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Workgroup Applied Analysis

Secretariat
Kollegiengebäude Mathematik (20.30)
Room 2.029

Address
Englerstrasse 2
76131 Karlsruhe
Germany

Dr. Kaori Nagato-Plum
kaori.nagatou@kit.edu




Office hours:
Mon - Fri 10:00 -- 12:00

Tel.: +49 721 608 42056

Fax.: +49 721 608 46214

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

Hard Sphere Dynamics

Kac Equation

Transport Equation