PhD Thesis, Théo Fradin

Théo Fradin defended his PhD thesis in June, 2026. His mathematical work was co-supervised by David Lannes and myself, and was dedicated to the variable-density Euler equations, with a particular focus on the shallow-water regime, density-stratified flows, stability/instability and well-posedness issues.

His research led to three manuscripts.

1. T. Fradin, Free-surface Euler equations with density variations, and shallow-water limit. preprint Arxiv

In this work, Théo provides a well-posedness theory for the inhomogeneous (incompressible) Euler equations with a free surface. His works extends results of Alvarez-Samaniego and Lannes ¹ and Tatsuo Iguchi ² (dedicated to irrotational, homogeneous flows) and Castro and Lannes³ (dedicated to homogeneous flows) in that it pays a specific attention to small vertical-to-horizontal aspect ratios (the shallow-water regime).

In my opinion it is to date the most natural entry to this problem. It does not rely on a specific reformulation such as the Zakharov formulation or Lagrangian variables, but uses standard tools such as elliptic estimates for the pressure reconstruction, and energy estimates together with Alinhac’s good unknown (adapted to free-interface problems) for the time-evolution.

His result sheds a light and brings quantitative information on the mathematical difficulties (and possible instability mechanisms) emerging from the presence of vorticity and density variations in the shallow-water regime.

2. T. Fradin, Well-posedness of the Euler equations in a stably stratified ocean in isopycnal coordinates, Ann. Inst. H. Poincaré Anal. Non Linéaire, 2024. preprint Arxiv, journal article

In this work, Théo obtains more favorable results by considering stably-stratified flows. His work builds a bridge between two works of his supervisors ^{4 5}, superseding the first by considering less stringent assumptions and the second by avoiding artifical thickness-diffusivity contributions.

At a technical level, the strategy makes use of semi-Lagrangian isopycnal coordinates as in ⁵, and one of its achievement is to avoid loss of derivatives stemming from this change of coordinates by a suitable use of Alinhac’s good unknown. I also learned from Théo’s work that the isopycnal coordinates are not suitable to study the well-posedness of the initial-value problem outside of stably-stratified flows (as in its first work), and in particular in the homogeneous framework.

3. T. Fradin, Numerical study of the sharp stratification limit towards bilayer models, Stud. Appl. Math., 2026 preprint Arxiv, journal article

In this work, Théo studies the limit of sharp stratification for stratified flows, when the (background) density and velocity profiles become piecewise constant. By a combination of rigorous and numerical results, Théo emphasizes the role of background shear velocity in the development of the so-called Kelvin-Helmholtz (high-frequency) instabilities.

I learned from this work that it is not possible to rigorously derive (in a smooth framework) the bilayer shallow-water equations from stably stratified equations, as laminar flows are inevitably destroyed by Kelvin-Helmholtz instabilities even (and in fact especially) in the shallow-water regime. From my understanding, small-scale turbulent effects will arise in the pycnocline.

It is possible that the effective contributions of these small-scale eddies may be described using the Gent and McWilliams thickness-diffusivity parameterization, as used in ⁵ (see this post) and in ⁶ where the sharp stratification limit was precisely obtained (see this post). However, to my knowledge, there is no rigorous result in that direction.