Section: New Results

Analysis of models in fluid mechanics

Well-posedness of multilayer Shallow Water-type equations

Participants : Emmanuel Audusse, Bernard Di Martino, Ethem Nayir, Yohan Penel.

The hyperbolicity of some 2-layer Shallow Water equations had been proven in  [26] , [23] , there are many open theoretical investigations to lead about these systems. In particular, E. Nayir proved the local well-posedness of the model derived in  [23] for periodic boundary conditions. Next steps will consist in extending this preliminary result to the whole space and proving the global existence of strong solutions. The existence of weak solutions will be studied from B. Di Martino's work. The hyperbolicity for $N$ layers must also be investigated.

As for numerical aspects, the use of Freshkiss3d will provide qualitative assessments for modelling issues (viscous tensor, source terms, variable density, interfacial velocities). It will also yield comparisons with theoretical results, in particular when the number of layers goes to infinity.

Non-hydrostatic models

Participants : Dena Kazerani, Jacques Sainte-Marie, Nicolas Seguin.

Together with Corentin Audiard from Univ. Pierre et Marie Curie, we investigated the structure of general non hydrostatic models for shallow water flows. This includes the Green–Naghdi equations and the model proposed by Bristeau et al. in [13] . D. Kazerani proved that such systems possess a symmetric structure based on the existence of an energy. The main difference with the well-known hyperbolic case is due to the presence of differential operators instead of matrices.