On the stability of global solutions to the three-dimensional Navier-Stokes equations
[Sur la stabilité de solutions globales aux équations de Navier-Stokes tridimensionnelles]
Journal de l’École polytechnique — Mathématiques, Tome 5 (2018), pp. 843-911.

On démontre un résultat de stabilité faible pour les équations de Navier-Stokes tridimensionnelles, incompressibles et homogènes. Plus précisément on étudie le problème suivant : si une suite de données initiales (u 0,n ) n , bornée dans un espace invariant d’échelle, converge faiblement vers une donnée u 0 qui engendre une solution globale régulière, est-ce que u 0,n engendre une solution globale régulière ? Une réponse affirmative à cette question en général aurait pour conséquence la régularité globale pour toute donnée initiale, via les exemples u 0,n =nϕ 0 (n·) ou u 0,n =ϕ 0 (·-x n ) avec |x n |. On introduit donc un nouveau concept de convergence faible (convergence faible remise à l’échelle) sous lequel on peut donner une réponse affirmative. La démonstration repose sur des décompositions en profils dans des espaces de régularité anisotrope, et leur propagation par les équations de Navier-Stokes.

We prove a weak stability result for the three-dimensional homogeneous incompressible Navier-Stokes system. More precisely, we investigate the following problem: if a sequence (u 0,n ) n of initial data, bounded in some scaling invariant space, converges weakly to an initial data u 0 which generates a global smooth solution, does u 0,n generate a global smooth solution? A positive answer in general to this question would imply global regularity for any data, through the following examples u 0,n =nϕ 0 (n·) or u 0,n =ϕ 0 (·-x n ) with |x n |. We therefore introduce a new concept of weak convergence (rescaled weak convergence) under which we are able to give a positive answer. The proof relies on profile decompositions in anisotropic spaces and their propagation by the Navier-Stokes equations.

Reçu le :
Accepté le :
Publié le :
DOI : https://doi.org/10.5802/jep.84
Classification : 35Q30,  42B37
Mots clés : Équations de Navier-Stokes, anisotropie, espaces de Besov, décompositions en profils
@article{JEP_2018__5__843_0,
     author = {Bahouri, Hajer and Chemin, Jean-Yves and Gallagher, Isabelle},
     title = {On the stability of global solutions to the three-dimensional {Navier-Stokes} equations},
     journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
     pages = {843--911},
     publisher = {Ecole polytechnique},
     volume = {5},
     year = {2018},
     doi = {10.5802/jep.84},
     zbl = {06988594},
     mrnumber = {3877168},
     language = {en},
     url = {http://archive.numdam.org/articles/10.5802/jep.84/}
}
TY  - JOUR
AU  - Bahouri, Hajer
AU  - Chemin, Jean-Yves
AU  - Gallagher, Isabelle
TI  - On the stability of global solutions to the three-dimensional Navier-Stokes equations
JO  - Journal de l’École polytechnique — Mathématiques
PY  - 2018
DA  - 2018///
SP  - 843
EP  - 911
VL  - 5
PB  - Ecole polytechnique
UR  - http://archive.numdam.org/articles/10.5802/jep.84/
UR  - https://zbmath.org/?q=an%3A06988594
UR  - https://www.ams.org/mathscinet-getitem?mr=3877168
UR  - https://doi.org/10.5802/jep.84
DO  - 10.5802/jep.84
LA  - en
ID  - JEP_2018__5__843_0
ER  - 
Bahouri, Hajer; Chemin, Jean-Yves; Gallagher, Isabelle. On the stability of global solutions to the three-dimensional Navier-Stokes equations. Journal de l’École polytechnique — Mathématiques, Tome 5 (2018), pp. 843-911. doi : 10.5802/jep.84. http://archive.numdam.org/articles/10.5802/jep.84/

[1] Auscher, P.; Dubois, S.; Tchamitchian, P. On the stability of global solutions to Navier-Stokes equations in the space, J. Math. Pures Appl. (9), Volume 83 (2004) no. 6, pp. 673-697 | Article | MR 2062638 | Zbl 1107.35096

[2] Bahouri, H.; Chemin, J.-Y.; Danchin, R. Fourier analysis and nonlinear partial differential equations, Grundlehren Math. Wiss., 343, Springer, Heidelberg, 2011 | MR 2768550 | Zbl 1227.35004

