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- Academic unit or major
- Graduate major in Mathematics
- Instructor(s)
- Takada Ryo Kagei Yoshiyuki
- Class Format
- Lecture (Face-to-face)
- Media-enhanced courses
- Day/Period(Room No.)
- Intensive ()
- Group
- -
- Course number
- MTH.E647
- Credits
- 2
- Academic year
- 2024
- Offered quarter
- 2Q
- Syllabus updated
- 2024/3/14
- Lecture notes updated
- -
- Language used
- Japanese
- Access Index
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Course description and aims
The main subject of this course is the mathematical analysis of nonlinear partial differential equations that describe the motion of incompressible fluids with rotation and stable stratification in geophysical fluid dynamics. In the first half, we explain the basic properties of oscillatory integrals. As applications, we study the dispersive and the space-time estimates for the linear propagators associated with rotation and stable stratification of the fluids. In the latter half, we learn the well-posedness of the initial value problem for the nonlinear equations, and study the asymptotic behavior of solutions in the fast rotation limit and the strongly stratified limit.
The goal of this course is to acquire fundamental knowledge and techniques for the mathematical analysis of nonlinear partial differential equations that arise in geophysical fluid dynamics.
Student learning outcomes
To learn some basic properties of oscillatory integrals.
To understand basic techniques for the mathematical analysis of PDEs concerning incompressible fluids with rotation and stable stratification.
Keywords
Rotating stably stratified fluids, Oscillatory integrals, Dispersion estimates, Navier-Stokes equations, Boussinesq 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: ・Basic properties of oscillatory integrals ・Derivation of the equations for rotating stably stratified fluids ・Linear solution formula ・Dispersion and space-time estimates for linear propagators ・Global solutions for the nonlinear problem ・Asymptotic behavior of solutions in the fast rotation limit and the strongly stratified limit | Details will be provided during each class session． |
Textbook(s)
None required.
Reference books, course materials, etc.
[1] J.-Y. Chemin, B. Desjardins, I. Gallagher, and E. Grenier, Mathematical geophysics. An introduction to rotating fluids and the Navier-Stokes equations, The Clarendon Press, Oxford University Press, Oxford, 2006.
[2] E. M. Stein, and R. Shakarchi, Functional analysis. Introduction to further topics in analysis, Princeton University Press, Princeton, NJ, 2011.
Assessment criteria and methods
Assignments (100%).
Related courses
- MTH.C305 ： Real Analysis I
- MTH.C306 ： Real Analysis II
- MTH.C351 ： Functional Analysis
Prerequisites (i.e., required knowledge, skills, courses, etc.)
None required.
