###### Kinetic Theory

Nonlinear approximation theory for Boltzmann equation

Quantum Boltzmann equation

Weak coercivity inequality (inspired by Control Theory)

Structure Preserving Scheme for the Kolmogorov-Fokker-Planck Equation (inspired by Control Theory)

1.  Nonlinear approximation theory for Boltzmann equation:

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. After the early work of Carleman, Discrete Velocity Models - DVMs has been developed as one of the main classes of deterministic algorithms to resolve the Boltzmann equation numerically. The second deterministic approximation is the Fourier Spectral Methods – FSMs. The major problem with deterministic methods like DVMs and FSMs that use a fixed discretization in the velocity space is that the velocity space is approximated by a finite region. Therefore in order to use both DVMs and FSMs, 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, developed by Professor Ron Devore, has become one of the most important theories in scientific computing. The theory for elliptic equations has been fully developed. In [14], I have introduced the first nonlinear approximation theory for the Boltzmann equation. Our nonlinear wavelet approximation is nontruncated and based on a nonlinear, adaptive spectral method associated with a new wavelet filtering technique and a new formulation of the equation. The approximation is proved to converge and perfectly preserve most of the properties of the homogeneous Boltzmann equation. The work is the first bridge between the two important theories: kinetic and nonlinear approximation. My nonlinear approximation solves the equation without having to impose non-physics conditions on the equation and could also be considered as an equivalent strategy with the Absorbing Boundary Conditions in PDEs theory for kinetic integral equations like Boltzmann equations, coagulation equations...

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 perfectly preserves the structure of the equation; therefore it is quite normal that most physical properties of the solution are reflected in the approximate solutions, while previous strategies could not.

My 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. Since the strategy is adaptive, it is much cheaper than previous ones. Moreover, it has a spectral accuracy.

2.  Quantum Boltzmann equation:

The quantum Boltzmann equation is a kinetic equation that describes the evolution of a non equilibrium spatially homogeneous distribution of quasiparticles in a dilute Bose gas below the Bose Einstein transition temperature. It is well known that the equation has a family of equilibria. The open question is to obtain an explicit convergence rate to equilibria of the equation. The relaxation of the density towards its corresponding equilibria has been considered previously by several authors, mainly in the physics literature. In my paper [15], I have obtained an explicit rate of convergence to equilibrium of the equation.

3.  Weak coercivity inequality: (inspired by Control Theory)

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. Three different approaches: Lyapunov functional technique, the pseudodifferential calculus, the macroscopic and microscopic decomposition have been developed to study this problem.

Inspired by Control Theory and the work of Professor Enrique Zuazua, 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.

4.  Structure Preserving Scheme for the Kolmogorov-Fokker-Planck Equation: (inspired by Control Theory)

We are interested in the following Kolmogorov-Fokker-Planck equation

As can see from its form, the solution of the equation does not only diffuse in the direction of $v$, by the effect of the diffusion operator $\partial_{v}^2$, but it is also diffuses in the direction of $x$, due to the transport equation $\partial_t f-v\,\partial_xf$. The solution to the Kolmogorov equation is known to decay polynomially in time, and it is our goal to preserve this decay rate through numerical schemes.

Inspired by the self-similarity technique in control theory, we propose in the paper [16] 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.