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1 December 2006 Birkhoff normal form for partial differential equations with tame modulus
D. Bambusi, B. Grébert
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Duke Math. J. 135(3): 507-567 (1 December 2006). DOI: 10.1215/S0012-7094-06-13534-2

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

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D. Bambusi. B. Grébert. "Birkhoff normal form for partial differential equations with tame modulus." Duke Math. J. 135 (3) 507 - 567, 1 December 2006. https://doi.org/10.1215/S0012-7094-06-13534-2

Information

Published: 1 December 2006
First available in Project Euclid: 10 November 2006

zbMATH: 1110.37057
MathSciNet: MR2272975
Digital Object Identifier: 10.1215/S0012-7094-06-13534-2

Subjects:
Primary: 37K55

Rights: Copyright © 2006 Duke University Press

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Vol.135 • No. 3 • 1 December 2006
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