Concentration compacité à la Kenig-Merle
Séminaire Bourbaki volume 2011/2012 exposés 1043-1058, Astérisque, no. 352 (2013), Exposé no. 1046, 26 p.
Le texte intégral des articles récents est réservé aux abonnés de la revue. Consultez le site de la revue.
@incollection{AST_2013__352__121_0,
     author = {Rapha\"el, Pierre},
     title = {Concentration compacit\'e \`a la Kenig-Merle},
     booktitle = {S\'eminaire Bourbaki volume 2011/2012 expos\'es 1043-1058},
     author = {Collectif},
     series = {Ast\'erisque},
     note = {talk:1046},
     publisher = {Soci\'et\'e math\'ematique de France},
     number = {352},
     year = {2013},
     zbl = {1294.35147},
     mrnumber = {3087344},
     language = {fr},
     url = {archive.numdam.org/item/AST_2013__352__121_0/}
}
Raphaël, Pierre. Concentration compacité à la Kenig-Merle, dans Séminaire Bourbaki volume 2011/2012 exposés 1043-1058, Astérisque, no. 352 (2013), Exposé no. 1046, 26 p. http://archive.numdam.org/item/AST_2013__352__121_0/

[1] H. Bahouri & P. Gérard - High frequency approximation of solutions to critical nonlinear wave equations, Amer. J. Math. 121 (1999) , n° 1, p. 131-175. | Article | MR 1705001 | Zbl 0919.35089

[2] V. Banica - Remarks on the blow-up for the Schrödinger equation with critical mass on a plane domain, Ann. Sc. Norm. Super. Pisa Cl. Sci. 3 (2004) , n° 1, p. 139-170. | EuDML 84523 | Numdam | MR 2064970 | Zbl 1170.35528

[3] H. Berestycki & T. Cazenave - Instabilité des états stationnaires dans les équations de Schrödinger et de Klein-Gordon non linéaires, C. R. Acad. Sci. Paris Sér. I Math. 293 (1981) , n° 9, p. 489-492. | MR 646873 | Zbl 0492.35010

[4] J. Bourgain - Global wellposedness of defocusing critical nonlinear Schrödinger equation in the radial case, J. Amer. Math. Soc. 12 (1999) , n° 1, p. 145-171. | Article | MR 1626257 | Zbl 0958.35126

[5] T. Cazenave & P. L. Lions - Orbital stability of standing waves for some nonlinear Schrödinger equations, Comm. Math. Phys. 85 (1982) , n° 4, p. 549-561. | Article | MR 677997 | Zbl 0513.35007

[6] T. Cazenave & F. B. Weissler - The Cauchy problem for the critical nonlinear Schrödinger equation in H s , Nonlinear Anal. 14 (1990) , n° 10, p. 807-836. | Article | MR 1055532 | Zbl 0706.35127

[7] D. Christodoulou & A. S. Tahvildar-Zadeh - On the regularity of spherically symmetric wave maps, Comm. Pure Appl. Math. 46 (1993), n° 7, p. 1041-1091. | Article | MR 1223662 | Zbl 0744.58071

[8] J. Colliander, M. Keel, G. Staffilani, H. Takaoka & T. Tao - Global well-posedness and scattering for the energy-critical nonlinear Schrödinger equation in 3 , Ann. of Math. 167 (2008), n° 3, p. 767-865. | Article | MR 2415387 | Zbl 1178.35345

[9] B. Dodson - Global well-posedness and scattering for the defocusing, L 2 -critical nonlinear Schrödinger equation when d3, J. Amer. Math. Soc. 25 (2012), n° 2, p. 429-463. | Article | MR 2869023 | Zbl 1236.35163

[10] B. Dodson, Global well-posedness and scattering for the defocusing, L 2 -critical non­linear Schrödinger equation when d=1, prépublication arXiv: 1010.0040. | Zbl 1341.35149

[11] B. Dodson, Global well-posedness and scattering for the defocusing, L 2 -critical non­linear Schrödinger equation when d=2, prépublication arXiv: 1006.1375.

[12] B. Dodson, Global well-posedness and scattering for the mass critical nonlinear Schrödinger equation with mass below the mass of the ground state, prépublication arXiv:1104.1114. | Article | MR 3406535

[13] T. Duyckaerts, C. E. Kenig & F. Merle - Universality of the blow-up profile for small type II blow-up solutions of energy-critical wave equation : the non-radial case, prépublication arXiv: 1003.0625. | Zbl 1282.35088

[14] T. Duyckaerts & F. Merle - Dynamics of threshold solutions for energy-critical wave equation, Int. Math. Res. Pap. IMRP (2008), p. Art ID rpn002, 67. | MR 2470571 | Zbl 1159.35043

