Lecturer

Prof. Dr. Mikaela Iacobelli

Coordinator

Dr. Ludovic Cesbron

Lectures

Mon 13-15 / HG G 26.3

First Lecture

17.02.2020

Office hours

Tue 09-11 / HG F 27.2

Thu 15-17 / HG F 27.2

Course Catalogue

401-3350-20L Topics in Partial Differential Equations

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 a 40min presentation on an assigned topic
• Attend at least 9 of the 13 lectures
• Submit the solutions of any exercise present in the part of the lecture notes associated with the assigned topic

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 topics and associated parts of the Lecture notes will be assigned to pairs of students two weeks before the presentation is due.

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