▶Japanese
Post to SNS
- Home
- > School of Science
- > Graduate major in Mathematics
- > Advanced topics in Analysis F1
- School of Science
-
School of Engineering
- Undergraduate major in Mechanical Engineering
- Undergraduate major in Systems and Control Engineering
- Undergraduate major in Electrical and Electronic Engineering
- Undergraduate major in Information and Communications Engineering
- Undergraduate major in Industrial Engineering and Economics
- Common courses
- School of Materials and Chemical Technology
- School of Computing
- School of Life Science and Technology
- School of Environment and Society
- 工学院，物質理工学院，環境・社会理工学院共通科目
- Liberal arts and basic science courses
- First-Year Courses
- School of Science
- School of Engineering
- School of Materials and Chemical Technology
- School of Computing
- School of Life Science and Technology
-
School of Environment and Society
- Department of Architecture and Building Engineering
- Department of Civil and Environmental Engineering
- Department of Transdisciplinary Science and Engineering
- Department of Social and Human Sciences
- Department of Innovation Science
- Graduate major in Technology and Innovation Management
- Common courses
- Liberal arts and basic science courses
- Academic unit or major
- Graduate major in Mathematics
- Instructor(s)
- Kagei Yoshiyuki
- Class Format
- Lecture
- Media-enhanced courses
- Day/Period(Room No.)
- Tue5-6(H119A)
- Group
- -
- Course number
- MTH.C506
- Credits
- 1
- Academic year
- 2019
- Offered quarter
- 4Q
- Syllabus updated
- 2019/3/18
- Lecture notes updated
- -
- Language used
- English
- Access Index
-
Course description and aims
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.
Student learning outcomes
・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 equtiations.
Keywords
Frechet derivatice, implicit function theorem, resolvent, spectrum, Lyapunov-Schmidt method, bifurcation analysis, incompressible 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. There will be some assignments.
Course schedule/Required learning
| Course schedule | Required learning | |
|---|---|---|
| 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. |
Textbook(s)
None required
Reference books, course materials, etc.
Details will be provided during each class.
Assessment criteria and methods
Attendance and Assignments.
Related courses
- MTH.C341 ： Differential Equations I
- MTH.C342 ： Differential Equations II
Prerequisites (i.e., required knowledge, skills, courses, etc.)
None
