Generating Unstable Dynamics in dispersive Hamiltonian fluids

Introduction

This project aims to prove rigorous results on the generation of unstable dynamics in the dispersive Hamiltonian water waves and geophysical fluid equations. I refer to the outstanding open problems of the formation of energy cascades from low to high frequencies, orbital instabilities, and extreme rare phenomena such as rogue waves.

Background

All these behaviors are well documented in experiments, real-world data, and numerical simulations, but still lack a rigorous mathematical proof. Accordingly, the project has three lines of research:

Research Lines

1. Energy cascades from low to high frequency modes: I plan to construct solutions of water waves and geophysical fluids equations in which energy shifts from low to high Fourier frequencies, induced by resonant interactions, provoking the growth of Sobolev norms. This will be attacked via a novel mechanism based on dispersive estimates in the frequency space, that I have pioneered in the linear setting.

2. Orbital instabilities of Stokes and Rossby waves: I plan to study the linear and nonlinear modulational instabilities of traveling solutions like the Stokes and Rossby waves in the higher dimensional setting, proving long-time conjectures in fluid dynamics and geophysics. This will be attacked by extending a program I have initiated in the one-dimensional setting.

3. Extreme phenomena formation: I plan to construct rogue wave solutions of the water wave equations, confirming (or disproving) two of the most renowned physical conjectures regarding their formation. This will be achieved by combining deterministic normal forms and probabilistic methods, like large deviation principles applied to PDEs.

Conclusion

These goals aim to open new paradigms to understand the long-time dynamics of dispersive Hamiltonian partial differential equations originating from fluids.