## Duke Mathematical Journal

### Birkhoff normal form for partial differential equations with tame modulus

#### 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

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.

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