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

Delort, J.-M.

doi:10.1090/S0002-9947-09-04747-3

Long-time Sobolev stability for small solutions of quasi-linear Klein-Gordon equations on the circle
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Trans. Amer. Math. Soc. 361 (2009), 4299-4365 Request permission

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.

References

Dario Bambusi, Birkhoff normal form for some nonlinear PDEs, Comm. Math. Phys. 234 (2003), no. 2, 253–285. MR 1962462, DOI 10.1007/s00220-002-0774-4
D. Bambusi, J.-M. Delort, B. Grébert, and J. Szeftel, Almost global existence for Hamiltonian semilinear Klein-Gordon equations with small Cauchy data on Zoll manifolds, Comm. Pure Appl. Math. 60 (2007), no. 11, 1665–1690. MR 2349351, DOI 10.1002/cpa.20181
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, DOI 10.1215/S0012-7094-06-13534-2
Jean-Michel 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 (French). MR 631751, DOI 10.24033/asens.1404
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, DOI 10.1007/BF02247885
Jean Bourgain, On the growth in time of higher Sobolev norms of smooth solutions of Hamiltonian PDE, Internat. Math. Res. Notices 6 (1996), 277–304. MR 1386079, DOI 10.1155/S1073792896000207
J. Colliander, M. Keel, G. Staffilani, H. Takaoka, and T. Tao, Resonant decompositions and the $I$-method for the cubic nonlinear Schrödinger equation on $\Bbb R^2$, Discrete Contin. Dyn. Syst. 21 (2008), no. 3, 665–686. MR 2399431, DOI 10.3934/dcds.2008.21.665
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 (French, with English summary). MR 1623424, DOI 10.1353/ajm.1998.0018
J.-M. Delort and J. Szeftel, Long-time existence for small data nonlinear Klein-Gordon equations on tori and spheres, Int. Math. Res. Not. 37 (2004), 1897–1966. MR 2056326, DOI 10.1155/S1073792804133321
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, DOI 10.1353/ajm.2006.0038
Jean-Marc Delort and Jérémie 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 (English, with English and French summaries). MR 2273861, DOI 10.5802/aif.2217
Benoît Grébert, Birkhoff normal form and Hamiltonian PDEs, Partial differential equations and applications, Sémin. Congr., vol. 15, Soc. Math. France, Paris, 2007, pp. 1–46 (English, with English and French summaries). MR 2352816
Lars Hörmander, Lectures on nonlinear hyperbolic differential equations, Mathématiques & Applications (Berlin) [Mathematics & Applications], vol. 26, Springer-Verlag, Berlin, 1997. MR 1466700
Vladimir A. Marchenko, Sturm-Liouville operators and applications, Operator Theory: Advances and Applications, vol. 22, Birkhäuser Verlag, Basel, 1986. Translated from the Russian by A. Iacob. MR 897106, DOI 10.1007/978-3-0348-5485-6
Jalal Shatah, Normal forms and quadratic nonlinear Klein-Gordon equations, Comm. Pure Appl. Math. 38 (1985), no. 5, 685–696. MR 803256, DOI 10.1002/cpa.3160380516