SHORT Course: Introduction to the mathematical analysis of incompressible Euler and Navier-Stokes equations

Monday 4 Nov 2024, 00:00 → Friday 7 Feb 2025, 23:59 Europe/Rome

Lecturers
Stefano Modena stefano.modena@gssi.it

Timetable and workload
Two lectures and one exercise session every week, as follows:

• Lectures: Mon, Wed 10.30-12.30.
• Exercises: Thu 10.30-12.30. 

Course description and outcomes
The course will provide an introduction to the rigorous mathematical analysis of incompressible Euler and Navier Stokes equations. In doing so, we will cover a broad range of tools commonly used in the analysis of partial differential equations.

Course content (most likely we will cover only part of the program):

1. Incompressible Euler equations.
   • Derivation and basic theory.
   • Local well posedness.
   • Blow-up criteria.
2. Incompressible Navier-Stokes equations. 
   • Derivation and basic theory.
   • Local well posedness of strong solutions.
   • Leary-Hopf weak solutions.
   • Mild solutions.
3. Advanced topics

Prerequisites

• Calculus
• ODE Theory
• Basic functional analysis (weak convergence, Ascoli-Arzelà Theorem, ...)
• Fourier Transform
• Lp and Sobolev Spaces
• Basic PDE Theory (Laplace, Poisson and heat equation)
• Theory of distributions

If necessary, afternoon classes will be organised for those who need to fill some gap in the prerequisites. 

References
We will mostly follow the book "The Mathematical Analysis of the Incompressible Euler and Navier-Stokes equations" by J. Bedrossian and V. Vicol (AMS Graduate Studies in Mathematics, 225).

Exam
Written and oral exam on the topics of the course.

Exercise01.pdf

Exercise02.pdf

Exercise03.pdf

Exercise04.pdf

Exercise05.pdf