Incompressible Euler equations: classical theory and convex integration

In this series of lectures I will deal with the incompressible Euler equations

where D ⊂ ℝⁿ, v : [0,T] × D → ℝⁿ is the velocity and p : [0,T] × D → ℝⁿ is the pressure of an incompressible nonviscous fluid moving inside a container with no external forces.

These equations, which are nothing but the Newton laws plus some additional structural hypotheses, are fundamental in the theory of fluid dynamics and were discovered by Euler in 1755. Notwithstanding this, many crucial questions concerning its solutions are still open and of great mathematical interest. In this course we will focus on the problem of existence and uniqueness, that surprisingly enough shows in this case two completely different behaviours at high and low regularity. Indeed, while in the class of classical solutions one can show (local in time) existence and uniqueness, in the class of weak solutions one can have, for a dense set of “wild” initial data, infinitely many solutions. In the first case, uniqueness can be easily deduced using the conservation of energy ∫|v(t)|², while for such “wild” initial data one can produce infinitely many solutions having the same energy.

The part of the theory dealing with smooth solutions is classical and will be addressed in the first part of the course. The second part follows from a technique called convex integration, which has been first applied in this context and then developed in a series of papers by De Lellis and Székelyhidi. In the last part of the course we will explore convex integration starting from problems of other nature, namely the isometric embedding problem and differential inclusions, and once we will have singled out its main features in these simpler examples we will see how it applies to the Euler equations. In between, we will also mention and spend some time on the vorticity formulation of the 2D Euler equations and the vortex sheet problem.

Date and time info
Monday 10.00 - 11.30 h

Keywords
incompressible Euler equations, vortex formulation, nonuniqueness, convex integration technique

Prerequisites
ODE's, Sobolev spaces, basic Fourier analysis

Audience
MSc students, PhD students, Postdocs

Language
English