Global existence for coupled Klein-Gordon equations with different speeds
[Solutions globales pour des équations de Klein-Gordon avec des vitesses différentes couplées]
Annales de l'Institut Fourier, Tome 61 (2011) no. 6, pp. 2463-2506.

Soit, en dimension 3, un système d’équations de Klein-Gordon dont les vitesses sont différentes, avec des termes non-linéaires quadratiques. On montre, pour des données suffisamment petites, regulières et localisées, qu’une solution globale existe et qu’elle disperse. La preuve repose sur la méthode des résonances en espace-temps. La structure des résonances du système se trouve être d’un type qui n’avait pas été étudié jusqu’ici, mais qui est générique dans un certain sens.

Consider, in dimension 3, a system of coupled Klein-Gordon equations with different speeds, and an arbitrary quadratic nonlinearity. We show, for data which are small, smooth, and localized, that a global solution exists, and that it scatters. The proof relies on the space-time resonance approach; it turns out that the resonant structure of this equation has features which were not studied before, but which are generic in some sense.

DOI : 10.5802/aif.2680
Classification : 35L70, 47H60
Keywords: Klein-Gordon, global existence, resonances
Mot clés : Klein-Gordon, existence globale, résonances
Germain, Pierre 1

1 Courant Institute of Mathematical Sciences New York University 251 Mercer Street New York, N.Y. 10012-1185 USA
@article{AIF_2011__61_6_2463_0,
     author = {Germain, Pierre},
     title = {Global existence for coupled {Klein-Gordon} equations with different speeds},
     journal = {Annales de l'Institut Fourier},
     pages = {2463--2506},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {61},
     number = {6},
     year = {2011},
     doi = {10.5802/aif.2680},
     zbl = {1255.35162},
     mrnumber = {2976318},
     language = {en},
     url = {http://archive.numdam.org/articles/10.5802/aif.2680/}
}
TY  - JOUR
AU  - Germain, Pierre
TI  - Global existence for coupled Klein-Gordon equations with different speeds
JO  - Annales de l'Institut Fourier
PY  - 2011
SP  - 2463
EP  - 2506
VL  - 61
IS  - 6
PB  - Association des Annales de l’institut Fourier
UR  - http://archive.numdam.org/articles/10.5802/aif.2680/
DO  - 10.5802/aif.2680
LA  - en
ID  - AIF_2011__61_6_2463_0
ER  - 
%0 Journal Article
%A Germain, Pierre
%T Global existence for coupled Klein-Gordon equations with different speeds
%J Annales de l'Institut Fourier
%D 2011
%P 2463-2506
%V 61
%N 6
%I Association des Annales de l’institut Fourier
%U http://archive.numdam.org/articles/10.5802/aif.2680/
%R 10.5802/aif.2680
%G en
%F AIF_2011__61_6_2463_0
Germain, Pierre. Global existence for coupled Klein-Gordon equations with different speeds. Annales de l'Institut Fourier, Tome 61 (2011) no. 6, pp. 2463-2506. doi : 10.5802/aif.2680. http://archive.numdam.org/articles/10.5802/aif.2680/

[1] Christodoulou, D. Global solutions of nonlinear hyperbolic equations for small initial data, Comm. Pure Appl. Math., Volume 39 (1986) no. 2, pp. 267-282 | DOI | MR | Zbl

[2] Coifman, R.; Meyer, Y. Au delà des opérateurs pseudo-différentiels, Astérisque, 57, Société Mathématique de France, Paris, 1978 | Numdam | MR | Zbl

[3] Delort, J.-M.; Fang, D. Almost global existence for solutions of semilinear Klein-Gordon equations with small weakly decaying Cauchy data, Comm. Partial Differential Equations, Volume 25 (2000) no. 11-12, p. 2119-1269 | DOI | MR | Zbl

[4] Delort, J.-M.; Fang, D.; Xue, R. Global existence of small solutions for quadratic quasilinear Klein-Gordon systems in two space dimensions, J. Funct. Anal., Volume 211 (2004) no. 2, pp. 288-323 | DOI | MR | Zbl

[5] Germain, P.; Masmoudi, N.; Shatah, J. Global solutions for 2D quadratic Schrödinger equations. (preprint)

[6] Germain, P.; Masmoudi, N.; Shatah, J. Global solutions for the gravity water waves equation in dimension 3 (preprint) | Zbl

[7] Germain, P.; Masmoudi, N.; Shatah, J. Global solutions for 3D quadratic Schrödinger equations, Int. Math. Res. Not. IMRN (2009) no. 3, pp. 414-432 | MR | Zbl

[8] Ginibre, J.; Velo, G. Time decay of finite energy solutions of the nonlinear Klein-Gordon and Schrödinger equations, Ann. Inst. H. Poincaré Phys. Théor., Volume 43 (1985) no. 4, pp. 399-442 | Numdam | MR | Zbl

[9] Hayashi, N.; Naumkin, P.; Wibowo, R. Nonlinear scattering for a system of nonlinear Klein-Gordon equations, J. Math. Phys., Volume 49 (2008) no. 10 | DOI | MR | Zbl

[10] Hörmander, L. Lectures on nonlinear hyperbolic differential equations, Mathématiques & Applications (Berlin), 26, Springer-Verlag, Berlin, 1997 | MR | Zbl

[11] Ibrahim, S.; Masmoudi, N.; Nakanishi, K. Scattering threshold for the focusing nonlinear Klein-Gordon equation (arXiv:1001.1474) | Zbl

[12] John, F. Blow-up for quasilinear wave equations in three space dimensions, Comm. Pure Appl. Math., Volume 34 (1981) no. 1, pp. 29-51 | DOI | MR | Zbl

[13] Katayama, S.; Yokoyama, K. Global small amplitude solutions to systems of nonlinear wave equations with multiple speeds, Osaka J. Math., Volume 43 (2006) no. 2, pp. 283-326 | MR | Zbl

[14] Klainerman, S. The null condition and global existence to nonlinear wave equations (Nonlinear systems of partial differential equations in applied mathematics Part 1 (Santa Fe, N.M., 1984), p. 293–326, Lectures in Appl. Math.), Volume 23, Amer. Math. Soc., Providence, RI, 1983 | MR | Zbl

[15] Klainerman, S. Global existence of small amplitude solutions to nonlinear Klein-Gordon equations in four space-time dimensions, Comm. Pure Appl. Math., Volume 38 (1985) no. 5, pp. 631-641 | DOI | MR | Zbl

[16] Ohta, M. Counterexample to global existence for systems of nonlinear wave equations with different propagation speeds, Funkcial. Ekvac., Volume 46 (2003) no. 3, pp. 471-477 | DOI | MR | Zbl

[17] Shatah, J. Normal forms and quadratic nonlinear Klein-Gordon equations, Comm. Pure Appl. Math., Volume 38 (1985) no. 5, pp. 685-696 | DOI | MR | Zbl

[18] Sideris, T.; Tu, S.-Y. Global existence for systems of nonlinear wave equations in 3D with multiple speeds, SIAM J. Math. Anal., Volume 33 (2001) no. 2, pp. 477-488 | DOI | MR | Zbl

[19] Tsutsumi, Y. Stability of constant equilibrium for the Maxwell-Higgs equations, Funkcial. Ekvac., Volume 46 (2003) no. 1, pp. 41-62 | DOI | MR | Zbl

[20] Yokoyama, K. Global existence of classical solutions to systems of wave equations with critical nonlinearity in three space dimensions, J. Math. Soc. Japan, Volume 52 (2000) no. 3, pp. 609-632 | DOI | MR | Zbl

Cité par Sources :