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.
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
|✔ Specialist skills||Intercultural skills||Communication skills||Critical thinking skills||Practical and/or problem-solving skills|
This is a standard lecture course. Occasionally I will give problems for reports.
|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||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.|
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.
- K. Masuda, Nonlinear mathematics (in Japanese), Asakura Shoten, 1985.
Attendance and Assignments.
Students are required to take Advanced topics in Analysis B1 (MTH.C406).