2021 Advanced courses in Analysis A

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Kagei Yoshiyuki 
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Syllabus updated
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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.


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.



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).



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