Lecture notes : Global Well-posedness, scattering and blow up for the energy-critical, focusing, non-linear Schrödinger and wave equations
Journées équations aux dérivées partielles (2007), article no. 1, 35 p.
@article{JEDP_2007____A1_0,
author = {Kenig, Carlos E.},
title = {Lecture notes : {Global} {Well-posedness,} scattering and blow up for the energy-critical, focusing, non-linear {Schr\"odinger} and wave equations},
journal = {Journ\'ees \'equations aux d\'eriv\'ees partielles},
eid = {1},
publisher = {Groupement de recherche 2434 du CNRS},
year = {2007},
doi = {10.5802/jedp.40},
language = {en},
url = {http://www.numdam.org/articles/10.5802/jedp.40/}
}
TY  - JOUR
AU  - Kenig, Carlos E.
TI  - Lecture notes : Global Well-posedness, scattering and blow up for the energy-critical, focusing, non-linear Schrödinger and wave equations
JO  - Journées équations aux dérivées partielles
PY  - 2007
DA  - 2007///
PB  - Groupement de recherche 2434 du CNRS
UR  - http://www.numdam.org/articles/10.5802/jedp.40/
UR  - https://doi.org/10.5802/jedp.40
DO  - 10.5802/jedp.40
LA  - en
ID  - JEDP_2007____A1_0
ER  - 
Kenig, Carlos E. Lecture notes : Global Well-posedness, scattering and blow up for the energy-critical, focusing, non-linear Schrödinger and wave equations. Journées équations aux dérivées partielles (2007), article  no. 1, 35 p. doi : 10.5802/jedp.40. http://www.numdam.org/articles/10.5802/jedp.40/

[1] Bahouri, H.; Gérard, P. High frequency approximation of solutions to critical nonlinear wave equations, American Journal of Mathematics, Volume 121 (1999), pp. 131-175 | MR 1705001 | Zbl 0919.35089

[2] Bahouri, Hajer; Shatah, Jalal Decay estimates for the critical semilinear wave equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, Volume 15 (1998) no. 6, pp. 783-789 | Numdam | MR 1650958 | Zbl 0924.35084

[3] Bourgain, J. Global wellposedness of defocusing critical nonlinear Schrödinger equations in the radial case, J. Amer. Math. Soc., Volume 12 (1999), pp. 145-171 | MR 1626257 | Zbl 0958.35126

[4] Cazenave, T.; Weissler, F. The Cauchy problem for the critical nonlinear Schrödinger equation in ${H}^{s}$, Nonlinear Analysis, Theory, Methods and Applications (1990), pp. 807-836 | MR 1055532 | Zbl 0706.35127

[5] Colliander, J.; Keel, M.; Staffilani, G.; Takaoka, H.; Tao, T. Global well-posedness and scattering for the energy-critical nonlinear Schrödinger equation in ${ℝ}^{3}$, Ann. of Math. (2), Volume 167 (2008) no. 3, pp. 767-865 | MR 2415387 | Zbl 1178.35345

[6] Foschi, Damiano Inhomogeneous Strichartz estimates, J. Hyperbolic Differ. Equ., Volume 2 (2005) no. 1, pp. 1-24 | MR 2134950 | Zbl 1071.35025

[7] Gerard, Patrick; Meyer, Yves; Oru, Frédérique Inégalités de Sobolev précisées, Séminaire sur les Équations aux Dérivées Partielles, 1996–1997, École Polytech., Palaiseau, 1997, pp. Exp. No. IV, 11 | Numdam | MR 1482810 | Zbl 1066.46501

[8] Giga, Yoshikazu; Kohn, Robert V. Nondegeneracy of blowup for semilinear heat equations, Comm. Pure Appl. Math., Volume 42 (1989) no. 6, pp. 845-884 | MR 1003437 | Zbl 0703.35020

[9] Ginibre, J.; Soffer, A.; Velo, G. The global Cauchy problem for the critical nonlinear wave equation, J. Funct. Anal., Volume 110 (1992) no. 1, pp. 96-130 | MR 1190421 | Zbl 0813.35054

[10] Ginibre, J.; Velo, G. Generalized Strichartz inequalities for the wave equation, Journal of Functional Analysis, Volume 1 (1995), pp. 50-68 | MR 1351643 | Zbl 0849.35064

[11] Glassey, R. T. On the blowing up of solutions to the Cauchy problem for nonlinear Schrödinger equations, J. Math. Phys., Volume 18 (1977) no. 9, pp. 1794-1797 | MR 460850 | Zbl 0372.35009

[12] Grillakis, Manoussos G. Regularity and asymptotic behaviour of the wave equation with a critical nonlinearity, Ann. of Math. (2), Volume 132 (1990) no. 3, pp. 485-509 | MR 1078267 | Zbl 0736.35067

[13] Hörmander, L. The Analysis of Linear Partial Differential Operators III, Grundlehren der mathematischen Wissenschaften, 274, Springer Verlag, 1985 | MR 781536 | Zbl 0601.35001

