An alternative approach to regularity for the Navier–Stokes equations in critical spaces

Kenig, Carlos E.; Koch, Gabriel S.

Annales de l'I.H.P. Analyse non linéaire, Tome 28 (2011) no. 2, pp. 159-187.

Dans cet exposé, nous présentons un point de vue différent sur les études récentes concernant la régularité des solutions des équations de Navier–Stokes dans les espaces critiques. En particulier, nous démontrons que les solutions faibles qui restent bornées dans lʼespace ${\dot{H}}^{\frac{1}{2}}$ ne deviennent pas singulières en temps fini. Ce résultat a été démontré dans un cas plus général par L. Escauriaza, G. Seregin et V. Šverák en utilisant une approche différente. Nous utilisons la méthode de « concentration-compacité » + « théorème de rigidité » utilisant des « éléments critiques » qui a été récemment développée par C. Kenig et F. Merle pour traiter les équations dispersives critiques. À la connaissance des auteurs, cʼest la première fois que cette méthode est appliquée à une équation parabolique.

In this paper we present an alternative viewpoint on recent studies of regularity of solutions to the Navier–Stokes equations in critical spaces. In particular, we prove that mild solutions which remain bounded in the space ${\dot{H}}^{\frac{1}{2}}$ do not become singular in finite time, a result which was proved in a more general setting by L. Escauriaza, G. Seregin and V. Šverák using a different approach. We use the method of “concentration-compactness” + “rigidity theorem” using “critical elements” which was recently developed by C. Kenig and F. Merle to treat critical dispersive equations. To the authorsʼ knowledge, this is the first instance in which this method has been applied to a parabolic equation.

[1] Seven Papers on Equations Related to Mechanics and Heat, American Mathematical Society Translations, Series 2 vol. 75, American Mathematical Society, Providence, RI (1968)

[2] J. Bourgain, Global Solutions of Nonlinear Schrödinger Equations, American Mathematical Society Colloquium Publications vol. 46, American Mathematical Society, Providence, RI (1999) | MR | Zbl

[3] J. Bourgain, Global wellposedness of defocusing critical nonlinear Schrödinger equation in the radial case, J. Amer. Math. Soc. 12 no. 1 (1999), 145-171 | MR | Zbl

[4] Jean Bourgain, Nataša Pavlović, Ill-posedness of the Navier–Stokes equations in a critical space in 3D, J. Funct. Anal. 255 no. 9 (2008), 2233-2247 | MR | Zbl

[5] L. Caffarelli, R. Kohn, L. Nirenberg, Partial regularity of suitable weak solutions of the Navier–Stokes equations, Comm. Pure Appl. Math. 35 no. 6 (1982), 771-831 | MR | Zbl

[6] Marco Cannone, Ondelettes, Paraproduits et Navier–Stokes, Diderot Editeur, Paris (1995) | MR | Zbl

[7] Marco Cannone, Harmonic analysis tools for solving the incompressible Navier–Stokes equations, Handbook of Mathematical Fluid Dynamics, vol. III, North-Holland, Amsterdam (2004), 161-244 | MR | Zbl

[8] Jean-Yves Chemin, Le système de Navier–Stokes incompressible soixante dix ans après Jean Leray, Actes des Journées Mathématiques à la Mémoire de Jean Leray, Sémin. Congr. vol. 9, Soc. Math. France, Paris (2004), 99-123 | MR | Zbl

[9] J. Colliander, M. Keel, G. Staffilani, H. Takaoka, T. Tao, Global well-posedness and scattering for the energy-critical nonlinear Schrödinger equation in ${ℝ}^3$, Ann. of Math. (2) 167 no. 3 (2008), 767-865 | MR | Zbl

[10] Raphaël Côte, Carlos E. Kenig, Frank Merle, Scattering below critical energy for the radial 4D Yang–Mills equation and for the 2D corotational wave map system, Comm. Math. Phys. 284 no. 1 (2008), 203-225 | MR | Zbl

[11] Hongjie Dong, Dapeng Du, On the local smoothness of solutions of the Navier–Stokes equations, J. Math. Fluid Mech. 9 no. 2 (2007), 139-152 | MR | Zbl

[12] L. Escauriaza, G. Seregin, V. Šverák, On backward uniqueness for parabolic equations, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 288 (Kraev. Zadachi Mat. Fiz. i Smezh. Vopr. Teor. Funkts. 32) (2002) 100–103, 272. | MR

[13] L. Escauriaza, G. Seregin, V. Šverák, Backward uniqueness for parabolic equations, Arch. Ration. Mech. Anal. 169 no. 2 (2003), 147-157 | MR | Zbl

