**Kinetic Theory **

**1.
**__Nonlinear
approximation theory for kinetic equations:__

Numerical
resolution methods for the Boltzmann equation plays a very important role in
the practical a theoretical study of the theory of rarefied gas. The main difficulty
in the approximation of the Boltzmann equation is due to the multidimensional
structure of the Boltzmann collision operator. The major problem with
deterministic numerical methods using to solve Boltzmann equation is that we
have to truncate the domain or to impose nonphysical conditions to keep the
supports of the solutions in the velocity space uniformly compact.

For the
last two decades, the nonlinear approximation theory, has become one of the
most important theories in scientific computing. The theory for elliptic
equations has been fully developed. In [14], [15], [16], I have tried to make
the connection between nonlinear approximation theory and kinetic theory.

Our
nonlinear wavelet approximation is nontruncated and based on an adaptive
spectral method associated with a new wavelet filtering technique and a new
formulation of the equation. The approximation is proved to converge and
preserve many properties of the homogeneous Boltzmann equation. The nonlinear
approximation solves the equation without having to impose non-physics
conditions on the equation. The reason that some physical properties of the
Boltzmann equation could not be preserved through classical Discrete Velocity
Models is the convolution structure is destroyed. One of the reasons making
Fourier basis not an ideal choice for spectral approximations is that it could
not preserve the structure of the collision operator, for example the ”coercivity” property of the ”gain” part of the
collision operator, which is due to the non-positivity of the projection of a
positive function onto its Fourier components as well as the effect of the
Gibbs phenomenon. With suitable wavelet basis, my approach preserves the
structure of the equation. The theory also gives a unified point of view for
the two available methods, Fourier Spectral Methods and Discrete Velocity
Models: these strategies could be considered as special cases of our method in
the sense that our approximation could produce spectral methods as well as
discrete velocity models by using different wavelet basis; moreover, they are
nonlinear and adaptive.

Inspired
by the above idea we also propose in the paper [18] a new strategy to design
structure preserving schemes for the Kolmogorov equation. We also present an
analysis for the operator splitting technique for the self-similar method and
numerical results for the described scheme. Numerically, the self-similarity
technique has a major benefit, in that, for long time simulation one need not
choose a large domain, since the solution maintains compact support for a well-chosen
initial domain. Additionally, the time scaling allows for fast time marching
and so simulations are more computationally efficient and less reliant on
artificial boundary conditions.

**2.
**__Weak
coercivity inequality__

The trend
to the equilibrium for kinetic equations was first systematically studied by L.
Desvillettes and C. Villani. The coercivity from the
damping or the collision is often degenerate and it is not trivial to see how
the whole dynamics dissipates and to establish the rate of convergence to the
equilibrium.

Inspired
by Control Theory, in the paper [13], I have introduced an observability
inequality or a weak coercive inequality to show the decay rate to the
equilibrium. My results then improve the previous ones for the Goldstein-Taylor
equation and related models as well as the linearized Boltzmann equation. In
particular, previous results for the Goldstein-Taylor equation assert only some
polynomial decay rates while my paper proves the exponential decay without
assuming too much on the cross-section. The technique is constructive, since
the constants in the decay rates could be obtained explicitly.

**3.
**__A
non-gradient Fokker-Planck equation in Computational Neuroscience__

In collaboration
with Jose-Antonio Carrillo and Simona Mancini [17], we have proven of the
exponential rate of convergence of the solution of a Fokker-Planck equation,
with a drift term not being the gradient of a potential function and endowed by
Robin type boundary conditions. This kind of problem arises, for example, in
the study of interacting neurons populations. Previous studies have numerically
shown that, after a small period of time, the solution of the evolution problem
exponentially converges to the stable state of the equation.