Hypocoercivity and convergence of kinetic equations

DATES: 14 - 18 November 2022 (5 sessions)
TIME: 09:00-11:00 (a total of 10 hours)
LOCATION: Maryam Mirzakhani Seminar Room at BCAM
ABSTRACT:
In kinetic equations, hypocoercivity and convergence of the solutions are always much concerned. In this course, we start by studying Fokker-Planck equation. We first introduce the entropy of kinetic Fokker-Planck equation (not always linear), which is Lyapunov functional, and use it to prove the hypocoercivity result. The entropy method is used like in [4] and the hypocoercivity result is based on [6]. In H1 point of view, we can prove the sharp rate of decaying towards the stationary state.
Next, based on the method well used in [3] for a large class of linear kinetic equations and a notion of scalar product adapted to the presence of a Poisson coupling, we can prove the hypocoercivity of linearized Vlasov-Poisson-Fokker- Planck equation (see [1]). Finally, for the Vlasov-Poisson equation, we introduce the decay estimate of the solution, which is studied in [7], and show that how the method can be used in other kinetic equations.

PREREQUISITES:
Partial differential equations, functional analysis, Fourier analysis

REFERENCES:
1] ADDALA, LANOIR, DOLBEAULT JEAN AND TAYEB, LAZHAR. L2-Hypocoercivity and large time asymptotics of the linearized Vlasov-Poisson-Fokker- Planck system. Journal of Statistical Physics, Vol.184, No.1 (2021), 1-34.
[2] DOLBEAULT, JEAN AND LI, XINGYU. Phi-Entropies: convexity, coercivity and hypocoercivity for Fokker-Planck and kinetic Fokker-Planck equations. MathematicalModels and methods in Applied Sciences, Vol. 28, No. 13 (2018), 2637 2666.
[3] DOLBEAULT, JEAN, MOUHOT, CLEMENT AND SCHMEISER, CHRISTIAN. Hypocoercivity for linear kinetic equations conserving mass.. Transcations of the AmericanMathematical Society, Vol. 367, No.6 (2015), 3807-3828.
[4] JUNGEL, ANSGAR EntropyMethods for Diffusive Partial Differential Equations. Springer, 2016.
[5] SCHMEISER, CHRISTIAN EntropyMethods.
https://homepage.univie.ac.at/christian.schmeiser/Entropy-course.pdf, .
[6] VILLANI, CEDRIC Hypocoercivity.Memoirs of the AmericanMathematical Society, 2009.
[7] WANG XUECHENG. Decay estimates for the 3D relativistic and non-relativistic Vlasov-Poisson systems. Kinetic and RelatedModels , 2022.


*Registration is free, but mandatory before 9 November 2022. To sign-up go to https://forms.gle/gB6M8Xj6cox9VC839 and fill the registration form

• BCAM_Course_2022_11_14