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- Common courses
- Liberal arts and basic science courses
- Academic unit or major
- Graduate major in Mathematics
- Instructor(s)
- Tonegawa Yoshihiro Maekawa Yasunori
- Class Format
- Lecture
- Media-enhanced courses
- Day/Period(Room No.)
- Intensive (H201)
- Group
- -
- Course number
- MTH.E439
- Credits
- 2
- Academic year
- 2016
- Offered quarter
- 3Q
- Syllabus updated
- 2016/12/14
- Lecture notes updated
- -
- Language used
- Japanese
- Access Index
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Course description and aims
The main topic of this course is stability analysis of stationary solutions for Navier-Stokes equations, basic equations of fluid mechanics. The Navier-Stokes equation is a non-linear partial differential equation proposed in the middle of the 19th century as an equation for describing the movement of incompressible, viscous fluids. Because it's a non-linear system and because of its non-locality, in many cases an exact analysis is difficult to achieve, with many problems remaining unsolved even now. This course will cover the existence and stability of stationary solutions for classes with Navier-Stokes equations for unbounded regions, including recent developments.
Students will learn about a typical analysis method for the existence and stability of solutions to non-linear partial differential equations, in this case the Navier-Stokes equation. Focusing on scale criticality for balancing linearity and nonlinearity, students will also gain an understanding of how intuition in fluid mechanics, and mathematical structure correspond, and how real analysis and functional analysis are useful.
Student learning outcomes
- Gain an understanding of standard discussions based on analysis of linearization operator for the existence and stability of a steady solution for Navier-Stokes equations.
- Gain an understanding of the relationship between scale invariance and behavior of solutions for Navier-Stokes equations.
- Gain an understanding of the mathematical structure of fluids around rotating bodies.
- Gain an understanding of distinctive mathematical properties of axisymmetric circulations.
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) | Details will be provided during each class session. |
| Class 3 | Some explicit solutions to the Navier-Stokes equations (2) | Details will be provided during each class session. |
| Class 4 | Time-periodic flows around a rotating obstacle in two dimensions (1) | Details will be provided during each class session. |
| Class 5 | Time-periodic flows around a rotating obstacle in two dimensions (2) | Details will be provided during each class session. |
| Class 6 | Time-periodic flows around a rotating obstacle in two dimensions (3) | Details will be provided during each class session. |
| Class 7 | Stability of steady circular flows in an exterior domain to the unit disk (1) | Details will be provided during each class session. |
| Class 8 | Stability of steady circular flows in an exterior domain to the unit disk (2) | Details will be provided during each class session. |
| Class 9 | Stability of steady circular flows in an exterior domain to the unit disk (3) | Details will be provided during each class session. |
| Class 10 | Stability of scale critical flows in the two-dimensional exterior domain (1) | Details will be provided during each class session. |
| Class 11 | Stability of scale critical flows in the two-dimensional exterior domain (2) | Details will be provided during each class session. |
| Class 12 | Stability of Burgers vortices: introduction | Details will be provided during each class session. |
| Class 13 | Stability of Burgers vortices in two dimensions | Details will be provided during each class session. |
| Class 14 | Stability of Burgers vortices in three dimensions (1) | Details will be provided during each class session. |
| Class 15 | Stability of Burgers vortices in three dimensions (2) | Details will be provided during each class session. |
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
- ZUS.F301 ： Foundations of Functional Analysis
Prerequisites (i.e., required knowledge, skills, courses, etc.)
None
