Asymptotics, Numerics, Analysis 
of the network 


Sites of F4  ENS Lyon + Bordeaux, Rennes, ClermontFerrand, Strasbourg 
URL  http://www.umpa.enslyon.fr/~cvillani/F4.html 
Team Organizer  C. Villani 
This team includes both senior scientists, and young researchers. Its area of expertise ranges from the kinetic to the hyperbolic points of view. The main members of the team in ClermontFerrand have been in the ENS Lyon before, while one of the young professors in Lyon has done his research in Rennes before. These links explain the mathematical coherence of the whole group.
The main quality of the group certainly lies in its competence in the theoretical analysis of PDE's (regularity/shocks, asymptotics, qualitative behavior), even though it has always regarded numerical simulation as a fundamental topic. Some of the members of the group are generally considered as being among the leading experts in the theoretical study of hyperbolic systems of conservation laws, nonlinear geometrical optics, boundary layers and Boltzmann equation. Over the last years they have shared this knowledge with other teams on both theoretical and applied problems.
The group is wellknown for its competence in the theoretical and numerical analysis of hyperbolic systems (topics C. 11 and D. 15 ). This includes both delicate questions in the classical theory (nonclassical shocks, stability of shock profiles...) and more original applications such as traffic flow (topic B. 10 ). Among current trends, we mention active studies of stability of boundary layers (topic C. 12 , in connection with topic D. 15 ) and of the incompressible limit for viscous flows. On these subjects there are collaborations with teams A1, I2, F3, D2. In the domain of fluid dynamic equations, the group has established some essential theoretical works about rotating fluids and atmospheric flows (topic C. 14 ), partly in collaboration with team F1. As for numerical methods, they have been worked out in the context of shallow water, simple models for oceanography, and boundary layers (Cf. topics A. 4 , D. 15 and D. 16 ), often based on finite volumes.
Kinetic formulations of various hyperbolic systems (topic A. 1 ) have been studied, partly in collaboration with team F2, and applied to study such topics as Helmholtz equations, or establish the besttodate results of regularization of hyperbolic systems of conservation laws.
In the domain of Boltzmann equation (topic B. 7 ), and more generally kinetic equations with collisions, the group has particular expertise for the regularizing effects of grazing collisions, entropy dissipation methods (linked to task A. 3 ), derivation of quantum Boltzmann equations in the low density limit (task 9 ). On these topics there is intense collaboration with the teams F2, S1 (grazing collisions), I3 and F2 again (entropy dissipation), F3, I1 and A1 (quantum Boltzmann). The group has also been working for quite a time on the asymptotic analysis in plasma physics, be it with or without collisions (Landau and Vlasov equations respectively; topic A. 2 ).
Part of the team is implied in a nationwide research group which is specialized in HamiltonJacobi equations, amplitude equations and geometrical optics (topic C. 13 ), and currently working on the propagation of nonlinear waves, multidimensional shocks and instabilities in geometrical optics.
Finally, connections between entropy methods and mass transportation (topic A. 3 ) date back to joint works between this group and teams D2 and E2. Applications to granular media (topic B. 10 ) with team I1, manyparticle systems (topic B. 8 ) with team D2, are being worked out currently.
The key scientific staff consists of the following members:
 C. Villani (TO) (ENS Lyon, 35%)  S. Benzoni (ENS Lyon, 75%);  E. Grenier (ENS Lyon, 25%)  D. Serre (SAB) (ENS Lyon, 40%);  C. Cheverry (Univ. Lyon I, 40%)  A. Heibig (INSA Lyon, 25%);  Y.J. Peng (ClermontFerrand, 30%)  F. Nier (Rennes, 30%)  E. Sonnendrucker (Strasbourg, 30%)
[1] E. Grenier, On the nonlinear instability of Euler and Prandtl equations , Comm. Pure Appl. Math. 53 , 2 (2000), 10671091.
[2] J. Francheteau, G. Métivier, Existence of weak shocks for multidimensional hyperbolic quasilinear systems , Astérisque 268, Soc. Math. France, Inst. Henri Poincaré (2000).