Hypocoercivity for perturbation theory and perturbation of hypocoercivity for confined Boltzmann-type collisional equations

• Published in: SeMA journal Vol. 80; no. 1; pp. 27 - 83
• Main Author: Briant, Marc
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 2023; Springer Nature B.V
• Subjects: Applications of Mathematics; Boltzmann transport equation; Coercivity; Kinetic equations; Kinetic theory; Linear equations; Linearization; Mathematical models; Mathematics; Mathematics and Statistics; Operators; Perturbation theory; Robustness
• ISSN: 2254-3902; 2281-7875
• DOI: 10.1007/s40324-021-00281-y

The main concept presented in this review is the hypocoercivity arising in several kinetic equations that are related to the Boltzmann equation. Let us emphasize that it only deals with collisional kernels which feature hard potential and angular cutoff, and when the space variable lives in a confined domain—either bounded or periodic. The review aims at:
Giving an overview of the use of hypocoercive properties for Boltzmann-type linear equations and their applications to nonlinear perturbative settings;
Describing the most recent techniques developed to recover the lack of coercivity of linear kinetic operators;
Showing the robustness of such methods with respect to coefficients and their applications to hydrodynamical limit;
Extending the previous points to what we shall see as a perturbation theory for hypocoercivity.
After giving a general presentation of different collisional models in kinetic theory—classical mono-species Boltzmann equation, Boltzmann system for multi-species mixtures and Boltzmann equation with a force—in Sect. 
1
, we motivate the four points above in Sect. 
2
which reviews Cauchy theories and situates hypocoercivity works in the litterature. The linearized operators which are at the heart of the present review are investigated in Sect. 
3
: both the results which have been obtained and information they taught us in recent models. Then Sect. 
4
explains hypocoercivity in natural spaces of linearization while Sect. 
5
deals with other functional spaces. Lastly, Sect. 
6
shows the robustness of hypocoercivity under different types of perturbation out of equilibrium.