Global existence for energy critical waves in 3-d domains

Burq, Nicolas; Lebeau, Gilles; Planchon, Fabrice

doi:10.1090/S0894-0347-08-00596-1

Global existence for energy critical waves in $3$-d domains
HTML articles powered by AMS MathViewer

by Nicolas Burq, Gilles Lebeau and Fabrice Planchon;

J. Amer. Math. Soc. 21 (2008), 831-845

DOI: https://doi.org/10.1090/S0894-0347-08-00596-1

Published electronically: January 31, 2008

PDF | Request permission

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.

References

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, DOI 10.1353/ajm.2004.0016
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 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 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 10.1155/S1073792896000153

Journées “Équations aux Dérivées Partielles” (Evian, 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 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 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, 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 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 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 10.1016/S0022-0396(02)00177-8
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 10.1090/S0894-0347-01-00375-7