[3] Bahouri, H.; Cohen, A.; Koch, G. A general wavelet-based profile decomposition in the critical embedding of function spaces, Confluentes Math., Volume 3 (2011) no. 3, pp. 387-411 | Article | MR 2847237 | Zbl 1231.42034

[4] Bahouri, H.; Gallagher, I. On the stability in weak topology of the set of global solutions to the Navier-Stokes equations, Arch. Rational Mech. Anal., Volume 209 (2013) no. 2, pp. 569-629 | Article | MR 3056618 | Zbl 1283.35061

[5] Bahouri, H.; Gérard, P. High frequency approximation of solutions to critical nonlinear wave equations, Amer. J. Math., Volume 121 (1999) no. 1, pp. 131-175 | Article | MR 1705001 | Zbl 0919.35089

[6] Bahouri, H.; Majdoub, M.; Masmoudi, N. On the lack of compactness in the 2D critical Sobolev embedding, J. Funct. Anal., Volume 260 (2011) no. 1, pp. 208-252 | Article | MR 2733577 | Zbl 1217.46017

[7] Bahouri, H.; Majdoub, M.; Masmoudi, N. Lack of compactness in the 2D critical Sobolev embedding, the general case, J. Math. Pures Appl. (9), Volume 101 (2014) no. 4, pp. 415-457 | Article | MR 3179749 | Zbl 1305.46024

[8] Bahouri, H.; Perelman, G. A Fourier approach to the profile decomposition in Orlicz spaces, Math. Res. Lett., Volume 21 (2014) no. 1, pp. 33-54 | Article | MR 3247037 | Zbl 1311.46030

[9] Bourdaud, G. La propriété de Fatou dans les espaces de Besov homogènes, Comptes Rendus Mathématique, Volume 349 (2011) no. 15-16, pp. 837-840 | Article | Zbl 1252.46021

[10] Bourgain, J.; Pavlović, N. Ill-posedness of the Navier-Stokes equations in a critical space in 3D, J. Funct. Anal., Volume 255 (2008) no. 9, pp. 2233-2247 | Article | MR 2473255 | Zbl 1161.35037

[11] Brezis, H.; Coron, J.-M. Convergence of solutions of H-systems or how to blow bubbles, Arch. Rational Mech. Anal., Volume 89 (1985) no. 1, pp. 21-56 | Article | MR 784102 | Zbl 0584.49024

[12] Chemin, J.-Y. Remarques sur l’existence globale pour le système de Navier-Stokes incompressible, SIAM J. Math. Anal., Volume 23 (1992) no. 1, pp. 20-28 | Article | Zbl 0762.35063

[13] Chemin, J.-Y. Théorèmes d’unicité pour le système de Navier-Stokes tridimensionnel, J. Anal. Math., Volume 77 (1999), pp. 27-50 | Article | Zbl 0938.35125

[14] Chemin, J.-Y.; Gallagher, I. Large, global solutions to the Navier-Stokes equations, slowly varying in one direction, Trans. Amer. Math. Soc., Volume 362 (2010) no. 6, pp. 2859-2873 | Article | MR 2592939 | Zbl 1189.35220

[15] Chemin, J.-Y.; Gallagher, I.; Mullaert, C. The role of spectral anisotropy in the resolution of the three-dimensional Navier-Stokes equations, Studies in phase space analysis with applications to PDEs (Progr. Nonlinear Differential Equations Appl.), Volume 84, Birkhäuser/Springer, New York, 2013, pp. 53-79 | Article | MR 3185890 | Zbl 1314.35080

[16] Chemin, J.-Y.; Gallagher, I.; Zhang, P. Sums of large global solutions to the incompressible Navier-Stokes equations, J. reine angew. Math., Volume 681 (2013), pp. 65-82 | MR 3181490

[17] Chemin, J.-Y.; Lerner, N. Flot de champs de vecteurs non lipschitziens et équations de Navier-Stokes, J. Differential Equations, Volume 121 (1995) no. 2, pp. 314-328 | Zbl 0878.35089

[18] Chemin, J.-Y.; Zhang, P. On the global wellposedness to the 3-D incompressible anisotropic Navier-Stokes equations, Comm. Math. Phys., Volume 272 (2007) no. 2, pp. 529-566 | MR 2300252 | Zbl 1132.35068

[19] Fujita, H.; Kato, T. On the Navier-Stokes initial value problem. I, Arch. Rational Mech. Anal., Volume 16 (1964), pp. 269-315 | Article | MR 166499 | Zbl 0126.42301

