2021 Faculty Courses School of Science Department of Mathematics Graduate major in Mathematics
Advanced topics in Analysis A1
- Academic unit or major
- Graduate major in Mathematics
- Instructor(s)
- Yoshiyuki Kagei
- Class Format
- Lecture
- Media-enhanced courses
- -
- Day of week/Period
(Classrooms) - 3-4 Tue
- Class
- -
- Course Code
- MTH.C405
- Number of credits
- 100
- Course offered
- 2021
- Offered quarter
- 1Q
- Syllabus updated
- Jul 10, 2025
- Language
- English
Syllabus
Course overview and goals
This course gives the theory of bifurcation and stability for the Navier-Stokes equations. In this course, basics of functional analysis and the standard bifurcation theory for partial differential equations are firstly explained; and an application of this theory is illustrated by considering stationary bifurcation problems for the incompressilble Navier-Stokes equations which is classified in a class of parabolic systems. Secondly, stationary bifurcation problems for compressilble Navier-Stokes equations is considered, which cannot be treated by the standard bifurcation theory. This course will be completed with "Advanced topics in Analysis B1" in the next quarter.
Course description and aims
The aim of this course is to learn some aspects of mathematical analysis of nonlinear partial differential equations through the bifurcation and stability analysis of the Navier-Stokes equations.
Keywords
Contraction mapping principle, implicit function theorem, Lyapunov-Schmidt method, bifurcation analysis, Navier-Stokes equations
Competencies
- Specialist skills
- Intercultural skills
- Communication skills
- Critical thinking skills
- Practical and/or problem-solving skills
Class flow
This is a standard lecture course. Occasionally I will give problems for reports.
Course schedule/Objectives
| Course schedule | Objectives | |
|---|---|---|
| Class 1 | Functional analysis and Sobolev spaces 1 | Details will be provided during each class session. |
| Class 2 | Functional analysis and Sobolev spaces | Details will be provided during each class session. |
| Class 3 | Contraction mapping principle and implicit function theorem | Details will be provided during each class session. |
| Class 4 | Navier-Stokes equations | Details will be provided during each class session. |
| Class 5 | Bifurcation from a simple eigenvalue | Details will be provided during each class session. |
| Class 6 | Bifurcation analysis of the incompressible Navier-Stokes equations 1 | Details will be provided during each class session. |
| Class 7 | Bifurcation analysis of the incompressible Navier-Stokes equations 2 | Details will be provided during each class session. |
| Class 8 | Other topics | Details will be provided during each class session. |
Study advice (preparation and review)
To enhance effective learning, students are encouraged to spend approximately 100 minutes preparing for class and another 100 minutes reviewing class content afterwards (including assignments) for each class.
They should do so by referring to textbooks and other course material.
Textbook(s)
None
Reference books, course materials, etc.
- K. Masuda, Nonlinear mathematics (in Japanese), Asakura Shoten, 1985.
Evaluation methods and criteria
Attendance and Assignments.
Related courses
- MTH.C406 ： Advanced topics in Analysis B1
Prerequisites
Students are required to take Advanced topics in Analysis B1 (MTH.C406).
Other
None