[14] L. Escauriaza, G.A. Seregin, V. Šverák, $L_{3,\infty }$-solutions of Navier–Stokes equations and backward uniqueness, Uspekhi Mat. Nauk 58 no. 2(350) (2003), 3-44 | MR | Zbl

[15] Luis Escauriaza, Francisco Javier Fernández, Unique continuation for parabolic operators, Ark. Mat. 41 no. 1 (2003), 35-60 | MR | Zbl

[16] Hiroshi Fujita, Kato Tosio, On the Navier–Stokes initial value problem. I, Arch. Ration. Mech. Anal. 16 (1964), 269-315 | MR | Zbl

[17] Giulia Furioli, Pierre G. Lemarié-Rieusset, Elide Terraneo, Unicité dans $L^3\left({ℝ}^3\right)$ et dʼautres espaces fonctionnels limites pour Navier–Stokes, Rev. Mat. Iberoamericana 16 no. 3 (2000), 605-667 | EuDML | MR | Zbl

[18] Giulia Furioli, Pierre Gilles Lemarié-Rieusset, Ezzedine Zahrouni, Ali Zhioua, Un théorème de persistance de la régularité en norme dʼespaces de Besov pour les solutions de Koch et Tataru des équations de Navier–Stokes dans ${𝐑}^3$, C. R. Acad. Sci. Paris Sér. I Math. 330 no. 5 (2000), 339-342 | MR | Zbl

[19] Isabelle Gallagher, Profile decomposition for solutions of the Navier–Stokes equations, Bull. Soc. Math. France 129 no. 2 (2001), 285-316 | EuDML | Numdam | MR | Zbl

[20] Isabelle Gallagher, Dragoş Iftimie, Fabrice Planchon, Non-explosion en temps grand et stabilité de solutions globales des équations de Navier–Stokes, C. R. Math. Acad. Sci. Paris 334 no. 4 (2002), 289-292 | MR | Zbl

[21] Isabelle Gallagher, Dragoş Iftimie, Fabrice Planchon, Asymptotics and stability for global solutions to the Navier–Stokes equations, Ann. Inst. Fourier (Grenoble) 53 no. 5 (2003), 1387-1424 | EuDML | Numdam | MR | Zbl

[22] Isabelle Gallagher, Gabriel S. Koch, Fabrice Planchon, A profile decomposition approach to the $L_t^{\infty }\left(L_x^3\right)$ Navier–Stokes regularity criterion, arXiv:1012.0145 (2010) | MR | Zbl

[23] Patrick Gérard, Description du défaut de compacité de lʼinjection de Sobolev, ESAIM Control Optim. Calc. Var. 3 (1998), 213-233 | EuDML | Numdam | MR

[24] Loukas Grafakos, Modern Fourier Analysis, Graduate Texts in Mathematics vol. 250, Springer, New York (2009) | MR | Zbl

[25] Kato Tosio, Strong $L^p$-solutions of the Navier–Stokes equation in ${𝐑}^m$, with applications to weak solutions, Math. Z. 187 no. 4 (1984), 471-480 | EuDML | MR | Zbl

[26] Carlos E. Kenig, 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 no. 3 (2006), 645-675 | MR | Zbl

[27] Carlos E. Kenig, Frank Merle, Global well-posedness, scattering and blow-up for the energy-critical focusing non-linear wave equation, Acta Math. 201 no. 2 (2008), 147-212 | MR | Zbl

[28] Carlos E. Kenig, Frank Merle, Scattering for $H^{1/2}$ bounded solutions to the cubic, defocusing NLS in 3 dimensions, Trans. Amer. Math. Soc. 362 no. 4 (2010), 1937-1962 | Zbl

[29] Sahbi Keraani, On the defect of compactness for the Strichartz estimates of the Schrödinger equations, J. Differential Equations 175 no. 2 (2001), 353-392 | MR | Zbl

[30] Sahbi Keraani, On the blow up phenomenon of the critical nonlinear Schrödinger equation, J. Funct. Anal. 235 no. 1 (2006), 171-192 | MR | Zbl

[31] R. Killip, T. Tao, M. Visan, The cubic nonlinear Schrödinger equation in two dimensions with radial data, J. Eur. Math. Soc. (JEMS) 11 no. 6 (2009), 1203-1258 | MR | Zbl

[32] Rowan Killip, Monica Visan, Xiaoyi Zhang, The mass-critical nonlinear Schrödinger equation with radial data in dimensions three and higher, Anal. PDE 1 no. 2 (2008), 229-266 | MR | Zbl

[33] Gabriel Koch, Profile decompositions for critical Lebesgue and Besov space embeddings, Indiana Univ. Math. J., in press, http://www.iumj.indiana.edu/IUMJ/forthcoming.php, arXiv:1006.3064, 2010. | MR

