Spring 2022

- Lecturers
- Prof. Dr. Mikaela Iacobelli, Dr. Alexandre Rege
- Coordinator
- Antoine Gagnebin
- Lectures
- Mon 2:15-4:00pm, ML J 37.1
- First Lecture
- 28.02.2022
- Office Hours
- TBD on zoom
- Zoom Meeting Room
- ID 570 479 7608
- Course Catalogue
- 401-4820-22L

This seminar will investigate the relation between "exact" microscopic models that govern large particle systems' evolution and a specific type of approximate models known in Statistical Mechanics as "mean-field equations". Roughly speaking, a mean-field equation is a model that describes the evolution of a typical particle subject to the collective interaction created by a large number of other, identical particles. The most famous example of mean-field equation is the Vlasov-Poisson equation.

The mathematical justification of the mean-field limit involves very different ideas and methods. This course aims to give an introductory description of the classical approaches to the problem of the mean-field limit in mathematical analysis.

In particular, the intent is to learn essential tools and techniques for studying Partial Differential Equations while applying them to 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

- Mean field limit

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

To obtain the credits for the seminar each student must:

- Give one presentation on assigned topics.
- Attend the lectures and follow your classmates' seminars.
- Submit the notes you prepared for your seminar so that they can be uploaded on this webpage (if you want).

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

Date | Content | Speaker | Reference |
---|---|---|---|

28.02 | Introduction to the seminar | Prof. Dr. Mikaela Iacobelli | |

07.03 | Tools for PDE's | Antoine Gagnebin | |

14.03 | Break | ||

21.03 | Transport Equations with Constant Coefficients Transport Equations with Variable Coefficients | Isabella Brovelli | Chap 2 Sect 2.1 & 2.2 [FG2013] |

28.03 | Conservative Transport and Weak Solutions | Adrian Dawid | Chap 2 Sect 2.3 [FG2013 ] |

04.04 | General formalism in classical mechanics Mean field characteristic flow | David Lenze | Chap 3 Sect 3.1 & 3.2 [FG2013] (Stop before Theorem 3.2.2) |

11.04 | Mean field characteristic flow The Monge-Kantorovich distance Dobrushin’s estimate | Fatime Rasiti | Chap 3 Sect 3.2, 3.3.1 & 3.3.2 [FG2013] (Start with Theorem 3.2.2) |

18.04 | Easter Monday | ||

25.04 | Sechseläuten afternoon off | ||

02.05 | The mean field limit On the choice of the initial data | Yuxiu Zhang | Chap 3 Sect 3.3.3 & 3.3.4 [FG2013] |

09.05 | |||

16.05 | |||

23.05 | |||

30.05 |

- [FG2013] Mean Field Kinetic Equations, F. Golse, 2013
- [FG2022] Mean-Field Limits in statistical dynamics, F. Golse, 2022
- [PEJ2016] A review of the mean field limits for Vlasov equations., P-E. Jabin, 2016

- 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