Habilitation

I defended my Habilitation in July, 2021. The memoir, entitled Many Models for Water Waves, attempts (and obviously fails) at providing a comprehensive overview of several standard and less-standard of models for the propagation of surface gravity waves, interfacial waves or internal waves.

The motivation for preparing this Habilitation was triggered some time ago as I realized that several distinct projects of my own, when pieced together and added to the existing litterature, could provide a somewhat consistent story. It took several years to complete the few “little things” that remained to be settled, and then the incentive of supervising a (second) PhD student to actually put an end to the overambitious project.

The memoir is an effort at unifying the theory on several (classes of) models, and can be thought as building upon the book of David Lannes.¹ However there is less emphasize at the water waves system (although the necessary results are quickly recalled), and more at the models derived from it. Its scope is also enlarged as it includes

• “improved” non-hydrostatic models such as the fully dispersive version of the Green-Naghdi system;²
• higher-order models, including
  • two families built from Friedrichs-type expansion, and in particular the ones described by Matsuno;³
  • several families built from variational principles, including “multilayer” models,⁴ the models of Isobe-Kakinuma,⁵ Klopman, van Groesen, and Dingemans,⁶ or Athanassoulis and Belibassakis;⁷
• models for interfacial waves,
  • in the hydrostatic framework, describing my results on the rigid-lid assumption and Boussinesq approximation
    for weak density contrasts;⁸
  • in the non-hydrostatic framework, extending the Green-Naghdi⁹ and Isobe-Kakinuma¹⁰ models to the bilayer framework;
• a small take at internal waves for stratified fluids in the hydrostatic framework.

For each of the models, I try to present the state of the art considering their rigorous justification as asymptotic models for water waves : at least their precision in the sense of consistency, and the complete justification when possible. Their Hamiltonian structure and conserved quantities are underlined. A modal analysis and discussion at solitary waves is provided when results are known. Finally, many open questions are put forward.

I also wanted to include a few words on numerical aspects, and specifically the Fourier pseudo-spectral approach (described here) which provides a very efficient way to treat and compare all these systems (in idealized situations) in a unified manner. Together with P. Navaro, we have implemented many of these models in a Julia package: WaterWaves1D.jl.