G.d.R CNRS 2900 "CHANT"


PRESENTATION



The presentation below is an excerpt of the scientific project
as proposed to the CNRS at the end of 2004.



     Inspired by the success of the European network HYKE, which had initiated connections between the "hyperbolic", "kinetic", and "viscosity solutions" communities, the network CHANT has approximately 250 members, from about 20 French laboratories. It is funded by the french CNRS.

     The "kinetic and hyperbolic community" can probably be characterized by some feature problems, such as : Vlasov and Boltzmann equations (their derivation from classical or quantum particles systems; the qualitative study of their solutions both for small and large times; their numerical resolution); viscosity solutions for Hamilton-Jacobi equations; caustic phenomenon; shocks; entropy conditions; stability for hyperbolic systems; kinetic formulation of conservations laws; fluid limit of kinetic problems. Moreover, the community is globally very concerned with the underlying numerical issues, and numerical simulation often backs up the modelling aspects. Last, the interest in actually using and developping various mathematical tools for the applications in other scientific areas, is also widely shared. For instance, refined methods of entropy estimates play a central role in the recent studies of models for biology or population dynamics. Similarly, Hamilton-Jacobi equations show up in the description of "fronts" propagation in chemotaxis.

     These general scientific considerations have guided the creation of the GdR CHANT.

     In the sequel, we identify and present more precisely some general themes of this network. Such an approach is necessarily rather arbitrary. We do not aim at giving a thorough overview.

 

Asymptotic analysis and hierarchy of models

Modelling, and classifying models, are an important source of motivation, as well as an active field of research in the community. Rather often, given a problem from physics, chemistry or biology, several models are available. They are more or less precise, and correspond to different levels of modelling, or different scales. Identifying scales, going from one model to another, deriving and justifying such models, are delicate tasks. They must be achieved with physicists, chemists or biologists. To give a famous typical example, we may consider the hierarchy of Newton, Boltzmann, and Navier-Stokes equations. At the particles level, the "natural" model is given by Newton's equations, that is, a system of ordinary differential equations. For large times or a large number of particles, Boltzmann equation becomes a more sensible model: this is a prototype of kinetic equations. Finally, for even larger times, and a more densified gas of particles, Navier-Stokes or Euler equations are the "correct" model: we get systems of "hyperbolic" type (that is, from fluid mechanics). Beyond this peculiar context, where underlying mathematical issues are very hard, note the methods: identifying and classifying models and regimes is a major concern of the community. In the same direction, we may also think of boundary layers in fluid dynamics, where recent progress make it now possible to simplify complete models such as Navier-Stokes equations, to models such as those used in oceanography. Also, the impact of geometrical optics has been important in various contexts such as fluid mechanics, wave propagation in nonlinear media, or numerical analysis of high-frequency waves. In a spirit close to geometrical optics, homogenization and Wigner transform are tools widely used by the hyperbolic and kinetic community. They provide a better understanding of the high-frequency behavior of a large range of models. Another well identified and still very open theme consists in modelling the transport of electrons in semiconductors. Mention for instance the derivation of "one particle" nonlinear Schrödinger equations, from linear models with N particles. Mention also the derivation of Boltzmann type equations from quantum models; the goal in that case is the description of "collisions" at a quantum level. Mention finally the derivation of fluid models for electronic transport, from kinetic models. More recently, the description of traffic flow, or the very flourishing domain of modelling for biology, gave rise to a variety of similar descriptions: microscopic description, at the level of vehicles or cells; mesoscopic description via kinetic models taking into account the collective behavior of individuals in position/velocity; macroscopic description, typically modelling the evolution of the average velocity of individuals.

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Coupled models and micro-macro models

In a context probably more open than in the previous section, the question of coupling different levels of modelling is also natural, and currently active. Indeed, for numerical reasons, one may have to split the study zone into several regions: a region described at a microscopic level (a very precise scale) and another region described at a macroscopic level (rougher model) As a typical case, consider the gas dynamics, for which one may have to couple a kinetic description (valid for rarefied gas) to an hydro-dynamical description. The coupling issue, in particular interface conditions, remains to be better understood. This is an important scientific scope. Another example is the transport in semiconductors. The goal is to couple quantum regions, described by Schrödinger type models, with classical regions, described by Vlasov type models. More generally and more recently, the notion of micro-macro models has emerged. In the context of mechanics of continuous media, the point is to integrate microscopic effects directly to macroscopic models. With such an approach, one can explain for instance the non-Newtonian behavior of certain fluids. This is a long term program, involving a fine understanding of models and scales, and a tough work as far as modelling, mathematical analysis and numerical analysis are concerned. Entropy and convexity methods naturally play a role in this framework, and make it possible to analyze such coupled systems. In a very different context, but in a similar spirit, people now try to derive models for chemistry, which take into account both elementary interactions between particles, and average behaviors of small molecules or even proteins. Here again, the task concerns modelling, mathematical analysis and numerical analysis.

