In this lecture, we will be concerned with the mathematical theory for the Euler and Navier-Stokes equations. It has two main topics.
In the first part of the lecture, we will study the existence and uniqueness of smooth solutions. In the sequel, we introduce weak solution concepts and focus on problems regarding existence, uniqueness and blow-ups.
In the second part, we concentrate on the Euler equations in two spatial dimensions and investigate the motion of vorticity fields. We show the stability of so-called Lamb-Oseen dipoles and infer a system of ordinary differential equations from the microscopic equations that describe the interaction of point vortices.
Conceptually, we will need a number of mathematical tools, that will be developed during the lecture. These include
- Compactness and weak convergence in Lp spaces
- Rearrangement inequalities
- Lp regularity estimates
This lecture is a continuation of a standard introductory course in partial differential equations.