine Ceremade UMR CNRS 7534 Place du Maréchal de Lattre de Tassigny 75775 Paris Cedex 16 (France) Karls Ruprecht Universität HeidelBerg Institut for Applied Mathematics Im Neuenheimer Feld 294 D-69120 Heildelberg (Germany)
@article{AIF_2012__62_5_1717_0,
     author = {Haspot, Boris},
     title = {Well-posedness for density-dependent incompressible fluids with {non-Lipschitz} velocity},
     journal = {Annales de l'Institut Fourier},
     pages = {1717--1763},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {62},
     number = {5},
     year = {2012},
     doi = {10.5802/aif.2734},
     mrnumber = {3025152},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/aif.2734/}
}
TY  - JOUR
AU  - Haspot, Boris
TI  - Well-posedness for density-dependent incompressible fluids with non-Lipschitz velocity
JO  - Annales de l'Institut Fourier
PY  - 2012
SP  - 1717
EP  - 1763
VL  - 62
IS  - 5
PB  - Association des Annales de l’institut Fourier
UR  - http://www.numdam.org/articles/10.5802/aif.2734/
DO  - 10.5802/aif.2734
LA  - en
ID  - AIF_2012__62_5_1717_0
ER  - 
%0 Journal Article
%A Haspot, Boris
%T Well-posedness for density-dependent incompressible fluids with non-Lipschitz velocity
%J Annales de l'Institut Fourier
%D 2012
%P 1717-1763
%V 62
%N 5
%I Association des Annales de l’institut Fourier
%U http://www.numdam.org/articles/10.5802/aif.2734/
%R 10.5802/aif.2734
%G en
%F AIF_2012__62_5_1717_0
Haspot, Boris. Well-posedness for density-dependent incompressible fluids with non-Lipschitz velocity. Annales de l'Institut Fourier, Tome 62 (2012) no. 5, pp. 1717-1763. doi : 10.5802/aif.2734. http://www.numdam.org/articles/10.5802/aif.2734/

[1] Abidi, Hammadi Équation de Navier-Stokes avec densité et viscosité variables dans l’espace critique, Université Paris VI (2000) (Ph. D. Thesis)

[2] Abidi, Hammadi; Paicu, M. Équation de Navier-Stokes avec densité et viscosité variables dans l’espace critique, Annales de l’Institut Fourier, Volume 57 (2007) no. 3, pp. 883-917 | DOI | Numdam | MR | Zbl

[3] Antontsev, S. N.; Kazhikhov, A. V.; Monakhov, V. N. Boundary value problems in mechanics of nonhomogeneous fluids, Studies in Mathematics and its Applications, 22, North-Holland Publishing Co., Amsterdam, 1990 (Translated from the Russian) | MR | Zbl

[4] Bahouri, Hajer; Chemin, Jean-Yves; Danchin, Raphaël Fourier analysis and nonlinear partial differential equations, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 343, Springer, Heidelberg, 2011 | MR | Zbl

[5] Bony, Jean-Michel Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires, Ann. Sci. École Norm. Sup. (4), Volume 14 (1981) no. 2, pp. 209-246 | Numdam | MR | Zbl

[6] Cannone, M.; Meyer, Y.; Planchon, F. 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 | Numdam | MR | Zbl

[7] Chemin, Jean-Yves; Gallagher, Isabelle; Paicu, Marius Global regularity for some classes of large solutions to the Navier-Stokes equations, Ann. of Math. (2), Volume 173 (2011) no. 2, pp. 983-1012 | DOI | MR | Zbl

[8] Danchin, Raphaël Erratum: “Local theory in critical spaces for compressible viscous and heat-conductive gases” [Comm. Partial Differential Equations 26 (2001), no. 7-8, 1183–1233; MR1855277 (2002g:76091)], Comm. Partial Differential Equations, Volume 27 (2002) no. 11-12, pp. 2531-2532 | MR | Zbl

[9] Danchin, Raphaël Density-dependent incompressible viscous fluids in critical spaces, Proc. Roy. Soc. Edinburgh Sect. A, Volume 133 (2003) no. 6, pp. 1311-1334 | DOI | MR | Zbl

