## Section: New Results

### Analysis of models in Fluid Mechanics

#### Weak solutions of multilayer Hydrostatic Flows

Participants : Bernard Di Martino, Boris Haspot, Yohan Penel.

We investigate the existence of global weak solutions for the multilayer model introduced by Audusse et al. [2] which is related to incompressible free surface flows. More precisely, in [22] we prove the global stability of weak solutions over the torus. We observe that this model admits the so called BD-entropy and a gain of integrability on the velocity in the spirit of the work of Mellet and Vasseur. The main difficulty comes from the terms describing the transfer of flux between the layers which are not taken into account in the immiscible case.

#### Hyperbolicity of the Layerwise Discretized Hydrostatic Euler equation: the bilayer case

Participants : Emmanuel Audusse, Edwige Godlewski, Martin Parisot.

*In collaboration with Nina Aguillon (UPMC)*.

Several model of free surface flows described in the literature are based on a layerwise discretization of the Euler equations. The question addressed in the current work is about the hyperbolicity of the layerwise discretized model. More precisely, we focus on the 2-layer case and we prove the well-posedness of the Riemann problem in two dimensional framework. Due to the mass exchange, the 2D Riemann problem is not a simple extension of the 1D Riemann problem.

#### Normal mode perturbation for the shallow water equations

Participants : Emmanuel Audusse, Albin Grataloup, Yohan Penel.

This work focuses on the shallow water equations for a fluid flow in subcritical regime with an arbitrary topography. A normal mode perturbation was performed around a 1D steady state in the 2D model. The resulting system of ODE was studied in terms of eigenvalues of the corresponding matrix and the derivation of a dispersion relation.

#### Global well-posdness of the Euler-Korteweg system for small irrotational data

Participant : Boris Haspot.

*In collaboration with C. Audiard (UPMC)*.

The Euler-Korteweg equations are a modification of the Euler equations that takes into account capillary effects. In the general case they form a quasi-linear system that can be recast as a degenerate Schrödinger type equation. We prove here that under a natural stability condition on the pressure, global well-posedness holds in dimension $d\ge 3$ for small irrotational initial data. The proof is based on a modified energy estimate, standard dispersive properties if $d\ge 5$, and a careful study of the nonlinear structure of the quadratic terms in dimensions 3 and 4 involving the theory of space time resonance.