Math Courses

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

by Stefano Modena (GSSI)

→ Europe/Rome
Description

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.

 

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

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

 

Tentative plan (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. 

 

Literature: 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