2025 (Current Year) Faculty Courses School of Science Department of Mathematics Graduate major in Mathematics
Advanced topics in Analysis C1
- Academic unit or major
- Graduate major in Mathematics
- Instructor(s)
- Yoshiyuki Kagei
- Class Format
- Lecture
- Media-enhanced courses
- -
- Day of week/Period
(Classrooms) - Class
- -
- Course Code
- MTH.C407
- Number of credits
- 100
- Course offered
- 2025
- Offered quarter
- 3Q
- Syllabus updated
- Mar 19, 2025
- Language
- English
Syllabus
Course overview and goals
This course gives the theory of bifurcation and stability for the compressible Navier-Stokes equations which are known to be the fundamental equations in the fluid mechanics. The compressible Navier-Stokes equations are classified in a class of quasilinear heyperbolic-parabolic systems and have provided fundamental issues in the field of partial differential equations such as existence, uniqueness, regularity and asymptotic behavior of solutions, and etc. In this course, the standard bifurcation theory for partial differential equations is 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 F" 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.
Course description and aims
・Understand differentiaion and implicit function theorem for nonlinear maps in infinite dimensional spaces.
・Understand the standard bifurcation theory.
・Understand resolvent and spectrum of linear operator.
・Understand fundamental properties of nonlinear parabolic equations.
Keywords
Frechet derivatice, implicit function theorem, resolvent, spectrum, Lyapunov-Schmidt method, bifurcation analysis, compressible 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. There will be some assignments.
Course schedule/Objectives
| Course schedule | Objectives | |
|---|---|---|
| Class 1 | The following topics will be covered in this order : -- Differentiation of nonlinear maps in infinite dimensional spaces: Frechet derivatives, Taylor expansions, etc. -- Contraction mapping principle -- Implicit function theorem -- Standard bifurcation theory -- Stability of bifurcating solutions -- Bifurcation and Stability analysis of the incompressible Navier-Stokes equations | 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 reviewing class content afterwards for each class.
Textbook(s)
none
Reference books, course materials, etc.
Details will be provided during each class
Evaluation methods and criteria
Assignments (100%).
Related courses
- MTH.C305 ： Real Analysis I
- MTH.C306 ： Real Analysis II
- MTH.C351 ： Functional Analysis
- MTH.C341 ： Differential Equations I
- MTH.C342 ： Differential Equations II
Prerequisites
Students are required to have understanding of basics of real analysis and functional analysis.