Mathematical Topics in Kinetic theory (Summer Semester 2018)
- Lecturer: Dr. Tobias Ried
- Classes: Lecture (0102700), Problem class (0102710)
- Weekly hours: 2+1
In this course we will introduce and discuss the basic questions in kinetic theory, and the methodical approaches to their solutions. In particular, we will focus on the following topics:
- Boltzmann equation: Cauchy problem and properties of solutions
- entropy and the H theorem
- equilibrium and convergence to equilibrium
Prerequisites: Functional Analysis
| Schedule | ||
|---|---|---|
| Lecture: | Wednesday 14:00-15:30 | SR 2.66 |
| Problem class: | Monday 11:30-13:00 | SR 3.68 |
| Lecturers | ||
|---|---|---|
| Lecturer, Problem classes | Dr. Tobias Ried | |
| Office hours: by appointment | ||
| Room 2.030/2.031 Kollegiengebäude Mathematik (20.30) | ||
| Email: tobias.ried@kit.edu | ||
Lectures
| Date | Topics (preliminary) | |
| 1 | 18.04. | Introduction, Hard Sphere Dynamics: Existence of Flow |
| 2 | 25.04. | Hard Sphere Dynamics: Existence of Flow, Liouville Equation, BBGKY Hierarchy |
| 3 | 02.05. | Hard Sphere Dynamics: BBGKY Hierarchy, Boltzmann Equation |
| 4 | 23.05 | Boltzmann Equation: Scattering, Collision Operator |
| 5 | 28.05. | Boltzmann Equation: Representations of the Boltzmann operator, Bobylev identity |
| 6 | 30.05. | Boltzmann Equation: Conserved quantities and Boltzmann H functional |
| 7 | 06.06. | Boltzmann Equation: Boltzmann H theorem |
| 8 | 13.06. | Boltzmann Equation: Boltzmann H theorem |
| 9 | 20.06. | Boltzmann Equation: Solutions of the homogeneous Boltzmann equation (Existence) |
| 10 | 27.06. | Boltzmann Equation: Solutions of the homogeneous Boltzmann equation (Conservation laws and H theorem) |
| 11 | 11.07. | Boltzmann Equation: Solutions of the homogeneous Boltzmann equation (H theorem) |
| 12 | 18.07. | Kac Equation: Chaoticity, Convergence to Boltzmann Equation |
Lecture Notes
Chapter 1
Chapter 2
Chapter 3
Chapter 4
Exercise Classes
| Date | Topics (preliminary) | |||
| 1 | 23.04. | Hamiltonian Dynamics, Liouville's Theorem, Hard Spheres | Solution | |
| 2 | 30.04. | Maxwell-Boltzmann Functional Equation | Solution | Original Article |
| 3 | 28.05. | Bobylev Identity | Solution | |
| 4 | 04.06. | Weak formulation of the Boltzmann operator | Solution | |
| 5 | 18.06. | Velocity averaging lemma | Solution | |
| 6 | 02.07. | Povzner's lemma and entropy of solutions of the Boltzmann equation | Solution | |
| 7 | 16.07. | Poincaré limit | Solution |
Projects
Throughout the lecture I propose some mini projects that can be presented as a seminar talk at the end of the semester.
| Project 1 | Derivation of the BBGKY hierarchy for hard spheres |
| Project 2 | Derivation of the Boltzmann kernel from classical scattering theory |
| Project 3 | Weak solutions of the homogeneous Boltzmann equation |
| Project 4 | Chaoticity in the Kac master equation |
| Project 5 | Convergence to equilibrium in the Kac equation in L^2 |
| Project 6 | Convergence to equilibrium in the Kac equation in relative entropy |
| Project 7 | Transport equation: method of characteristics and DiPerna-Lions theory |
Examination
Online registration (KIT Campus System) for the oral exam is now open until 31 July 2018.
Exams for the lecture will take place in August.
References
General Introduction/Boltzmann Equation
- Carlo Cercignani, The Boltzmann Equation and Its Applications, Applied Mathematical Sciences Volume 67, Springer New York (1988).
- Cédric Villani, A review of mathematical topics in collisional kinetic theory, Handbook of Mathematical Fluid Dynamics (Vol. 1), edited by S. Friedlander and D. Serre, Elsevier Science (2002).
Hard Sphere Dynamics
- Isabelle Gallagher, Laure Saint-Raymond, Benjamin Texier, From Newton to Boltzmann: Hard Spheres and Short-range Potentials, Zurich Lectures in Advanced Mathematics, EMS Publishing House (2014), see also ArXiv:1208.5753
- Roger Keith Alexander, The infinite hard-sphere system. Lawrence Berkeley National Laboratory. LBNL Report #: LBL-4801 (1975).
Kac Equation
- Eric Carlen, Maria Carvalho, Michael Loss, Kinetic Theory and the Kac Master Equation, Entropy & the Quantum II, Contemporary Mathematics 552, 1-20 (2011).
Transport Equation
- R.J. DiPerna, P.L. Lions, Ordinary differential equations, transport theory and Sobolev spaces, Inventiones mathematicæ 98, 511-547 (1989).