Hamiltonian Dynamics, Normal Forms and Water Waves

Descripción del proyecto

Análisis de las propiedades cualitativas de las ondas en los fluidos

La capacidad de predecir el comportamiento de las ondas en los fluidos es uno de los principales desafíos de las matemáticas. El equipo del proyecto HamDyWWa, financiado con fondos europeos, pretende cambiar esta situación. Con ese fin, desarrollará el teorema de Kolmogorov-Arnold-Moser, así como métodos de la teoría de la forma normal para ecuaciones diferenciales parciales en espacios de dimensiones superiores. Uno de los principales focos de atención será el movimiento oscilatorio y las propiedades de estabilidad para las ecuaciones diferenciales parciales de dinámica de fluidos y mecánica cuántica, mediante técnicas del teorema de Kolmogorov-Arnold-Moser, la teoría de la forma normal, así como el análisis armónico y microlocal.

Objetivo

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. 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). 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, the water waves equation being a prominent example. These novel techniques are based on a combination of pseudo-differential and para-differential calculus, with the classical perturbative techniques and they allowed to make significant advances of the KAM and normal form theory for one-dimensional PDEs. On the other hand, many challenging problems remain open and the purpose of this proposal is to investigate some of them. The main goal of this project is to develop KAM and normal form methods for PDEs in higher space dimension, with a particular focus on equations arising from fluid dynamics, like Euler, Navier-Stokes and 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 dimension (like the Euler equation in 3d), which makes me confident that the proposed project is feasible.