## Section: New Results

### Analysis of models in Fluid Mechanics

#### Analysis of the Riemann problem for a shallow water model with two velocities

Participants : Emmanuel Audusse, Edwige Godlewski, Martin Parisot.

*In collaboration with N. Aguillon.*

The question addressed in [24] is the hyperbolicity of a shallow water model with two velocities. The model is written in a nonconservative form and the analysis of its eigenstructure shows the possibility that two eigenvalues coincide. A definition of the nonconservative product is given which enables us to analyse the resonance and coalescence of waves. Eventually, we prove the well-posedness of the two dimensional Riemann problem with initial condition constant by half-plane.

#### Different formulations of an elliptic problem issued from geophysics

Participants : Cindy Guichard, Ani Miraçi, Yohan Penel, Jacques Sainte-Marie.

A simplified problem coming from [33] involving pressure and velocity unknowns is studied. Some weak formulations (conform or mixed) are derived and their well-posedness is analysed. These weak formulations are then discretised in a finite element framework with suitable discrete spaces.