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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.

* Although important questions of well-posedness in fluid dynamics remain open, several new methods have been developed during the last five years:
Evans functions for the study of wave instability work efficiently for shock profiles and boundary layers for various perturbations (parabolic, relaxation, numerical): see for instance the papers by Gardner, Zumbrun, Serre, Benzoni & al.
Liapunov-type functionals can be used to study the well-posedness of the Cauchy problem: see for instance the papers by Liu, Bressan and Yong.
Green's function for proving that spectral (linear) stability implies full (nonlinear) stability: see for instance the papers by Howard, Zumbrun, Grenier and Rousset.

The analysis of different asymptotic limits for a model with several relevant scales and regimes leads to a multiscale analysis . This asymptotic study will be further developed to the investigation of hybrid methods by coupling different regimes of the same model. The integration of various ideas, methods and techniques in this area will be significant. Useful techniques in this area are: diffusive limits, hydrodynamic limits, relaxation limits, semi-classical limits, kinetic-fluid coupling, quantum-kinetic coupling, spherical harmonics expansions, ...

Numerical analysis and simulation will be one of the main issues of this proposal.

*The numerical simulation and numerical analysis of problems included in this proposal will be one of the main issues in the proposed research. The various problems which are treated allow up to compare different concepts: Monte-Carlo and random methods (an expertise for coagulation-fragmentation Boltzmann equation is well established), multiresolution and particle methods, finite volume and finite element approaches. Our goal is to cover the convergence analysis ( by means of functional analysis, wave interaction methods, probabilistic representation) as well as the actual implementation for model problems and sometimes in industrial codes. Due to the large number of methods and problems, we shall emphasize systematic comparisons by developing benchmark problems and including numerical comparison sessions in our network conferences.

We are now going to list new trends, which are probably not as well established as the previous points. The main issue is that applied mathematicians want attack physically complex problems with high relevance for applications, which is the main motivation of the research in the field of kinetic and hyperbolic equations. They also want to take advantage of physical and numerical intuition.

° We are going to consider better approximations of conservation laws , for instance Navier-Stokes instead of artificial viscosity, or physically relevant relaxation.

° Balance laws , especially for combustion problems, will be more prominent.

° Regularization effects for kinetic equations in a spatially inhomogeneous framework, hypoellipticity properties in relationship with averaging lemmas, moment estimates have to be better understood.

° Multi-dimensional hyperbolic problems will be more frequently considered.

° The study of concentration phenomena will be one of the key axes of our study for many systems which arise from various physical models, but whose common point is to lead to clustering of particles. This concerns Bose condensation phenomena, phase transitions, the study of aerosols, or granular flows.



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