The network covers a wide range of problems inkinetic theory and hyperbolic equations or systems. We shall list below techniques which have already proven theirrobustness and efficiency, including tools which are the result of recent progress (*), and new trends or newand promising mathematical ideas (ø).
* For the study of quantum systems(Schrödinger equations, models for atoms, molecules andcrystals, Helmholtz equations) and their large time, semiclassical, andthermodynamical limits, we shall use Wigner's formalism(Wigner transform and Wigner or semi-classicalmeasures, Husimi transform), Strichartz' estimates,pseudo-differential calculus and variational methods.
* In the study of the long-time behavior ofdissipative systems, the entropy dissipation method has been very successful in various contexts such as homogeneous kinetic equations and degenerate parabolic equations. Morerecently, applications have started to be found in the studyof coupled systems of equations, granular media, thin fluidequations,
° An emerging related trend is the use of mass transportation methods , including Wasserstein distance, Monge-Kantorovich problems, Monge-Ampère equations, displacement interpolation, ... After their appearance in a fluid mechanical context, these methods are now beginning to be applied to systems of interacting particles and may provide new estimates.
* As regards the derivation of asymptotic regimes ( e.g. hydrodynamics from kinetic, either in a classical or in a quantum regime), moment expansions have proven robust and flexible. In connection with entropy dissipation techniques, and by analogy with probabilistic results, preliminary results have been obtained both in the context of particle systems and of hydrodynamic limits. This approach has to be investigated further.
* Symmetries, defect measures, connections with the theory of parabolic equation for singular collision operators, and analogies with probabilistic results on nonlinear unbounded Poisson processes have been used for the study of the homogeneous Boltzmann equation . Regularization effects and application to hydrodynamic limits will be studied further. Theoretical justifications of particle methods are also expected.