[15] T. Duyckaerts & F. Merle, Dynamic of threshold solutions for energy-critical NLS, Geom. Funct. Anal. 18 (2009), n° 6, p. 1787-1840. | Article | MR 2491692 | Zbl 1232.35150

[16] T. Duyckaerts & S. Roudenko - Threshold solutions for the focusing 3D cubic Schrödinger equation, Rev. Mat. Iberoam. 26 (2010), n° 1, p. 1-56. | Article | MR 2662148 | Zbl 1195.35276

[17] P. Gérard - Description du défaut de compacité de l'injection de Sobolev, ESAIM Control Optim. Calc. Var. 3 (1998), p. 213-233. | Article | EuDML 90520 | Numdam | MR 1632171 | Zbl 0907.46027

[18] B. Gidas, W. M. Ni & L. Nirenberg - Symmetry and related properties via the maximum principle, Comm. Math. Phys. 68 (1979), n° 3, p. 209-243. | Article | MR 544879 | Zbl 0425.35020

[19] J. Ginibre & G. Velo - Existence of solutions and scattering theory for the nonlinear Schrödinger equation, in Proceedings of the International Conference on Operator Algebras, Ideals, and their Applications in Theoretical Physics (Leipzig, 1977), Teubner, 1978, p. 320-334. | MR 528288 | Zbl 0402.35077

[20] J. Ginibre & G. Velo, On a class of nonlinear Schrödinger equations. I. The Cauchy problem, general case, J. Funct. Anal. 32 (1979), n° 1, p. 1-32. | Article | MR 533218 | Zbl 0396.35028

[21] M. G. Grillakis - Regularity and asymptotic behaviour of the wave equation with a critical nonlinearity, Ann. of Math. 132 (1990), n° 3, p. 485-509. | Article | MR 1078267 | Zbl 0736.35067

[22] T. Hmidi & S. Keraani - Remarks on the blowup for the L 2 -critical nonlinear Schrödinger equations, SIAM J. Math. Anal. 38 (2006), n° 4, p. 1035-1047. | Article | MR 2274472 | Zbl 1122.35135

[23] L. Iskauriaza, G. A. Serëgin & V. Shverak - L 3, -solutions of Navier-Stokes equations and backward uniqueness, Uspekhi Mat. Nauk 58 (2003), n° 2(350), p. 3-44. | MR 1992563 | Zbl 1064.35134

[24] C. E. Kenig & F. Merle - Global well-posedness, scattering and blow-up for the energy-critical, focusing, non-linear Schrödinger equation in the radial case, Invent. Math. 166 (2006), n° 3, p. 645-675. | Article | MR 2257393 | Zbl 1115.35125

[25] C. E. Kenig & F. Merle, Global well-posedness, scattering and blow-up for the energy-critical focusing non-linear wave equation, Acta Math. 201 (2008), n° 2, p. 147-212. | Article | MR 2461508 | Zbl 1183.35202

[26] C. E. Kenig & F. Merle, Nondispersive radial solutions to energy supercritical non-linear wave equations, with applications, Amer. J. Math. 133 (2011), n° 4, p. 1029-1065. | Article | MR 2823870 | Zbl 1241.35136

[27] S. Keraani - On the defect of compactness for the Strichartz estimates of the Schrödinger equations, J. Differential Equations 175 (2001), n° 2, p. 353-392. | Article | MR 1855973 | Zbl 1038.35119

[28] R. Killip, T. Tao & M. Visan - The cubic nonlinear Schrödinger equation in two dimensions with radial data, J. Eur. Math. Soc. (JEMS) 11 (2009), n° 6, p. 1203-1258. | Article | MR 2557134 | Zbl 1187.35237

[29] R. Killip & M. Visan - The focusing energy-critical nonlinear Schrödinger equation in dimensions five and higher, Amer. J. Math. 132 (2010), n° 2, p. 361-424. | Article | MR 2654778 | Zbl 1208.35138

[30] R. Killip, M. Visan & X. Zhang - The mass-critical nonlinear Schrödinger equation with radial data in dimensions three and higher, Anal. PDE 1 (2008), n° 2, p. 229-266. | Article | MR 2472890 | Zbl 1171.35111

[31] J. Krieger & W. Schlag - Concentration compactness for critical wave maps, EMS Monographs in Math., European Mathematical Society (EMS), Zurich, 2012. | MR 2895939

[32] J. Krieger, W. Schlag & D. Tataru - Renormalization and blow up for charge one equivariant critical wave maps, Invent. Math. 171 (2008), n° 3, p. 543-615. | Article | MR 2372807 | Zbl 1139.35021

[33] M. K. Kwong - Uniqueness of positive solutions of Δu-u+u p =0 in 𝐑 n , Arch. Rational Mech. Anal. 105 (1989), n° 3, p. 243-266. | MR 969899 | Zbl 0676.35032

