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 , , , 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  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 , 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 , 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.