**
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.

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.
Top

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.
Top

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**.
Top

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).
Top

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.
Top

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.
Top