Quasi-invariant modified Sobolev norms for semi linear reversible PDEs

Abstract

We consider a general class of infinite dimensional reversible differential systems. Assuming a non resonance condition on the linear frequencies, we construct for such systems almost invariant pseudo norms that are closed to Sobolev-like norms. This allows us to prove that if the Sobolev norm of index s of the initial data z 0 is sufficiently small (of order ǫ) then the Sobolev norm of the solution is bounded by 2ǫ during very long time (of order ǫ −r with r arbitrary). It turns out that this theorem applies to a large class of reversible semi linear PDEs including the non linear Schrödinger equation on the d-dimensional torus. We also apply our method to a system of coupled NLS equations which is reversible but not Hamiltonian.We also notice that for the same class of reversible systems we can prove a Birkhoff normal form theorem that in turn implies the same bounds on the Sobolev norms. Nevertheless the technics that we use to prove the existence of quasi invariant pseudo norms is much more simple and direct.

Almost global existence of solutions for capillarity-gravity water waves equations with periodic spatial boundary conditions

“…The concept of reversibility was introduced in KAM theory by Moser [57], see also Arnold [12] and [21] for further developments, and then it has also been used to prove normal form stability results, see for example the exponential estimates in [38], [39] near an elliptic equilibrium. Concerning PDEs we refer, for KAM results, to Zhang, Gao, Yuan [67] for reversible derivative Schrödinger equations and Berti, Biasco, Procesi [15] for reversible derivative wave equations, and to Faou-Grébert [35] and Fang-Han-Zhang [34] for normal form results for semi-linear reversible PDEs.…”

“…First we shall perform a series of non-linear para-differential changes of variables, similar to the transformations used in Alazard-Baldi [1] and Berti-Montalto [16] to reduce the linearized equations (which arise in a Nash-Moser iteration to prove the existence of periodic and quasi-periodic solutions) to constant coefficients. Then we shall develop a normal form method parallel to those used by Bambusi, Delort, Grébert and Szeftel [13], [14], [30], and Faou-Grébert [35] for semi-linear PDEs. The modified energy E s is explicitly constructed in (4.4.29) (with q = N − 1).…”

“…In many instances, the Hamiltonian nature of the vector field P provides that property. Here, we shall instead exploit another classical structure, namely reversibility, that for semilinear PDEs has been used in [34,35]. For the model equation (0.2.1), considering the involution S : u → ū acting on the space of functions even in…”

KAM Theory for Partial Differential Equations

“…Here the Hamiltonian or reversible structure of the PDE plays a role. A systematic approach for Hamiltonian semilinear PDEs has been realized in [13,14,61], see [87] for reversible PDEs. This approach fails in case of quasi-linear PDEs because the Birkhoff transformations could not be well defined (it is the similar difficulty 2 which appears in the reducibility scheme).…”

Long time existence for fully nonlinear NLS with small Cauchy data on the circle

“…We quote for instance the papers by Bambusi [4], Bambusi-Grebert [6] and by Delort-Szeftel [12,13]. Regarding BNF theory for reversible PDEs we mention [15] by Grebert-Faou. The paper [5] regards long time existence of solutions for the semi-linear Klein-Gordon equation on Zoll manifolds, here are collected all the ideas of the preceding (and aforementioned) literature.…”