S4B2 - Graduate Seminar in Analysis and PDEs - Topics in Incompressible Fluid Dynamics

Winter term 2024/2025

Dr. Dimitri Cobb

Organisational details

Overview

This seminar series are centered on the study of the Euler and Navier-Stokes equations, and the properties of their solutions. We will first start by covering the classical question of existence and uniqueness of solutions for a given initial datum. This will involve getting familiar with important tools from harmonic analysis. However, the analysis of these PDEs goes much deeper than the (already fascinating) existence/uniqueness problems: we will also examine some important properties of the solutions. For example, we will study the asymptotic behavior of solutions at large space scales, mixing properties and the growth of higher order norms, point vortex dynamics, or the distribution of kinetic energy in the solutions. Other related models such as magnetohydrodynamics will also recieve some attention.

See here for more information, including a list of literature recommendations.

Prerequisites: basic knowledge of Fourier series and transform, functional analysis, Sobolev spaces and ODEs. No prior experience in physics is required.

Schedule of talks

This is an approximate schedule. The order of the talks may vary depending on the topics that are chosen.

October 10- Preliminary meeting

October 17- Weak Leray Solutions for the Navier-Stokes Equations Taken

October 24- Strong Solutions for the Navier-Stokes Equations in Sobolev Spaces Taken

October 31- Strong Solutions for the Navier-Stokes Equations in Lebesgue Spaces Taken

November 7- Hexagonal Structure in the 2D Navier-Stokes Flow Taken

November 14- No Seminar (Rescheduled to...)

November 21- Yudovich Solutions for the 2D Euler Equations Taken

November 28- Transport and Instability for Perfect Fluids Taken

December 5- No seminar (rescheduled to...)

December 12- Point Vortex Solutions for the 2D Euler Equations Taken

December 19- Uniqueness for the Vortex-Wave System when the Vorticity is Constant Near the Point Vortex TBA

December 26- Christmas break

January 2- Christmas break

January 9- Maximal Regularity and the 2D inviscid MHD Equations TBA

January 16- Infinite Energy Solutions of the Two-Dimensional Navier-Stokes Equations TBA

January 23- Stability of Vortices in Ideal Fluids TBA