Section: New Results

Analysis of models in Fluid Mechanics

Proving the existence of global weak solutions is a difficult problem for Navier-Stokes type equations, particularly in case of a degenerate viscosity (viscosity term can vanish if density or thickness goes to zero). In some recent works, Vasseur and Yu [46] , have proved this existence for 2D shallow water equations. For the multilayer model, a collaboration with Boris Haspot (Univ. Paris-Dauphine) lead to stability results for the system with a focus on the difficulty to construct a sequence of approximate solutions that conserve all a priori estimates.

The existence and uniqueness of strong solutions of the multilayer model proposed in [41] was previously proven in the case of boundary conditions. We extended this result to an unbounded domain for short times, overcoming the issue of integrability often barely evoked in similar investigations. Current works deal with the long time existence by a continuation process which requires a particular care of the short time solution at the end of its existence interval.

In [21] , we study a family of linear hyperbolic systems whose solution must satisfy a constraint (e.g. a simplified model of river flows taking risk of flooding into account). We analyse relaxed models based on a penalisation. This theoretical approach could be used to derive numerical methods.

In [44] , we carry out an analysis of 1st-order entropy-satisfying finite volume schemes for hyperbolic systems. More precisely, we investigate the numerical dissipation on unstructured meshes under relevant stability conditions. This results in a minimal convergence order towards smooth solutions.

In [31] , we consider the Cauchy problem for the Green-Naghdi equations with viscosity, for small initial data. It is well-known that adding a second order diffusion term to a hyperbolic system leads to the existence of global smooth solutions, as soon as the hyperbolic system is symmetrizable and the so-called Kawashima-Shizuta condition is satisfied. In a previous work, we have proved that the Green-Naghdi equations can be written in a symmetric form, using the associated Hamiltonian. This system being dispersive, in the sense that it involves third order derivatives, the symmetric form is based on symmetric differential operators. We use this structure for an appropriate change of variable to prove that adding viscosity effects through a second order term leads to global existence of smooth solutions, for small data. We also deduce that constant solutions are asymptotically stable.