[34] P.-L. Lions - The concentration-compactness principle in the calculus of variations. The locally compact case. I, Ann. Inst. H. Poincaré Anal. Non Linéaire 1 (1984), n° 2, p. 109-145. | Article | EuDML 78069 | Numdam | MR 778970 | Zbl 0541.49009

[35] Y. Martel & F. Merle - A Liouville theorem for the critical generalized Korteweg-de Vries equation, J. Math. Pures Appl. 79 (2000), n° 4, p. 339-425. | Article | MR 1753061 | Zbl 0963.37058

[36] Y. Martel & F. Merle, Blow up in finite time and dynamics of blow up solutions for the L 2 ­critical generalized KdV equation, J. Amer. Math. Soc. 15 (2002), n° 3, p. 617-664. | Article | MR 1896235 | Zbl 0996.35064

[37] Y. Martel & F. Merle, Stability of blow-up profile and lower bounds for blow-up rate for the critical generalized KdV equation, Ann. of Math. 155 (2002), n° 1, p. 235-280. | Article | MR 1888800 | Zbl 1005.35081

[38] H. Matano & F. Merle - On nonexistence of type II blowup for a supercritical nonlinear heat equation, Comm. Pure Appl. Math. 57 (2004), n° 11, p. 1494-1541. | Article | MR 2077706 | Zbl 1112.35098

[39] H. Matano & F. Merle, Classification of type I and type II behaviors for a supercritical nonlinear heat equation, J. Funct. Anal. 256 (2009), n° 4, p. 992-1064. | Article | MR 2488333 | Zbl 1178.35084

[40] H. Matano & F. Merle, Threshold and generic type I behaviors for a supercritical nonlinear heat equation, J. Funct. Anal. 261 (2011), n° 3, p. 716-748. | Article | MR 2799578 | Zbl 1223.35088

[41] F. Merle - On uniqueness and continuation properties after blow-up time of self-similar solutions of nonlinear Schrödinger equation with critical exponent and critical mass, Comm. Pure Appl. Math. 45 (1992), n° 2, p. 203-254. | Article | MR 1139066 | Zbl 0767.35084

[42] F. Merle, Existence of blow-up solutions in the energy space for the critical generalized KdV equation, J. Amer. Math. Soc. 14 (2001), n° 3, p. 555-578. | Article | MR 1824989 | Zbl 0970.35128

[43] F. Merle & P. Raphael - On universality of blow-up profile for L 2 critical nonlinear Schrödinger equation, Invent. Math. 156 (2004), n° 3, p. 565-672. | Article | MR 2061329 | Zbl 1067.35110

[44] F. Merle & P. Raphael, Profiles and quantization of the blow up mass for critical nonlinear Schrödinger equation, Comm. Math. Phys. 253 (2005), n° 3, p. 675-704. | Article | MR 2116733 | Zbl 1062.35137

[45] F. Merle & P. Raphaël - Blow up of the critical norm for some radial L 2 super critical nonlinear Schrödinger equations, Amer. J. Math. 130 (2008), n° 4, p. 945-978. | Article | MR 2427005 | Zbl 1188.35182

[46] F. Merle, P. Raphaël & I. Rodnianski - Blow up dynamics for smooth equivariant solutions to the energy critical Schrödinger map, C. R. Math. Acad. Sci. Paris 349 (2011), nos 5-6, p. 279-283. | Article | MR 2783320 | Zbl 1213.35139

[47] F. Merle, P. Raphaël & J. Szeftel - Stable self-similar blow-up dynamics for slightly L 2 super-critical NLS equations, Geom. Funct. Anal. 20 (2010), n° 4, p. 1028-1071. | Article | MR 2729284 | Zbl 1204.35153

[48] F. Merle, P. Raphaël & J. Szeftel - The instability of Bourgain Wang solutions for the mass critical NLS, à paraître dans Amer. J. Math. | Zbl 1294.35145

[49] F. Merle & L. Vega - Compactness at blow-up time for L 2 solutions of the critical nonlinear Schrödinger equation in 2D, Int. Math. Res. Not. 1998 (1998), n° 8, p. 399-425. | Article | MR 1628235 | Zbl 0913.35126

[50] F. Merle & H. Zaag - Optimal estimates for blowup rate and behavior for nonlinear heat equations, Comm. Pure Appl. Math. 51 (1998), n° 2, p. 139-196. | Article | MR 1488298 | Zbl 0899.35044

[51] F. Merle & H. Zaag, Determination of the blow-up rate for the semilinear wave equation, Amer. J. Math. 125 (2003), n° 5, p. 1147-1164. | Article | MR 2004432 | Zbl 1052.35043

[52] F. Merle & H. Zaag, Existence and universality of the blow-up profile for the semilinear wave equation in one space dimension, J. Funct. Anal. 253 (2007), n° 1, p. 43-121. | Article | MR 2362418 | Zbl 1133.35070

