A quasi-linear Birkhoff normal forms method. Application to the quasi-linear Klein-Gordon equation on $\mathbb{S}^1$

A quasi-linear Birkhoff normal forms method. Application to the quasi-linear Klein-Gordon equation on

𝕊^{1}

Delort, Jean-Marc

Astérisque, no. 341 (2012) , 119 p.

MR Zbl | 4 citations dans Numdam

@book{AST_2012__341__1_0,
     author = {Delort, Jean-Marc},
     title = {A quasi-linear {Birkhoff} normal forms method. {Application} to the quasi-linear {Klein-Gordon} equation on $\mathbb{S}^1$},
     series = {Ast\'erisque},
     year = {2012},
     publisher = {Soci\'et\'e math\'ematique de France},
     number = {341},
     mrnumber = {2952065},
     zbl = {1243.35123},
     language = {en},
     url = {https://www.numdam.org/item/AST_2012__341__1_0/}
}

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T3  - Astérisque
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PB  - Société mathématique de France
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%T A quasi-linear Birkhoff normal forms method. Application to the quasi-linear Klein-Gordon equation on $\mathbb{S}^1$
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Delort, Jean-Marc. A quasi-linear Birkhoff normal forms method. Application to the quasi-linear Klein-Gordon equation on $\mathbb{S}^1$. Astérisque, no. 341 (2012), 119 p. https://www.numdam.org/item/AST_2012__341__1_0/

Bibliographie
Cité par

[1] D. Bambusi - "Birkhoff normal form for some nonlinear PDEs", Comm. Math. Phys. 234 (2003), p. 253-285. | MR | Zbl | DOI

[2] D. Bambusi, J.-M. Delort, B. Grébert & J. Szeftel - "Almost global existence for Hamiltonian semilinear Klein-Gordon equations with small Cauchy data on Zoll manifolds", Comm. Pure Appl. Math. 60 (2007), p. 1665-1690. | MR | Zbl | DOI

[3] D. Bambusi & B. Grébert - "Birkhoff normal form for partial differential equations with tame modulus", Duke Math. J. 135 (2006), p. 507-567. | MR | Zbl | DOI

[4] J.-M. Bony - "Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires", Ann. Sci. École Norm. Sup. 14 (1981). p. 209-246. | MR | Zbl | EuDML | Numdam | DOI

[5] J. Bourgain - "Construction of approximative and almost periodic solutions of perturbed linear Schrödinger and wave equations", Geom. Funct. Anal. 6 (1996), p. 201-230. | MR | Zbl | EuDML | DOI

[6] J. Bourgain, Green's function estimates for lattice Schrödinger operators and applications, Annals of Math. Studies, vol. 158, Princeton Univ. Press, 2005. | MR | Zbl

[7] W. Craig - "Problèmes de petits diviseurs dans les équations aux dérivées partielles", Panoramas et Synthèses 9 (2000). | MR | Zbl

[8] W. Craig & C. E. Wayne - "Newton's method and periodic solutions of nonlinear wave equations", Comm. Pure Appl. Math. 46 (1993), p. 1409-1498. | MR | Zbl | DOI

[9] J.-M. Delort - "Temps d'existence pour l'équation de Klein-Gordon semi-linéaire à données petites périodiques", Amer. J. Math. 120 (1998), p. 663-689. | MR | Zbl | DOI

[10] J.-M. Delort, "Existence globale et comportement asymptotique pour l'équation de Klein-Gordon quasi linéaire à données petites en dimension $1$ ", Ann. Sci. École Norm. Sup. 34 (2001), p. 1-61. | MR | Zbl | EuDML | Numdam | DOI

J.-M. Delort, "Existence globale et comportement asymptotique pour l'équation de Klein-Gordon quasi linéaire à données petites en dimension $1$ ", erratum: Ann. Sci. École Norm. Sup. 39 (2006), p. 335-345. | MR | Zbl | EuDML | Numdam | DOI

[11] J.-M. Delort, "Long-time Sobolev stability for small solutions of quasi-linear Klein- Gordon equations on the circle", Trans. Amer. Math. Soc. 361 (2009), p. 4299- 4365. | MR | Zbl | DOI

[12] J.-M. Delort & J. Szeftel - "Long-time existence for small data nonlinear Klein-Gordon equations on tori and spheres", Int. Math. Res. Not. 2004 (2004), p. 1897-1966. | MR | Zbl | DOI

[13] J.-M. Delort & J. Szeftel, "Long-time existence for semi-linear Klein-Gordon equations with small Cauchy data on Zoll manifolds", Amer. J. Math. 128 (2006), p. 1187-1218. | MR | Zbl | DOI

[14] L. H. Eliasson & S. B. Kuksin - "KAM for the nonlinear Schrödinger equation", Ann. of Math. 172 (2010), p. 371-435. | MR | Zbl | DOI

[15] B. Grébert - "Birkhoff normal form and Hamiltonian PDEs", Sémin. Congr. 15 (2007), p. 1-46. | MR | Zbl

[16] S. Klainerman - "Global existence of small amplitude solutions to nonlinear Klein-Gordon equations in four space-time dimensions", Comm. Pure Appl. Math. 38 (1985), p. 631-641. | MR | Zbl | DOI

[17] S. B. Kuksin - Nearly integrable infinite-dimensional Hamiltonian systems, Lecture Notes in Math., vol. 1556, Springer, 1993. | MR | Zbl

[18] S. B. Kuksin, Analysis of Hamiltonian PDEs, Oxford Lecture Series in Mathematics and its Applications, vol. 19, Oxford Univ. Press, 2000. | MR | Zbl

[19] T. Ozawa, K. Tsutaya & Y. Tsutsumi - "Global existence and asymptotic behavior of solutions for the Klein-Gordon equations with quadratic nonlinearity in two space dimensions", Math. Z. 222 (1996), p. 341-362. | MR | Zbl | EuDML | DOI

[20] J. Shatah - "Normal forms and quadratic nonlinear Klein-Gordon equations", Comm. Pure Appl. Math. 38 (1985), p. 685-696. | MR | Zbl | DOI

[21] C. E. Wayne - "Periodic and quasi-periodic solutions of nonlinear wave equations via KAM theory", Comm. Math. Phys. 127 (1990), p. 479-528. | MR | Zbl | DOI