Duke Mathematical Journal

Birkhoff normal form for partial differential equations with tame modulus

D. Bambusi and B. Grébert

Full-text: Access denied (no subscription detected) We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

We prove an abstract Birkhoff normal form theorem for Hamiltonian partial differential equations (PDEs). The theorem applies to semilinear equations with nonlinearity satisfying a property that we call tame modulus. Such a property is related to the classical tame inequality by Moser. In the nonresonant case we deduce that any small amplitude solution remains very close to a torus for very long times. We also develop a general scheme to apply the abstract theory to PDEs in one space dimensions, and we use it to study some concrete equations (nonlinear wave (NLW) equation, nonlinear Schrödinger (NLS) equation) with different boundary conditions. An application to an NLS equation on the $d$-dimensional torus is also given. In all cases we deduce bounds on the growth of high Sobolev norms. In particular, we get lower bounds on the existence time of solutions

Article information

Source
Duke Math. J. Volume 135, Number 3 (2006), 507-567.

Dates
First available in Project Euclid: 10 November 2006

Permanent link to this document
http://projecteuclid.org/euclid.dmj/1163170201

Digital Object Identifier
doi:10.1215/S0012-7094-06-13534-2

Mathematical Reviews number (MathSciNet)
MR2272975

Zentralblatt MATH identifier
1110.37057

Subjects
Primary: 37K55: Perturbations, KAM for infinite-dimensional systems

Citation

Bambusi, D.; Grébert, B. Birkhoff normal form for partial differential equations with tame modulus. Duke Math. J. 135 (2006), no. 3, 507--567. doi:10.1215/S0012-7094-06-13534-2. http://projecteuclid.org/euclid.dmj/1163170201.


Export citation

References

  • D. Bambusi, Nekhoroshev theorem for small amplitude solutions in nonlinear Schrödinger equations, Math. Z. 230 (1999), 345--387.
  • —, On long time stability in Hamiltonian perturbations of non-resonant linear PDEs, Nonlinearity 12 (1999), 823--850.
  • —, An averaging theorem for quasilinear Hamiltonian PDEs, Ann. Henri Poincaré 4 (2003), 685--712.
  • —, Birkhoff normal form for some nonlinear PDEs, Comm. Math. Phys. 234 (2003), 253--285.
  • —, ``Birkhoff normal form for some quasilinear Hamiltonian PDEs'' in XIVth International Congress on Mathematical Physics (Lisbon, 2003), World Sci., Hackensack, N.J., 2005, 273--280.
  • D. Bambusi and B. GréBert, Forme normale pour NLS en dimension quelconque, C. R. Math. Acad. Sci. Paris 337 (2003), 409--414.
  • G. Benettin, L. Galgani, and A. Giorgilli, A proof of Nekhoroshev's theorem for the stability times in nearly integrable Hamiltonian systems, Celestial Mech. 37 (1985), 1--25.
  • J. Bourgain, Construction of quasi-periodic solutions for Hamiltonian perturbations of linear equations and applications to nonlinear PDE, Internat. Math. Res. Notices 1994, no. 11, 475--497.
  • —, Construction of approximative and almost periodic solutions of perturbed linear Schrödinger and wave equations, Geom. Funct. Anal. 6 (1996), 201--230.
  • —, Quasi-periodic solutions of Hamiltonian perturbations of $2D$ linear Schrödinger equations, Ann. of Math. (2) 148 (1998), 363--439.
  • —, On diffusion in high-dimensional Hamiltonian systems and PDE, J. Anal. Math. 80 (2000), 1--35.
  • —, Remarks on stability and diffusion in high-dimensional Hamiltonian systems and partial differential equations, Ergodic Theory Dynam. Systems 24 (2004), 1331--1357.
  • —, Green's Function Estimates for Lattice Schrödinger Operators and Applications, Ann. of Math. Stud. 158, Princeton Univ. Press, Princeton, 2005.
  • W. Craig, Problèmes de petits diviseurs dans les équations aux dérivées partielles, Panor. et Synthèses 9, Soc. Math. France, Montrouge, 2000.
  • W. Craig and C. E. Wayne, Newton's method and periodic solutions of nonlinear wave equations, Comm. Pure Appl. Math. 46 (1993), 1409--1498.
  • J.-M. Delort and J. Szeftel, Long-time existence for small data nonlinear Klein-Gordon equations on tori and spheres, Int. Math. Res. Not. 2004, no. 37, 1897--1966.
  • —, Long-time existence for semi-linear Klein-Gordon equations with small Cauchy data on Zoll manifolds, to appear in Amer. J. Math.
  • T. Kappeler and B. Mityagin, Estimates for periodic and Dirichlet eigenvalues of the Schrödinger operator, SIAM J. Math. Anal. 33 (2001), 113--152.
  • T. Kappeler and J. PöSchel, KdV & KAM, Ergeb. Math. Grenzgeb. (3) 45, Springer, Berlin, 2003.
  • S. Klainerman, On ``almost global'' solutions to quasilinear wave equations in three space dimensions, Comm. Pure Appl. Math. 36 (1983), 325--344.
  • S. B. Kuksin, Hamiltonian perturbations of infinite-dimensional linear systems with an imaginary spectrum, Funct. Anal. Appl. 21 (1987), 192--205.
  • —, Nearly Integrable Infinite-Dimensional Hamiltonian Systems, Lecture Notes in Math. 1556, Springer, Berlin, 1993.
  • —, Analysis of Hamiltonian PDEs, Oxford Lecture Ser. Math. Appl. 19, Oxford Univ. Press, New York, 2000.
  • S. B. Kuksin and J. PöSchel, Invariant Cantor manifolds of quasi-periodic oscillations for a nonlinear Schrödinger equation, Ann. of Math. (2) 143 (1996), 149--179.
  • V. A. Marchenko, Sturm-Liouville Operators and Applications, Oper. Theory Adv. Appl. 22, Birkhäuser, Basel, 1986.
  • N. V. Nikolenko, The method of Poincaré normal forms in problems of integrability of equations of evolution type, Russian Math. Surveys 41, no. 5 (1986), 63--114.
  • J. PöSchel and E. Trubowitz, Inverse Spectral Theory, Pure Appl. Math. 130, Academic Press, Boston, 1987.
  • C. E. Wayne, Periodic and quasi-periodic solutions of nonlinear wave equations via KAM theory, Comm. Math. Phys. 127 (1990), 479--528.
  • J. Xu, J. You, and Q. Qiu, Invariant tori for nearly integrable Hamiltonian systems with degeneracy, Math. Z. 226 (1997), 375--387.