2023 Students Enrolled in or before 2015 School of Science Mathematics
Advanced courses in Analysis A
- Academic unit or major
- Mathematics
- Instructor(s)
- Kai Koike
- Class Format
- Lecture (Face-to-face)
- Media-enhanced courses
- -
- Day of week/Period
(Classrooms) - 3-4 Mon (M-112(H117))
- Class
- -
- Course Code
- ZUA.C331
- Number of credits
- 100
- Course offered
- 2023
- Offered quarter
- 1Q
- Syllabus updated
- Jul 8, 2025
- Language
- English
Syllabus
Course overview and goals
This lecture is an introduction to the mathematical analysis of nonlinear partial differential equations (NLPDEs). NLPDEs are abundant in nature and each of them form a unique landscape. However, there are certain principles underlying many theories. And in this lecture, through analyzing specific NLPDEs, we will learn these ideas.
Note that this lecture is followed by Advanced topics in Analysis B1 (MTH.C406).
Course description and aims
To understand important methods and ideas in the mathematical analysis of nonlinear partial differential equations.
Keywords
Nonlinear partial differential equations, Sobolev spaces, A priori estimates, energy estimates
Competencies
- Specialist skills
- Intercultural skills
- Communication skills
- Critical thinking skills
- Practical and/or problem-solving skills
Class flow
This is a standard lecture course. Problems for reports are given occasionally.
Course schedule/Objectives
| Course schedule | Objectives | |
|---|---|---|
| Class 1 | Introduction | To be able to list typical PDEs, explain what sort of phenomena are there, and what do we expect to prove mathematically. |
| Class 2 | Linear diffusion equation | To be able to analyze solutions to the linear diffusion equation using Fourier transform, energy method, and maximum principles. |
| Class 3 | Linear parabolic equation | To be able to use the notion of Sobolev spaces, weak solutions, and weak compactness to solve linear parabolic equations. |
| Class 4 | Global-in-time solution to nonlinear diffusion equation: Examples from ODE | To be able to construct global-in-time solutions to nonlinear ODEs using the idea of a priori estimates. |
| Class 5 | Global-in-time solution to nonlinear diffusion equation: Energy method and a priori estimates (1) | To be able to use the energy method to derive a priori estimates and solve nonlinear diffusion equations. |
| Class 6 | Global-in-time solution to nonlinear diffusion equation: Energy method and a priori estimates (2) | Same as above. |
| Class 7 | Asymptotic behavior of solutions to viscous Burgers' equation | To be able to analyze the asymptotic behavior of solutions to viscous Burgers' equation using the techniques learned so far. |
| Class 8 | Other topics | Details will be provided in the class. |
Study advice (preparation and review)
To enhance effective learning, by referring to textbooks and other course materials, students are encouraged to spend approximately 100 minutes preparing for class and another 100 minutes reviewing class contents afterwards (including assignments) for each class.
Textbook(s)
None required.
Reference books, course materials, etc.
[1] Lawrence C. Evans, Partial Differential Equations, American Mathematical Society, 2010
[2] A. Matsumura and K. Nishihara, Global-in-time Solutions of Nonlinear Differential Equations, Nihon-hyoron-sha, 2004 (Japanese)
[3] Stanley Farlow, Partial Differential Equations Scientists Engineers, Dover Publications, 1993
Evaluation methods and criteria
Evaluation is based on attendance and assignments.
Related courses
- MTH.C406 ： Advanced topics in Analysis B1
Prerequisites
Students are required to take Advanced topics in Analysis B1 (MTH.C406).