Hamiltonian Dynamics, Normal Forms and Water Waves

Introduction

KAM and normal form methods are very powerful tools for analyzing the dynamics of nearly integrable finite dimensional Hamiltonian systems. In the last decades, the extension of these methods to infinite dimensional systems, like Hamiltonian PDEs (partial differential equations), has attracted the interest of many outstanding mathematicians like Bourgain, Craig, Kuksin, Wayne, and many others.

Techniques and Applications

These techniques provide some tools for describing the phase space of nearly integrable PDEs. More precisely, they give a way to construct special global solutions (like periodic and quasi-periodic solutions) and to analyze stability issues close to equilibria or close to special solutions (like solitons).

Recent Developments

In the last seven years, I developed new methods for proving the existence of quasi-periodic solutions of quasi-linear, one-dimensional PDEs. This is an important step towards treating many of the fundamental equations from physics since most of these equations are quasi-linear. In particular, this is the case for the equations in fluid dynamics, with the water waves equation being a prominent example.

Novel Techniques

These novel techniques are based on a combination of pseudo-differential and para-differential calculus, along with classical perturbative techniques. They have allowed for significant advances in the KAM and normal form theory for one-dimensional PDEs.

Future Challenges

On the other hand, many challenging problems remain open, and the purpose of this proposal is to investigate some of them.

Project Goals

The main goal of this project is to develop KAM and normal form methods for PDEs in higher space dimensions, with a particular focus on equations arising from fluid dynamics, such as:

1. Euler equations
2. Navier-Stokes equations
3. Water waves equations

By extending the novel approach developed for PDEs in one space dimension, I have already obtained some preliminary results on PDEs in higher space dimensions (like the Euler equation in 3D), which makes me confident that the proposed project is feasible.