2025 (Current Year) Faculty Courses School of Science Department of Mathematics Graduate major in Mathematics
Special lectures on advanced topics in Mathematics L
- Academic unit or major
- Graduate major in Mathematics
- Instructor(s)
- Shota Sakamoto / Hideyuki Miura
- Class Format
- Lecture
- Media-enhanced courses
- -
- Day of week/Period
(Classrooms) - Class
- -
- Course Code
- MTH.E536
- Number of credits
- 200
- Course offered
- 2025
- Offered quarter
- 3Q
- Syllabus updated
- Mar 19, 2025
- Language
- Japanese
Syllabus
Course overview and goals
This is an introductory course for the mathematical analysis of the Boltzmann equation, known as the fundamental equation for the dynamics of dilute gases. In particular, we are interested in the so-called non cutoff case. This equation is, mathematically speaking, a non linear integro-differential equation, hence the equation contains nonlinearity, integral operator, and partial derivatives. Such a combination is unique to this equation, so particular ideas and technique are required to analyse this equation. First, we introduce the more classical cutoff case as a starter, then the non cutoff case for comparison.
The study of the Boltzmann equation is not closed by itself; for instance, they have strong relation with the study of hydrodynamical limit, which reveals the relation between solutions to kinetic and fluid equations, and abstract non local integro-differential equations. This course can serve as a starter to study these topics from kinetic side, when students are interested in these topics.
Course description and aims
Be familiar with the Boltzmann equation
Understand properties of the Boltzmann equation
Be able to estimate linear and non linear terms of the Boltzmann equation as integral operators
Understand the technical differences between the cutoff and the non cutoff Boltzmann equation
Keywords
Boltzmann equation, Hypocoercivity, Cancellation lemma, Regularizing effect
Competencies
- Specialist skills
- Intercultural skills
- Communication skills
- Critical thinking skills
- Practical and/or problem-solving skills
Class flow
This is a standard lecture course. There will be some assignments.
Course schedule/Objectives
| Course schedule | Objectives | |
|---|---|---|
| Class 1 | The following topics will be lectured in order: 1. Introduction to the Boltzmann equation 2. Cauchy problem of the cutoff Boltzmann equation around equilibrium 3. The non cutoff Boltzmann equation and regularizing effect 4. Cauchy problem of the non cutoff Boltzmann equation around equilibrium | Details will be provided during each class session. |
Study advice (preparation and review)
To enhance effective learning, students are encouraged to prepare for and review the class content (including assignments).
They should do so by referring to course material.
Textbook(s)
None required
Reference books, course materials, etc.
Will be given during the class.
Evaluation methods and criteria
Assignments (100%).
Related courses
- MTH.C341 ： Differential Equations I
- MTH.C342 ： Differential Equations II
Prerequisites
None