2024 Faculty Courses School of Science Department of Mathematics Graduate major in Mathematics
Special lectures on advanced topics in Mathematics E
- Academic unit or major
- Graduate major in Mathematics
- Instructor(s)
- Ryo Takada / Yoshiyuki Kagei
- Class Format
- Lecture (Face-to-face)
- Media-enhanced courses
- -
- Day of week/Period
(Classrooms) - Intensive
- Class
- -
- Course Code
- MTH.E435
- Number of credits
- 200
- Course offered
- 2024
- Offered quarter
- 2Q
- Syllabus updated
- Mar 14, 2025
- Language
- Japanese
Syllabus
Course overview and goals
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.
Course description and aims
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
- 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: ・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 |
Study advice (preparation and review)
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.
Evaluation methods and criteria
Assignments (100%).
Related courses
- MTH.C305 ： Real Analysis I
- MTH.C306 ： Real Analysis II
- MTH.C351 ： Functional Analysis
Prerequisites
None