[10] Danchin, Raphaël Local and global well-posedness results for flows of inhomogeneous viscous fluids, Adv. Differential Equations, Volume 9 (2004) no. 3-4, pp. 353-386 | MR | Zbl

[11] Danchin, Raphaël Fourier analysis method for PDE’s, 2005 (Preprint)

[12] Danchin, Raphaël On the uniqueness in critical spaces for compressible Navier-Stokes equations, NoDEA Nonlinear Differential Equations Appl., Volume 12 (2005) no. 1, pp. 111-128 | DOI | MR | Zbl

[13] Danchin, Raphaël The inviscid limit for density-dependent incompressible fluids, Ann. Fac. Sci. Toulouse Math. (6), Volume 15 (2006) no. 4, pp. 637-688 | DOI | Numdam | MR | Zbl

[14] Danchin, Raphaël Well-posedness in critical spaces for barotropic viscous fluids with truly not constant density, Comm. Partial Differential Equations, Volume 32 (2007) no. 7-9, pp. 1373-1397 | DOI | MR | Zbl

[15] Danchin, Raphaël; Paicu, Marius Les théorèmes de Leray et de Fujita-Kato pour le système de Boussinesq partiellement visqueux, Bull. Soc. Math. France, Volume 136 (2008) no. 2, pp. 261-309 | Numdam | MR | Zbl

[16] Desjardins, Benoît Regularity results for two-dimensional flows of multiphase viscous fluids, Arch. Rational Mech. Anal., Volume 137 (1997) no. 2, pp. 135-158 | DOI | MR | Zbl

[17] Fujita, Hiroshi; Kato, Tosio On the Navier-Stokes initial value problem. I, Arch. Rational Mech. Anal., Volume 16 (1964), pp. 269-315 | DOI | MR | Zbl

[18] Germain, Pierre Weak-strong uniqueness for the isentropic compressible Navier-Stokes system, J. Math. Fluid Mech., Volume 13 (2011) no. 1, pp. 137-146 | DOI | MR

[19] Haspot, B. Existence of global strong solutions in critical spaces for barotropic viscous fluids, Archive for Rational Mechanics and Analysis, Volume 202 (2011) no. 2, pp. 427-460 | DOI | MR

[20] Haspot, B. Existence of strong solutions in critical spaces for barotropic viscous fluids in larger spaces, Journal of Differential Equations, Volume 251 (2011) no. 8, pp. 2262-2295 | DOI | MR | Zbl

[21] Itoh, S.; Tani, A. Solvability of nonstationnary problems for nonhomogeneous incompressible fluids and the convergence with vanishing viscosity, Tokyo Journal of Mathematics, Volume 22 (1999), pp. 17-42 | DOI | MR | Zbl

[22] Koch, Herbert; Tataru, Daniel Well-posedness for the Navier-Stokes equations, Adv. Math., Volume 157 (2001) no. 1, pp. 22-35 | DOI | MR | Zbl

[23] Ladyženskaja, O. A.; Solonnikov, V. A. The unique solvability of an initial-boundary value problem for viscous incompressible inhomogeneous fluids, J. Soviet Math., Volume 9 (1978), pp. 697-749 | DOI | Zbl

[24] Lemarié-Rieusset, P. G. Recent developments in the Navier-Stokes problem, Chapman & Hall/CRC Research Notes in Mathematics, 431, Chapman & Hall/CRC, Boca Raton, FL, 2002 | DOI | MR | Zbl

[25] Lions, Pierre-Louis Mathematical topics in fluid mechanics. Vol. 1, Oxford Lecture Series in Mathematics and its Applications, 3, The Clarendon Press Oxford University Press, New York, 1996 (Incompressible models, Oxford Science Publications) | MR | Zbl

[26] Meyer, Yves Wavelets, paraproducts, and Navier-Stokes equations, Current developments in mathematics, 1996 (Cambridge, MA), Int. Press, Boston, MA, 1997, pp. 105-212 | MR | Zbl

Cité par Sources :