### 2021　Advanced courses in Analysis A

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Mathematics
Instructor(s)
Kagei Yoshiyuki
Course component(s)
Lecture    (ZOOM)
Day/Period(Room No.)
Tue3-4(H119A)
Group
-
Course number
ZUA.C331
Credits
1
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.

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).

None