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- Academic unit or major
- Mathematics
- Instructor(s)
- Tonegawa Yoshihiro Maekawa Yasunori
- Class Format
- Lecture
- Media-enhanced courses
- Day/Period(Room No.)
- Intensive (H201)
- Group
- -
- Course number
- ZUA.E339
- Credits
- 2
- Academic year
- 2016
- Offered quarter
- 3Q
- Syllabus updated
- 2016/4/27
- Lecture notes updated
- -
- Language used
- Japanese
- Access Index
-
Course description and aims
The main subject of this course is the stability analysis of stationary solutions to the Navier-Stokes equations for viscous incompressible flows. The Navier-Stokes equations are famous nonlinear partial differential equations, which are the most fundamental equations in the fluid dynamics. This course is an introduction to the recent progress on the existence and the stability of some stationary solutions to the Navier-Stokes equations in unbounded domains.
This course is designed to learn a typical approach for the stability problem in the nonlinear partial differential equations, through several topics from mathematical fluid mechanics. This course will also highlight the importance and difficulty of the analysis when the linearity and the nonlinearity are balanced in view of scaling, and will introduce some recent results in the study of the Navier-Stokes equations which overcome this difficulty with the aid of the real analysis and the functional analysis together with the knowledge from the fluid dynamics.
Student learning outcomes
・Understand a typical argument in the study of the existence and stability of stationary solutions to the Navier-Stokes equations based on the analysis of the linearized operators.
・Understand the relation between the scale invariance of the Navier-Stokes equations and the asymptotic behavior of solutions.
・Understand mathematical structures of the flows around a rotating obstacle.
・Understand typical features of the axisymmetric circular flows.
Keywords
Navier-Stokes equations, vorticity fields, spectrum and resolvent of linear operators, existence and stability of stationary solutions, scaling invariance and asymptotic behavior of solutions, axisymmetric circular flows
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 | Introduction to the Navier-Stokes equations. | Details will be provided during each class session. |
| Class 2 | Some explicit solutions to the Navier-Stokes equations (1) | |
| Class 3 | Some explicit solutions to the Navier-Stokes equations (2) | |
| Class 4 | Time-periodic flows around a rotating obstacle in two dimensions (1) | |
| Class 5 | Time-periodic flows around a rotating obstacle in two dimensions (2) | |
| Class 6 | Time-periodic flows around a rotating obstacle in two dimensions (3) | |
| Class 7 | Stability of steady circular flows in an exterior domain to the unit disk (1) | |
| Class 8 | Stability of steady circular flows in an exterior domain to the unit disk (2) | |
| Class 9 | Stability of steady circular flows in an exterior domain to the unit disk (3) | |
| Class 10 | Stability of scale critical flows in the two-dimensional exterior domain (1) | |
| Class 11 | Stability of scale critical flows in the two-dimensional exterior domain (2) | |
| Class 12 | Stability of Burgers vortices: introduction | |
| Class 13 | Stability of Burgers vortices in two dimensions | |
| Class 14 | Stability of Burgers vortices in three dimensions (1) | |
| Class 15 | Stability of Burgers vortices in three dimensions (2) |
Textbook(s)
none required
Reference books, course materials, etc.
「Navier-Stokes houteishiki no suri」 Hisashi Okamoto (Tokyo Univ. Press, 2009); 「Nonlinear partial differential equations. Asymptotic behavior of solutions and self-similar solutions. Progress in Nonlinear Differential Equations and their Applications, 79」 M.-H. Giga, Y. Giga, and J. Saal (Birkh{\"a}user, Boston, 2010)
Assessment criteria and methods
Assignments (100%).
Related courses
- MTH.C351 ： Functional Analysis
Prerequisites (i.e., required knowledge, skills, courses, etc.)
none in particular