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Entropy methods

The main feature of entropy methods is to measure the rate of return to equilibrium in nonlinear models, when most of the time a linear analysis yields only partial results. These are global methods: one can measure the return to equilibrium even for initial data far from the equilibrium. These are also, in many cases, quantitative methods: they yield not only an estimate for the first negative eigenvalue, but also a sharp rate of convergence. Finally, they emphasize a structure of gradient flow for nonlinear drift-diffusion equations. Identifying a "right" geometry has also motivated a more general theory of gradients flows in a Riemannian context. Moreover, these methods provide an explicit rate of convergence, which is often sharp (possibly combining spectral methods), and which can be adapted to spatially inhomogeneous cases. Lately, these methods have known an important success, and have been set up for a large variety of collisional kinetic models. The fine theoretical understanding of these tools has made it possible to solve (partly) some important conjectures, in the context of Boltzmann or Fokker-Planck equations. It has also led to deep links with mass transportation issues. The same approach gave a finer understanding of more recent models that appeared in biology: coagulation-fragmentation, dynamics of populations. Finally, we mention the very active domain of modelling of granular media.

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Hyperbolic systems

There has been much progress lately concerning for instance the understanding of the Cauchy problem for conservation laws having polyconvex entropies (as well as non convex: Maxwell equations, or the model of Born-Infeld, enter this framework), the study of strong oscillations along a linearly degenerate field, and the analysis of multidimensional viscous shocks. Fine questions concerning stabilities can now be tackled. This theme enters the more general framework of stability issues, and bifurcation in infinite dimension phenomena, which came up to the community from different approaches. Also, we wish to emphasize the role of geometrical optics in this context, which provides and studies completely models of shock profiles. Progress has also been made recently in the numerical analysis of hyperbolic systems. Without trying to give a complete picture, we want to underline that the hyperbolic/kinetic interaction has yielded major progress, via kinetic formulation and approximation of conservation laws, and their modern counterpart, relaxation. We think for instance of the important work achieved around shallow water equations (degenerate systems with source terms).

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Hamilton-Jacobi equations

Hamilton-Jacobi equations are naturally present in the high-frequency analysis of wave equations: it is the eikonal equation. They also appear for instance in the description of fronts propagation in combustion, in the dynamics of populations or in chemotaxis. One may also think of Allen-Cahn and Ginzburg-Landau theories. They have a proper geometrical meaning. The recent effort has been focused on the description of such fronts, and the asymptotic behavior of solutions of drift-diffusion equations, or even the asymptotic behavior of the solutions of the Hamilton-Jacobi equations themselves. From a more geometric point of view, these equations provide a natural framework to study the motion of interfaces (level-sets approach). One may think for instance of mean curvature motion, image processing (the front is then the border of the object under study), or the interface between two fluids, in a multi-phase flow. The homogenization of Hamilton-Jacobi equations is also a very active domain. Last, from the numerical viewpoint, much progress has been done, both in the direction of discretizing Hamilton-Jacobi equations, and in the direction of detecting caustics, or other related questions.

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Kinetic equations

Besides return to equilibrium and regularity theory for Boltzmann equation, and the tremendous developments around the kinetic formulation of conservation laws mentioned above, new applications of kinetic formulations have grown in the Ginzburg-Landau theory. They are also at the origin of a fine regularity theory for scalar conservation laws, which does not use averaging lemmas. The understanding of the regularity of solutions to systems of Vlasov type is also a very active aspect. Finally, we refer to the previous sections for a presentation of the theme of quantum kinetic theory. Similarly, and more recently, kinetic modelling for biology has known a growing success. From a numerical point of view, we wish to mention recent progress made around the discretization of Vlasov equations (Vlasov-Maxwell, Vlasov-Poisson). This is a very delicate issue, since it is posed in the phase space, henceforth in dimension 4 or 6. This problem has known a lot of evolution lately, thanks to adaptive codes which by-passed the classical difficulty of "concentration" for the standard particular methods.

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