[34] Gabriel Koch, Nikolai Nadirashvili, Gregory A. Seregin, Vladimír Šverák, Liouville theorems for the Navier–Stokes equations and applications, Acta Math. 203 no. 1 (2009), 83-105 | MR | Zbl

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

[36] Joachim Krieger, Wilhelm Schlag, Concentration Compactness for Critical Wave Maps, Monographs of the European Mathematical Society, in press, http://www.math.upenn.edu/~kriegerj/Papers.html, arXiv:0908.2474. | MR

[37] O.A. Ladyzenskaja, V.A. Solonnikov, N.N. UralʼCeva, Linear and Quasi-Linear Equations of Parabolic Type, American Mathematical Society, Providence, RI (1968) | MR

[38] P.G. Lemarié-Rieusset, Recent Developments in the Navier–Stokes Problem, Chapman & Hall/CRC Research Notes in Mathematics vol. 431, Chapman & Hall/CRC, Boca Raton, FL (2002) | MR | Zbl

[39] Fanghua Lin, A new proof of the Caffarelli–Kohn–Nirenberg theorem, Comm. Pure Appl. Math. 51 no. 3 (1998), 241-257 | MR | Zbl

[40] Changxing Miao, Xu. Guixiang, Lifeng Zhao, Global well-posedness and scattering for the energy-critical, defocusing Hartree equation in ${ℝ}^{1+n}$, arXiv:0707.3254 | Zbl

[41] Changxing Miao, Guixiang Xu, Lifeng Zhao, On the blow-up phenomenon for the mass-critical focusing Hartree equation in ${ℝ}^4$, Colloq. Math. 119 no. 1 (2010), 23-50 | EuDML | MR | Zbl

[42] Changxing Miao, Guixiang Xu, Lifeng Zhao, Global well-posedness and scattering for the energy-critical, defocusing Hartree equation for radial data, J. Funct. Anal. 253 no. 2 (2007), 605-627 | MR | Zbl

[43] Changxing Miao, Guixiang Xu, Lifeng Zhao, Global well-posedness and scattering for the mass-critical Hartree equation with radial data, J. Math. Pures Appl. (9) 91 no. 1 (2009), 49-79 | MR | Zbl

[44] Changxing Miao, Guixiang Xu, Lifeng Zhao, Global well-posedness, scattering and blow-up for the energy-critical, focusing Hartree equation in the radial case, Colloq. Math. 114 no. 2 (2009), 213-236 | EuDML | MR | Zbl

[45] Sylvie Monniaux, Uniqueness of mild solutions of the Navier–Stokes equation and maximal $L^p$-regularity, C. R. Acad. Sci. Paris Sér. I Math. 328 no. 8 (1999), 663-668 | MR | Zbl

[46] J. Nečas, M. Růžička, V. Šverák, On Lerayʼs self-similar solutions of the Navier–Stokes equations, Acta Math. 176 no. 2 (1996), 283-294 | MR | Zbl

[47] E. Ryckman, M. Visan, Global well-posedness and scattering for the defocusing energy-critical nonlinear Schrödinger equation in ${ℝ}^{1+4}$, Amer. J. Math. 129 no. 1 (2007), 1-60 | MR | Zbl

[48] V.A. Solonnikov, Estimates for solutions of a non-stationary linearized system of Navier–Stokes equations, Trudy Mat. Inst. Steklov. 70 (1964), 213-317 | MR | Zbl

[49] E.M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton, NJ (1970) | MR | Zbl

[50] Terence Tao, Global regularity of wave maps III. Large energy from ${ℝ}^{1+2}$ to hyperbolic spaces, arXiv:0805.4666

[51] Terence Tao, Global regularity of wave maps VI. Minimal-energy blowup solutions, arXiv:0906.2833 | Zbl

[52] Terence Tao, Monica Visan, Xiaoyi Zhang, Global well-posedness and scattering for the defocusing mass-critical nonlinear Schrödinger equation for radial data in high dimensions, Duke Math. J. 140 no. 1 (2007), 165-202 | MR | Zbl

[53] Terence Tao, Monica Visan, Xiaoyi Zhang, Minimal-mass blowup solutions of the mass-critical NLS, Forum Math. 20 no. 5 (2008), 881-919 | MR | Zbl

[54] T.-P. Tsai, On Lerayʼs self-similar solutions of the Navier–Stokes equations satisfying local energy estimates, Arch. Ration. Mech. Anal. 143 no. 1 (1998), 29-51 | MR | Zbl

[55] Monica Visan, The defocusing energy-critical nonlinear Schrödinger equation in higher dimensions, Duke Math. J. 138 no. 2 (2007), 281-374 | MR | Zbl