MATH-647 / 2 credits
Teacher: Colombo Maria
Remark: Fall semester
Only this year
This topics course focuses on recent and classical fundamental results on the Euler and Navier-Stokes equations, such as global existence of weak solutions, (non)uniqueness results, blow-ups, partial regularity, anomalous dissipation.
The course regards the mathematical theory of the three-dimensional Euler and NavierâStokes equations. It provides self-contained proofs of some of the most significant results in the area, many of which can only be found in research papers. The first part of the course is dedicated to Elgindi's result on blow-up of solutions of 3D Euler. The second part of the course, include a selection of topics between the following list: the existence of global-in-time LerayâHopf weak solutions for Navier-Stokes and the local existence of strong solutions; the conditional local regularity results of Prodi-Serrin; the partial regularity results of Caffarelli, Kohn, and Nirenberg; nonuniqueness results obtained via convex integration or instability; anomalous dissipation.
Navier-Stokes equations, Euler equations
Solid foundations in all analysis courses, including PDEs I and II and functional analysis I and II, is expected.
By the end of the course, the student must be able to:
- Work out / Determine with certain solutions of the Navier-Stokes equations, assessing their uniqueness, expected regularity and blow up properties.
Books "The Navier-Stokes equations in the 21st century" by Lemarrié-Rieussiet, "The incomprehessible Euler and Navier-Stokes equations" by Bedrossian and Vicol.
In the programs
- Exam form: Oral presentation (session free)
- Subject examined: Topics on the Euler and Navier-Stokes equations
- Lecture: 22 Hour(s)
- Exercises: 12 Hour(s)