### 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).