Topics in Non-Collisional Kinetic Theory Spring 2021

Lecturers: Ludovic Cesbron, Mikaela Iacobelli
Coordinator: Antoine Gagnebin
Lectures: Mon 2:15-4:00pm, online
First Lecture: 01.03.2021
Office Hours: Tue. 10-12am, Thur. 2:15-4:15pm
Zoom Meeting Room: ID 914 4700 2409

Syllabus

This class is an introduction to the mathematical study of models from the Kinetic Theory of Plasmas. The intent is to learn essential tools and techniques for the study of Partial Differential Equations while applying them to some of the most important equations of Kinetic Theory: Vlasov equations.

Content of the course:

Transport equations
- Characteristic method
- Weak solutions to conservative transport equations
Kinetic theory of Plasmas
- Mean field limit
  - From particles model to Vlasov-Poisson
  - Dobrushin’s stability theorem
- The Cauchy problem for Vlasov-Poisson

Prerequisites

Required: Notions in functional analysis, differential equations and Lebesgue integration
Optional: Distribution theory, Sobolev spaces, notions in elliptic PDEs

Rules of the Seminar Class

To obtain the credits for the seminar each student must:

Give two 40min presentations on assigned topics. One during the first half of the semester and one during the second half.
Submit the visual support you prepared for your seminar (pdf, beamer, written notes...) so that they can be uploaded on this webpage
Attend the lectures and follow your classmates' seminars

Students are encouraged to work in pairs in order for the two presentations given during a Lecture to be coordinated and complementary. We strongly advise the students to do a mock presentation with their partner to ensure that the time limit of 40 minutes is respected and that the two presentations are clear and well-structured.

The first half of the topics and associated parts of the Lecture notes will be assigned to pairs of students on the first lecture.

Exercise classes

Date	Content	Speaker	Reference
01.03	Tools for PDEs	A. Gagnebin	Functional Analysis II, Prof. Carlotto
08.03	Introduction to Kinetic Theory	Dr. L. Cesbron	Chapter 1 [FG2013]
15.03	Transport Equations with Constant Coefficients Transport Equations with Variable Coefficients	M. Planincic A. Aydin	Chap 2 Sect 2.1-2.2 [FG2013]
22.03	Conservative Transport and Weak Solutions (Part 1) Conservative Transport and Weak Solutions (Part 2)	C. Naiaretti J. Junné	Chap 2 Sect 2.3 [FG2013 ]
29.03	General formalism in classical mechanics Mean field characteristic flow	A. Hirschi	Chap 3 Sect 3.1-3.2 [FG2013]
12.04	Dobrushin’s stability estimate and the mean field limit	A. Gagnebin	Chap 3 Sect 3.3 [FG2013]
03.05	The Vlasov-Poisson Equation and Elementary Inequalities	A. Gagnebin	Chap 4 Sect 4.1 [FG2013]
10.05	The approximate Vlasov-Poisson system	J. Junné	Chap 4 Sect 4.2.1 [FG2013]
17.05	Convergence to the Vlasov-Poisson system	M. Planincic	Chap 4 Sect 4.2.2 [FG2013]
31.06	Propagation of Moments in Dimension 2 + Step 1 of C1 Regularity Propagation of C1 Regularity in Dimensions 2 and 3	C. Naiaretti A. Aydin	Chap 4 Sect 4.3 & 4.5 [FG2013]

Elementary Inequalities

Literature

Lecture Notes

[FG2013] Mean Field Kinetic Equations, F. Golse, 2013

Extended Litterature

The Cauchy Problem in Kinetic Theory, by R.T. Glassey, Society for Industrial and Applied Mathematics, 1996
On the Dynamics of Large Particle Systems in the Mean Field Limit, by F. Golse , arxiv preprint 1301.5494, 2013
Partial Differential Equations, L.C. Evans, American Mathematical Society, 2010
Partial Differential Equations in Action, S. Salsa, Springer International Publishing, 2015
Functional Analysis, T. Bühler and D.A. Salamon, American Mathematical Society, 2018
Topics in Optimal Transport, C. Villani, American Mathematical Society, 2003