2. Scientific originality

Recent years have seen a very fruitful exchange of ideas between the fields of kinetic equations and hyperbolic systems, a trend that we will strongly emphasize within the proposed network. The subject of kinetic and relaxation methods for conservation laws has developed rapidly in the past ten years. In this area methods from hyperbolic and kinetic theory interact very effectively, with a rich interplay between modeling, analytical, and computational concepts. The approach is very natural in problems involving multiple scales and produced new algorithms for hyperbolic and diffusion equations. We expect further advances, leading to applications in the computation of diffusive limits in kinetic equations and to the design of efficient algorithms for rarefied gases.

The ideas for rarefied gases have successfully been transferred to charged particle systems like the electron gas in semiconductors and plasmas. The understanding and rigorous derivation of time dependent model hierarchies via asymptotic analysis , i. e., by identifying the relevant parameters and scaling limits, has been and will continue to be fostered by most significant contributions of this community. The whole hierarchy of nonlinear relativistic (quantum) transport from the Dirac-Maxwell and Vlasov-Einstein systems to the Vlasov-Poisson equation down to incompressible Euler equations will be tackled by the new methods developed in the course of the previous networks.

The use of entropy and entropy production methods or logarithmic-Sobolev type inequalities has in many cases clarified the long-time behaviour of kinetic and parabolic equations. This technique will yield many more results for equations such as fourth order diffusion equations or spatially inhomogeneous kinetic equations. The connection between entropy methods and the classical problems of mass transportation will be a key issue in kinetic theory in the future.

While well-posedness for multiphase flows is a widely open problem, various results on the modeling and analysis of complex flows such as sprays are being obtained by a new approach, namely mixed kinetic-hyperbolic equations and Eulerian-Lagrangian methods .

Many results on the Boltzmann kernel have given a good understanding of the properties of the spatially homogeneous Boltzmann equation. However, these results, concerning the smoothness of the solutions or their asymptotic behaviour for large times, still need to be transferred to the space-dependent equation. We expect breakthroughs in this area in the next few years. Many numerical schemes have recently been proposed to simulate collisions, such as spectral schemes, or deterministic particle methods. Well-posedness and/or global regularity remain major open problems for the Boltzmann equation as well as for the Vlasov-Maxwell system, strictly hyperbolic conservation laws, or various equations from fluid and continuum mechanics.

The interplay between the microscopic description of gases as finite particle systems, the kinetic description and the fluid description via Euler or Navier-Stokes equations has received a lot of attention lately, and many results for example on the passage from Boltzmann equation to incompressible fluid equations have been obtained. Bringing together specialists of kinetic and hyperbolic equations should stimulate further progress in this exciting field.

To derive from quantum mechanics a "quantum Boltzmann equation" which includes scattering effects is an open and very challenging problem. However, considerable progress has been made concerning Boltzmann-type equations for quantum particles, as they are usually used in the physics literature, especially for Fermi-Dirac particles. But this field is largely open as well, and questions of well-posedness, trend to equilibrium, Landau limits, fluid limits, and Bose particles need to be investigated. Some of the most interesting questions are strongly related with other fields like electron transport in crystals, oscillatory phenomena induced by phonon scattering, and in particular Bose-Einstein condensation . Numerical evidence, based on "simple" kinetic models for quantum particles, indicates that singularity formation in finite time is an important mechanism here, and some simpler models exhibit condensation in infinite time. Therefore, we will systematically study kinetic models exhibiting blow-up in finite time and quantify this phenomenon which is relevant also for another case of phase transitions , namely for coagulation-fragmentation . Coagulation models begin to be understood, but a precise description of the process of gelation is essentially lacking, and the combination of coagulation and fragmentation effects is even less understood.

Recently, very encouraging progress on the derivation of kinetic models for granular media has been made. We will emphasize the study of the qualitative behaviour of such models such as the trend to equilibrium and will base the development of efficient numerical methods on this analysis. As another area where kinetic models are becoming important we mention the study of thin magnetic films which formally leads to a dimensionally reduced equation which is closely related to the eikonal equation. A recent kinetic reformulation of the problem provides a powerful tool to address the crucial issues of compactness and selection of a natural solution.

The last few years have seen progress at several fronts in the study of hyperbolic systems , manifested by a multitude of new perspectives. We mention the advances achieved for the problems of uniqueness and continuous dependence and the development of the Bressan-Liu-Yang functional. We expect further progress on continuous dependence, structure, blow-up issues for bounded variation solutions, and possibly on the challenging problem of zero-viscosity limits. The kinetic formulation has offered a new perspective for studying propagation of oscillations. Ideas from geometrical optics and semi-classical limits provide a framework for studying the problem of wave propagation in inhomogeneous media. We expect that for the same problem the emerging field of conservation laws and Hamilton-Jacobi equations with randomness will play an important role. Other issues concern the quantitative understanding of the effect of diffusion matrices or other approximating mechanisms such as relaxation or diffusion-dispersion on the selection of solutions. Finally, there is renewed interest in the structure of special multi-dimensional systems and on the transfer of expertise from the calculus of variations to evolution systems.

Recently, there has been interest in the stability of boundary layers , and we expect progress in this direction, and in the analysis of over-/under-compressive shocks and their stability, in the derivation of kinetic relations from the diffusive structure of approximating systems, and on the stability of wave structures like detonation waves or magneto-hydrodynamic shocks. A challenging classical problem is to provide a rigorous analysis of the regular reflection of a 2-D gas dynamics shock wave past a wedge. The dynamics of complex materials leads to conservation laws which involve both shocks and phase boundaries . There is a longstanding debate which, if any, extra conditions need to be prescribed at phase boundaries. New approaches (such as Evans functions) to analyze profiles of regularized equations which capture additional physics offer the possibility to resolve this controversy.

The Wigner transform and the ensuing relation of multiphase geometrical optics with kinetic equations offers an approach that complements WKB expansions, viscosity solutions, and wave tracing. There are parallels to the structure of the equations of pressureless gas, and connections with issues of discontinuous transport equations. We expect a fruitful interaction of these ideas. Progress in multiphase geometrical optics has a multitude of applications in practical problems such as underwater acoustics and damage assessment.

The development of numerical methods for kinetic and hyperbolic problems has always gone hand-in-hand with a thorough analysis of the solution properties. In particular, the stable computation of singular limits requires a preliminary asymptotic analysis. Examples for this influx of analytical knowledge into algorithm design can be found in the computation of shocks, conservation laws with stiff source terms, finite element schemes via relaxation, and kinetic simulations close to hydrodynamic regimes, with applications for example in semiconductor simulations, multiphase flows, detonations, shallow water waves, and porous media.