[14] Kapitanski, Lev Global and unique weak solutions of nonlinear wave equations, Math. Res. Lett., Volume 1 (1994) no. 2, pp. 211-223 | MR 1266760 | Zbl 0841.35067

[15] Keel, M.; Tao, T. Endpoint Strichartz estimates, Amer. Jour. of Math. (1998), pp. 955-980 | MR 1646048 | Zbl 0922.35028

[16] Kenig, Carlos E.; Merle, Frank Global well-posedness, scattering and blow-up for the energy-critical, focusing, non-linear Schrödinger equation in the radial case, Invent. Math., Volume 166 (2006) no. 3, pp. 645-675 | MR 2257393 | Zbl 1115.35125

[17] Kenig, Carlos E.; Merle, Frank Global well-posedness, scattering and blow-up for the energy-critical, focusing, non-linear wave equation, To appear, Acta Math. (2008) | MR 2461508 | Zbl 1183.35202

[18] Kenig, Carlos E.; Merle, Frank Scattering for the ${\stackrel{˙}{H}}^{1/2}$ bounded solutions to the cubic defocusing NLS in $3$ dimensions, To appear, Trans. A.M.S. (2008) | MR 2574882

[19] Kenig, Carlos E.; Ponce, Gustavo; Vega, Luis Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle, Comm. Pure Appl. Math., Volume 46 (1993) no. 4, pp. 527-620 | MR 1211741 | Zbl 0808.35128

[20] Keraani, Sahbi 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

[21] Krieger, J.; Schlag, W.; Tataru, D. Slow blow-up solutions for the ${H}^{1}\left({ℝ}^{3}\right)$ critical focusing semi-linear wave equation in ${ℝ}^{3}$, arxiv:math.AP/0711.1818 (2007) | MR 2494455

[22] Levine, Howard A. Instability and nonexistence of global solutions to nonlinear wave equations of the form $P{u}_{tt}=-Au+ℱ\left(u\right)$, Trans. Amer. Math. Soc., Volume 192 (1974), pp. 1-21 | MR 344697 | Zbl 0288.35003

[23] Merle, Frank; Zaag, Hatem Determination of the blow-up rate for a critical semilinear wave equation, Math. Ann., Volume 331 (2005) no. 2, pp. 395-416 | MR 2115461 | Zbl 1136.35055

[24] Raphaël, Pierre Existence and stability of a solution blowing up on a sphere for an ${L}^{2}$-supercritical nonlinear Schrödinger equation, Duke Math. J., Volume 134 (2006) no. 2, pp. 199-258 | MR 2248831 | Zbl 1117.35077

[25] Ryckman, E.; Visan, M. Global well-posedness and scattering for the defocusing energy-critical nonlinear Schrödinger equation in ${ℝ}^{1+4}$, Amer. J. Math., Volume 129 (2007) no. 1, pp. 1-60 | MR 2288737 | Zbl 1160.35067

[26] Shatah, Jalal; Struwe, Michael Well-posedness in the energy space for semilinear wave equations with critical growth, Internat. Math. Res. Notices (1994) no. 7, pp. 303ff., approx. 7 pp. (electronic) | MR 1283026 | Zbl 0830.35086

[27] Shatah, Jalal; Struwe, Michael Geometric wave equations, Courant Lecture Notes in Mathematics, 2, New York University Courant Institute of Mathematical Sciences, New York, 1998 | MR 1674843 | Zbl 0993.35001

[28] Strichartz, Robert S. Restrictions of Fourier transforms to quadratic surfaces and decay of solutions of wave equations, Duke Math. J., Volume 44 (1977) no. 3, pp. 705-714 | MR 512086 | Zbl 0372.35001

[29] Struwe, Michael Globally regular solutions to the ${u}^{5}$ Klein-Gordon equation, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), Volume 15 (1988) no. 3, p. 495-513 (1989) | EuDML 84039 | Numdam | MR 1015805 | Zbl 0728.35072

[30] Tao, Terence Global well-posedness and scattering for the higher-dimensional energy-critical nonlinear Schrödinger equation for radial data, New York J. Math., Volume 11 (2005), p. 57-80 (electronic) | EuDML 127869 | MR 2154347 | Zbl 1119.35092

[31] Trudinger, Neil S. Remarks concerning the conformal deformation of Riemannian structures on compact manifolds, Ann. Scuola Norm. Sup. Pisa (3), Volume 22 (1968), pp. 265-274 | EuDML 83458 | Numdam | MR 240748 | Zbl 0159.23801

[32] Vilela, M. C. Inhomogeneous Strichartz estimates for the Schrödinger equation, Trans. Amer. Math. Soc., Volume 359 (2007) no. 5, p. 2123-2136 (electronic) | MR 2276614 | Zbl 1196.35074

[33] Visan, Monica The defocusing energy-critical nonlinear Schrödinger equation in higher dimensions, Duke Math. J., Volume 138 (2007) no. 2, pp. 281-374 | MR 2318286 | Zbl 1131.35081

Cité par Sources :