Birkhoff normal form for partial differential equations with tame modulus

Abstract

We prove an abstract Birkhoff normal form theorem for Hamiltonian Partial Differential Equations. The theorem applies to semilinear equations with nonlinearity satisfying a property that we call of 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 (NLW,NLS) with different boundary conditions. An application to a nonlinear Schrödinger 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.

Normal Forms for Semilinear Quantum Harmonic Oscillators

“…[Wan08]). Actually in the case studied in [Bou96,Bam03,BG06], the linear modes (i.e. the eigenfunctions of the linear part) are localized around the exponentials e ikx , i.e.…”

“…The proof that there exists a set F k ⊂ W k whose measure equals 1 such that if m = (m j ) j∈N ∈ F k then the frequency vector (ω j ) j≥1 is non resonant is exactly the same as the proof of Theorem 5.7 in [Gré07]. So we do not repeat it here (see also [BG06]). …”

On long time existence for small solutions of semi-linear Klein-Gordon equations on the torus

“…over an interval of length c N −N for any N ) for equation (1.1.2) on T 1 . This has been done by Bourgain [5], Bambusi [1], Bambusi-Grébert [3]. Such an almost global theorem has been proved in higher dimensions as well by Bambusi, Delort, Grébert and Szeftel [2] for equation (1.1.2) on the sphere S d (or more generally on a Zoll manifold).…”

“…The aim of this paper is to study long-time existence problems for semi-linear Klein-Gordon equations of type This problem has been studied in dimension 1 by Bourgain [5], Bambusi [1], Bambusi-Grébert [3]. They showed that one has then almost global existence: for any N , if the data are in H s+1 × H s Mathematics Subject Classification: 35L70.…”

Bounded almost global solutions for non hamiltonian semi-linear Klein-Gordon equations with radial data on compact revolution hypersurfaces

“…Alternatively one could make use of inequalities obtained in more elementary ways by Bambusi [1] and Bambusi and Grébert [2].…”