Transactions of the American Mathematical Society

Long-time Sobolev stability for small solutions of quasi-linear Klein-Gordon equations on the circle

Author(s): J.-M. Delort
Journal: Trans. Amer. Math. Soc. 361 (2009), 4299-4365.
MSC (2000): Primary 35L70, 35S50
Posted: March 13, 2009
Retrieve article in: PDF

Abstract: We prove that higher Sobolev norms of solutions of quasi-linear Klein-Gordon equations with small Cauchy data on $\mathbb{S}^1$ remain small over intervals of time longer than the ones given by local existence theory. This result extends previous ones obtained by several authors in the semi-linear case. The main new difficulty one has to cope with is the loss of one derivative coming from the quasi-linear character of the problem. The main tool used to overcome it is a global paradifferential calculus adapted to the Sturm-Liouville operator with periodic boundary conditions.

1.: D. Bambusi: Birkhoff normal form for some nonlinear PDEs, Comm. Math. Phys. 234 (2003), no. 2, 253-285. MR 1962462 (2003k:37121)
2.: D. Bambusi, J.-M. Delort, B. Grébert and J. Szeftel: Almost global existence for Hamiltonian semi-linear Klein-Gordon equations with small Cauchy data on Zoll manifolds, Comm. Pure Appl. Math. 60 (2007), no. 11, 1665-1690. MR 2349351
3.: D. Bambusi and B. Grébert: Birkhoff normal form for partial differential equations with tame modulus, Duke Math. J. 135 (2006), no. 3, 507-567. MR 2272975 (2007j:37124)
4.: J.-M. Bony: Calcul symbolique et propagation des singularités pour les équations aux dérivées partielles non linéaires, Ann. Sci. École Norm. Sup. (4) 14 (1981), no. 2, 209-246. MR 631751 (84h:35177)
5.: J. Bourgain: Construction of approximative and almost periodic solutions of perturbed linear Schrödinger and wave equations, Geom. Funct. Anal. 6 (1996), no. 2, 201-230. MR 1384610 (97f:35013)
6.: J. Bourgain: On the growth in time of higher Sobolev norms of smooth solutions of Hamiltonian PDE, Internat. Math. Res. Notices (1996), no. 6, 277-304. MR 1386079 (97k:35016)
7.: J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao: Resonant decompositions and the -method for cubic nonlinear Schrödinger equation on $\mathbb{R}^2$ , Discrete Contin. Dyn. Syst. 21 (2008), 665-686. MR 2399431
8.: J.-M. Delort: Temps d'existence pour l'équation de Klein-Gordon semi-linéaire à données petites périodiques, Amer. J. Math. 120 (1998), no. 3, 663-689. MR 1623424 (99d:35108)
9.: J.-M. Delort and J. Szeftel: Long-time existence for small data nonlinear Klein-Gordon equations on tori and spheres, Internat. Math. Res. Notices (2004), no. 37, 1897-1966. MR 2056326 (2005k:35281)
10.: J.-M. Delort and J. Szeftel: Long-time existence for semi-linear Klein-Gordon equations with small Cauchy data on Zoll manifolds, Amer. J. Math. 128 (2006), no. 5, 1187-1218. MR 2262173 (2007i:58039)
11.: J.-M. Delort and J. Szeftel: Bounded almost global solutions for non Hamiltonian semi-linear Klein-Gordon equations with radial data on compact revolution hypersurfaces, Ann. Inst. Fourier (Grenoble) 56 (2006), no. 5, 1419-1456. MR 2273861 (2008c:35191)
12.: B. Grébert: Birkhoff normal form and Hamiltonian PDEs, Partial differential equations and applications, 1-46, Sémin. Congr. 15, Soc. Math. France, Paris, 2007. MR 2352816
13.: L. Hörmander: Lectures on nonlinear hyperbolic differential equations, Mathématiques & Applications 26, Springer-Verlag, Berlin, (1997), viii+289 pp. MR 1466700 (98e:35103)
14.: V. Marchenko: Sturm-Liouville operators and applications, Operator Theory: Advances and Applications, 22. Birkhäuser Verlag, Basel (1986), xii+367 pp. MR 897106 (88f:34034)
15.: J. Shatah: Normal forms and quadratic nonlinear Klein-Gordon equations, Comm. Pure Appl. Math. 38 (1985), 685-696. MR 803256 (87b:35160)

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 35L70, 35S50

J.-M. Delort
Affiliation: Université Paris 13, Institut Galilée, CNRS, UMR 7539, Laboratoire Analyse, Géométrie et Applications, 99, Avenue J.-B. Clément, F-93430 Villetaneuse, France
Email: delort@math.univ-paris13.fr

DOI: 10.1090/S0002-9947-09-04747-3
PII: S 0002-9947(09)04747-3
Keywords: Quasi-linear Klein-Gordon equation, long-time stability, paradifferential calculus
Received by editor(s): September 19, 2007
Posted: March 13, 2009
Additional Notes: This work was partially supported by the ANR project Equa-disp.
Copyright of article: Copyright 2009, American Mathematical Society
The copyright for this article reverts to public domain after 28 years from publication.

			Journals Home Search Author Info Subscribe Tech Support Help
ISSN 1088-6850(e) ISSN 0002-9947(p)

Previous issue \| Table of contents \| Next issue Articles in press \| Recently posted articles \| Most recent issue \| All issues Previous Article \| Next Article


	Comments: webmaster@ams.org © Copyright 2009, American Mathematical Society Privacy Statement		Search the AMS