Special courses on advanced topics in Mathematics E

Academic unit or major

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

ZUA.E335

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

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.C305 ： Real Analysis I
• MTH.C351 ： Functional Analysis

Prerequisites

None