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- Academic unit or major
- Mathematics
- Instructor(s)
- Kagei Yoshiyuki
- Class Format
- Lecture
- Media-enhanced courses
- Day/Period(Room No.)
- Tue3-4(H119A)
- Group
- -
- Course number
- ZUA.C331
- Credits
- 1
- Academic year
- 2021
- Offered quarter
- 1Q
- Syllabus updated
- 2021/3/19
- Lecture notes updated
- -
- Language used
- English
- Access Index
-
Course description and aims
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 courses in Analysis B" in the next quarter.
Student learning outcomes
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 that will be developed
| ✔ 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/Required learning
| Course schedule | Required learning | |
|---|---|---|
| Class 1 | Functional analysis and Sobolev spaces 1 | Details will be provided during each class session. |
| Class 2 | Functional analysis and Sobolev spaces 2 | |
| Class 3 | Contraction mapping principle and implicit function theorem | |
| Class 4 | Navier-Stokes equations | |
| Class 5 | Bifurcation from a simple eigenvalue | |
| Class 6 | Bifurcation analysis of the incompressible Navier-Stokes equations 1 | |
| Class 7 | Bifurcation analysis of the incompressible Navier-Stokes equations 2 | |
| Class 8 | Other topics |
Out-of-Class Study Time (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.
Assessment criteria and methods
Attendance and Assignments.
Related courses
- ZUA.C332 ： Advanced courses in Analysis B
Prerequisites (i.e., required knowledge, skills, courses, etc.)
Students are required to take Advanced courses in Analysis B (ZUA.C332).
Other
None
