Global existence for energy critical waves in $3$-d domains
Authors:
Nicolas Burq, Gilles Lebeau and Fabrice Planchon
Journal:
J. Amer. Math. Soc. 21 (2008), 831-845
MSC (2000):
Primary 35L05, 35L70
DOI:
https://doi.org/10.1090/S0894-0347-08-00596-1
Published electronically:
January 31, 2008
MathSciNet review:
2393429
Full-text PDF Free Access
Abstract | References | Similar Articles | Additional Information
Abstract: We prove that the defocusing quintic wave equation, with Dirichlet boundary conditions, is globally well posed on $H^1_0(\Omega ) \times L^2( \Omega )$ for any smooth (compact) domain $\Omega \subset \mathbb {R}^3$. The main ingredient in the proof is an $L^5$ spectral projector estimate, obtained recently by Smith and Sogge, combined with a precise study of the boundary value problem.
-
An05 Ramona Anton. Strichartz inequalities for lipschitz metrics on manifolds and nonlinear Schrödinger equation on domains, 2005. Preprint, arXiv:arXiv:math.AP/0512639.
- N. Burq, P. Gérard, and N. Tzvetkov, Strichartz inequalities and the nonlinear Schrödinger equation on compact manifolds, Amer. J. Math. 126 (2004), no. 3, 569–605. MR 2058384
- Jacques Chazarain and Alain Piriou, Introduction à la théorie des équations aux dérivées partielles linéaires, Gauthier-Villars, Paris, 1981 (French). MR 598467
- Jacques Chazarain and Alain Piriou, Introduction to the theory of linear partial differential equations, Studies in Mathematics and its Applications, vol. 14, North-Holland Publishing Co., Amsterdam-New York, 1982. Translated from the French. MR 678605
- Michael Christ and Alexander Kiselev, Maximal functions associated to filtrations, J. Funct. Anal. 179 (2001), no. 2, 409–425. MR 1809116, DOI https://doi.org/10.1006/jfan.2000.3687
- Manoussos G. Grillakis, Regularity and asymptotic behaviour of the wave equation with a critical nonlinearity, Ann. of Math. (2) 132 (1990), no. 3, 485–509. MR 1078267, DOI https://doi.org/10.2307/1971427
- Sergiu Klainerman and Matei Machedon, Remark on Strichartz-type inequalities, Internat. Math. Res. Notices 5 (1996), 201–220. With appendices by Jean Bourgain and Daniel Tataru. MR 1383755, DOI https://doi.org/10.1155/S1073792896000153 Leproc06 Gilles Lebeau. Estimation de dispersion pour les ondes dans un convexe. In Journées “Équations aux Dérivées Partielles” (Evian, 2006), 2006.
- Gerd Mockenhaupt, Andreas Seeger, and Christopher D. Sogge, Local smoothing of Fourier integral operators and Carleson-Sjölin estimates, J. Amer. Math. Soc. 6 (1993), no. 1, 65–130. MR 1168960, DOI https://doi.org/10.1090/S0894-0347-1993-1168960-6
- Jeffrey Rauch, I. The $u^{5}$ Klein-Gordon equation. II. Anomalous singularities for semilinear wave equations, Nonlinear partial differential equations and their applications. Collège de France Seminar, Vol. I (Paris, 1978/1979) Res. Notes in Math., vol. 53, Pitman, Boston, Mass.-London, 1981, pp. 335–364. MR 631403
- Jalal Shatah and Michael Struwe, Regularity results for nonlinear wave equations, Ann. of Math. (2) 138 (1993), no. 3, 503–518. MR 1247991, DOI https://doi.org/10.2307/2946554
- Jalal Shatah and Michael Struwe, Well-posedness in the energy space for semilinear wave equations with critical growth, Internat. Math. Res. Notices 7 (1994), 303ff., approx. 7 pp.}, issn=1073-7928, review=\MR{1283026}, doi=10.1155/S1073792894000346,.
- Hart F. Smith and Christopher D. Sogge, On the critical semilinear wave equation outside convex obstacles, J. Amer. Math. Soc. 8 (1995), no. 4, 879–916. MR 1308407, DOI https://doi.org/10.1090/S0894-0347-1995-1308407-1
- Hart F. Smith and Christopher D. Sogge, On the $L^p$ norm of spectral clusters for compact manifolds with boundary, Acta Math. 198 (2007), no. 1, 107–153. MR 2316270, DOI https://doi.org/10.1007/s11511-007-0014-z
- Michael Struwe, Globally regular solutions to the $u^5$ Klein-Gordon equation, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 15 (1988), no. 3, 495–513 (1989). MR 1015805
- Terence Tao, Local well-posedness of the Yang-Mills equation in the temporal gauge below the energy norm, J. Differential Equations 189 (2003), no. 2, 366–382. MR 1964470, DOI https://doi.org/10.1016/S0022-0396%2802%2900177-8
- Daniel Tataru, Strichartz estimates for second order hyperbolic operators with nonsmooth coefficients. III, J. Amer. Math. Soc. 15 (2002), no. 2, 419–442. MR 1887639, DOI https://doi.org/10.1090/S0894-0347-01-00375-7
Retrieve articles in Journal of the American Mathematical Society with MSC (2000): 35L05, 35L70
Retrieve articles in all journals with MSC (2000): 35L05, 35L70
Additional Information
Nicolas Burq
Affiliation:
Laboratoire de Mathématiques, Université Paris Sud, UMR 8628 du C.N.R.S., Bât 425, 91405 Orsay Cedex, France and Institut Universitaire de France
MR Author ID:
315457
Email:
Nicolas.burq@math.u-psud.fr
Gilles Lebeau
Affiliation:
Laboratoire J.-A. Dieudonné, UMR 6621 du C.N.R.S, Université de Nice - Sophia Antipolis, Parc Valrose 06108 Nice Cedex 02, France and Institut Universitaire de France
Email:
lebeau@math.unice.fr
Fabrice Planchon
Affiliation:
Laboratoire Analyse, Géométrie & Applications, UMR 7539 du C.N.R.S, Institut Galilée, Université Paris 13, 99 avenue J.B. Clément, F-93430 Villetaneuse, France
Email:
fab@math.univ-paris13.fr
Keywords:
Wave equation,
Dirichlet boundary conditions.
Received by editor(s):
July 27, 2006
Published electronically:
January 31, 2008
Additional Notes:
The third author was partially supported by A.N.R. grant ONDE NON LIN
Article copyright:
© Copyright 2008
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.
