Course Information
| Instructor | Dr. Changhui Tan |
| Lectures | T Th 11:40am - 12:55pm, at LeConte 315 |
| Office | LeConte 427 |
| Office Hours | T 9:30am - 11:30am, or by appointment |
| Textbooks | There is no required textbook for this topic course. The material are based on the lecture notes. |
| Syllabus | Click Here |
Course Description
- The course will focus on the mathematical theory for partial differential equations that arise from fluid dynamics, including the Euler equations, the Navier-Stokes equations, and beyond. Topics include the analysis of weak and strong solutions, energy estimates, local and global well-posedness, and asymptotic behaviors.
- Upon successfully completing this course, students will be familiar with modern analytical tools that can be used to treat PDEs in fluid dynamics and beyond. Selected analytical methods include
- A priori energy estimates for local and global wellposedness.
- Commutator estimates and interpolation inequalities.
- The method of modulus of continuity.
- Method of characteristics and critical thresholds.
- These advanced techniques can be used to analyze a large class of time-dependent PDEs.
Tentative Lecture Schedule
- Week 1: Introduction to incompressible and compressible Euler equations.
- Week 2: Strong and weak solutions, Sobolev spaces.
- Week 3-4: 2D incompressible Euler equations - Global regularity.
- Week 5-7: Energy estimates and propagation of regularity.
- Week 8: 1D damped Burgers equation - Critical threshold.
- Week 9-12: 1D fractal Burgers equation - Modulus of continuity.
- Week 13-14: 1D compressible Euler equations - pressureless versus with pressure.
Grades
- The grade will be based on class participation, occasional homework assignments, and a possible final project.
Contact Information
- Contact me at This email address is being protected from spambots. You need JavaScript enabled to view it. if you have any questions.