[20] Gallagher, I. Profile decomposition for solutions of the Navier-Stokes equations, Bull. Soc. math. France, Volume 129 (2001) no. 2, pp. 285-316 | Article | Numdam | MR 1871299 | Zbl 0987.35120

[21] Gallagher, I.; Iftimie, D.; Planchon, F. Asymptotics and stability for global solutions to the Navier-Stokes equations, Ann. Inst. Fourier (Grenoble), Volume 53 (2003) no. 5, pp. 1387-1424 | Article | Numdam | MR 2032938 | Zbl 1038.35054

[22] Gallagher, I.; Koch, G. S.; Planchon, F. A profile decomposition approach to the L t (L x 3 ) Navier-Stokes regularity criterion, Math. Ann., Volume 355 (2013) no. 4, pp. 1527-1559 | Article | Zbl 1291.35180

[23] Gérard, P. Microlocal defect measures, Comm. Partial Differential Equations, Volume 16 (1991) no. 11, pp. 1761-1794 | Article | MR 1135919 | Zbl 0770.35001

[24] Gérard, P. Description du défaut de compacité de l’injection de Sobolev, ESAIM Contrôle Optim. Calc. Var., Volume 3 (1998), pp. 213-233

[25] Germain, P. The second iterate for the Navier-Stokes equation, J. Funct. Anal., Volume 255 (2008) no. 9, pp. 2248-2264 | Article | MR 2473256 | Zbl 1173.35097

[26] Giga, Y.; Miyakawa, T. Solutions in L r of the Navier-Stokes initial value problem, Arch. Rational Mech. Anal., Volume 89 (1985) no. 3, pp. 267-281 | Article | MR 786550

[27] Gui, G.; Huang, J.; Zhang, P. Large global solutions to 3-D inhomogeneous Navier-Stokes equations slowly varying in one variable, J. Funct. Anal., Volume 261 (2011) no. 11, pp. 3181-3210 | Article | MR 2835995 | Zbl 1229.35180

[28] Gui, G.; Zhang, P. Stability to the global large solutions of 3-D Navier-Stokes equations, Adv. in Math., Volume 225 (2010) no. 3, pp. 1248-1284 | Article | MR 2673730

[29] Guillot, J.; Šverák, V. Numerical investigations of non-uniqueness for the Navier-Stokes initial value problem in borderline spaces (2017) (arXiv:1704.00560)

[30] Hmidi, T.; Keraani, S. Blowup theory for the critical nonlinear Schrödinger equations revisited, Internat. Math. Res. Notices (2005) no. 46, pp. 2815-2828 | Article | Zbl 1126.35067

[31] Iftimie, D. The resolution of the Navier-Stokes equations in anisotropic spaces, Rev. Mat. Iberoamericana, Volume 15 (1999) no. 1, pp. 1-36 | Article | MR 1681635 | Zbl 0923.35119

[32] Jaffard, S. Analysis of the lack of compactness in the critical Sobolev embeddings, J. Funct. Anal., Volume 161 (1999) no. 2, pp. 384-396 | Article | MR 1674639 | Zbl 0922.46030

[33] Jia, H.; Šverák, V. Minimal L 3 -initial data for potential Navier-Stokes singularities (2012) (arXiv:1201.1592)

[34] Jia, H.; Šverák, V. Minimal L 3 -initial data for potential Navier-Stokes singularities, SIAM J. Math. Anal., Volume 45 (2013) no. 3, pp. 1448-1459 | Article | MR 3056752 | Zbl 1294.35065

[35] Kato, T. Strong L p -solutions of the Navier-Stokes equation in R m , with applications to weak solutions, Math. Z., Volume 187 (1984) no. 4, pp. 471-480 | Article | MR 760047

[36] Kenig, C. E.; Koch, G. S. An alternative approach to regularity for the Navier-Stokes equations in critical spaces, Ann. Inst. H. Poincaré Anal. Non Linéaire, Volume 28 (2011) no. 2, pp. 159-187 | Article | Numdam | MR 2784068 | Zbl 1220.35119

[37] Kenig, C. E.; Merle, F. Global well-posedness, scattering and blow-up for the energy-critical focusing non-linear wave equation, Acta Math., Volume 201 (2008) no. 2, pp. 147-212 | MR 2461508 | Zbl 1183.35202

[38] Keraani, S. On the defect of compactness for the Strichartz estimates of the Schrödinger equations, J. Differential Equations, Volume 175 (2001) no. 2, pp. 353-392 | MR 1855973 | Zbl 1038.35119