[53] F. Merle & H. Zaag, Existence and classification of characteristic points at blow-up for a semilinear wave equation in one space dimension, Amer. J. Math. 134 (2012), n° 3, p. 581-648. | Article | MR 2931219 | Zbl 1252.35204

[54] N. Mizoguchi - Rate of type II blowup for a semilinear heat equation, Math. Ann. 339 (2007), n° 4, p. 839-877. | Article | MR 2341904 | Zbl 1172.35420

[55] K. Nakanishi & W. Schlag - Global dynamics above the ground state energy for the focusing nonlinear Klein-Gordon equation, J. Differential Equations 250 (2011), n° 5, p. 2299-2333. | Article | MR 2756065 | Zbl 1213.35307

[56] K. Nakanishi & W. Schlag, Global dynamics above the ground state energy for the cubic NLS equation in 3D, Calc. Var. Partial Differential Equations 44 (2012), nos 1-2, p. 1-45. | Article | MR 2898769 | Zbl 1237.35148

[57] G. Perelman - On the blow up phenomenon for the critical nonlinear Schrödinger equation in 1D, in Nonlinear dynamics and renormalization group (Montreal, QC, 1999), CRM Proc. Lecture Notes, vol. 27, Amer. Math. Soc., 2001, p. 147-164. | Article | MR 1826598 | Zbl 1055.35116

[58] F. Planchon - Existence globale et scattering pour les solutions de masse finie de l'équation de Schrödinger cubique en dimension deux (d'après Benjamin Dodson, Rowan Killip, Terence Tao, Monica Vişan et Xiaoyi Zhang), Sém. Bourbaki (2010/11), exp. n° 1042, Astérisque 348 (2012), p. 425-447. | MR 3051205 | Zbl 1296.35176

[59] P. Raphaël - Existence and stability of a solution blowing up on a sphere for an L 2 -supercritical nonlinear Schrödinger equation, Duke Math. J. 134 (2006), n° 2, p. 199-258. | Article | MR 2248831 | Zbl 1117.35077

[60] P. Raphaël, Stability and blow up for the non linear Schrödinger equation, in Clay summer school on nonlinear evolution equations, Zurich, 2008, .

[61] P. Raphaël & I. Rodnianski - Stable blow up dynamics for critical corotational wave maps and the equivariant Yang Mills problems, à paraître dans Publ. Math. IHÉS. | Zbl 1284.35358

[62] P. Raphaël & J. Szeftel - Standing ring blow up solutions to the N-dimensional quintic nonlinear Schrödinger equation, Comm. Math. Phys. 290 (2009), n° 3, p. 973-996. | Article | MR 2525647 | Zbl 1184.35295

[63] I. Rodnianski & J. Sterbenz - On the formation of singularities in the critical O(3)σ-model, Ann. of Math. 172 (2010), n° 1, p. 187-242. | Article | MR 2680419 | Zbl 1213.35392

[64] J. Sterbenz & D. Tataru - Energy dispersed large data wave maps in 2+1 dimensions, Comm. Math. Phys. 298 (2010), n° 1, p. 139-230. | Article | MR 2657817 | Zbl 1218.35129

[65] J. Sterbenz & D. Tataru, Regularity of wave-maps in dimension 2+1, Comm. Math. Phys. 298 (2010), n° 1, p. 231-264. | Article | MR 2657818 | Zbl 1218.35057

[66] R. S. Strichartz - Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations, Duke Math. J. 44 (1977), n° 3, p. 705-714. | Article | MR 512086 | Zbl 0372.35001

[67] M. Struwe - On the evolution of harmonic mappings of Riemannian surfaces, Comment Math. Helv. 60 (1985), n° 4, p. 558-581. | Article | EuDML 140031 | MR 826871 | Zbl 0595.58013

[68] M. Struwe, Globally regular solutions to the u 5 Klein-Gordon equation, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 15 (1988), n° 3, p. 495-513. | EuDML 84039 | Numdam | MR 1015805 | Zbl 0728.35072

[69] M. Struwe, Equivariant wave maps in two space dimensions, Comm. Pure Appl. Math. 56 (2003), n° 7, p. 815-823. | Article | MR 1990477 | Zbl 1033.53019

[70] T. Tao - Global regularity of wave maps III-VII, prépublication arXiv:0908.0776.

[71] T. Tao, M. Visan & X. Zhang - Global well-posedness and scattering for the defocusing mass-critical nonlinear Schrödinger equation for radial data in high dimensions, Duke Math. J. 140 (2007), n° 1, p. 165-202. | Article | MR 2355070 | Zbl 1187.35246

[72] M. I. Weinstein - Nonlinear Schrödinger equations and sharp interpolation estimates, Comm. Math. Phys. 87 (1982/83), n° 4, p. 567-576. | Article | MR 691044 | Zbl 0527.35023