[39] Koch, G. S. Profile decompositions for critical Lebesgue and Besov space embeddings, Indiana Univ. Math. J., Volume 59 (2010) no. 5, pp. 1801-1830 | Article | MR 2865431 | Zbl 1230.46030

[40] Koch, H.; Tataru, D. Well-posedness for the Navier-Stokes equations, Adv. in Math., Volume 157 (2001) no. 1, pp. 22-35 | MR 1808843 | Zbl 0972.35084

[41] 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 | MR 1938147 | Zbl 1034.35093

[42] Leray, J. Essai sur le mouvement d’un liquide visqueux emplissant l’espace, Acta Math., Volume 63 (1933), pp. 193-248 | Article

[43] Leray, J. Étude de diverses équations intégrales non linéaires et de quelques problèmes que pose l’hydrodynamique, J. Math. Pures Appl., Volume 12 (1933), pp. 1-82 | Zbl 0006.16702

[44] Lions, P.-L. The concentration-compactness principle in the calculus of variations. The limit case. I, Rev. Mat. Iberoamericana, Volume 1 (1985) no. 1, pp. 145-201 | Article | MR 834360 | Zbl 0704.49005

[45] Lions, P.-L. The concentration-compactness principle in the calculus of variations. The limit case. II, Rev. Mat. Iberoamericana, Volume 1 (1985) no. 2, pp. 45-121 | Article | MR 850686 | Zbl 0704.49006

[46] Merle, F.; Vega, L. Compactness at blow-up time for L 2 solutions of the critical nonlinear Schrödinger equation in 2D, Internat. Math. Res. Notices (1998) no. 8, pp. 399-425 | Article | Zbl 0913.35126

[47] Meyer, Y. Wavelets, paraproducts, and Navier-Stokes equations, Current developments in mathematics, 1996 (Cambridge, MA), Int. Press, Boston, MA, 1997, pp. 105-212 | Zbl 0926.35115

[48] Paicu, M. Équation anisotrope de Navier-Stokes dans des espaces critiques, Rev. Mat. Iberoamericana, Volume 21 (2005) no. 1, pp. 179-235 | Article | Zbl 1110.35060

[49] Planchon, F. Asymptotic behavior of global solutions to the Navier-Stokes equations in R 3 , Rev. Mat. Iberoamericana, Volume 14 (1998) no. 1, pp. 71-93 | Article | MR 1639283

[50] Poulon, E. Behaviour of Navier-Stokes solutions with data in H s with 1/2<s<3/2 (in progress)

[51] Runst, T.; Sickel, W. Sobolev spaces of fractional order, Nemytskij operators, and nonlinear partial differential equations, De Gruyter Series in Nonlinear Analysis and Applications, 3, Walter de Gruyter & Co., Berlin, 1996 | MR 1419319 | Zbl 0873.35001

[52] Rusin, W.; Šverák, V. Minimal initial data for potential Navier-Stokes singularities, J. Funct. Anal., Volume 260 (2011) no. 3, pp. 879-891 | Article | MR 2737400 | Zbl 1206.35199

[53] Struwe, M. A global compactness result for elliptic boundary value problems involving limiting nonlinearities, Math. Z., Volume 187 (1984) no. 4, pp. 511-517 | Article | MR 760051 | Zbl 0535.35025

[54] Tartar, L. H-measures, a new approach for studying homogenisation, oscillations and concentration effects in partial differential equations, Proc. Edinburgh Math. Soc., Volume 115 (1990) no. 3-4, pp. 193-230 | Article | MR 1069518 | Zbl 0774.35008

[55] Tintarev, Kyril; Fieseler, K.-H. Concentration compactness. Functional-analytic grounds and applications, Imperial College Press, London, 2007 | Zbl 1118.49001

[56] Triebel, H. Theory of function spaces, Monographs in Mathematics, 78, Birkhäuser Verlag, Basel, 1983 | MR 781540 | Zbl 0546.46028

[57] Triebel, H. Interpolation theory, function spaces, differential operators, Johann Ambrosius Barth, Heidelberg, 1995 | Zbl 0830.46028

[58] Weissler, F. B. The Navier-Stokes initial value problem in L p , Arch. Rational Mech. Anal., Volume 74 (1980) no. 3, pp. 219-230 | Article | MR 591222 | Zbl 0454.35072

Cité par Sources :