Reading Group “Analysis of PDEs with applications to hydrodynamics”

Prof Sergey Zelik

Department of Mathematics of University of Surrey, Zhejiang Normal University (China)

International Centre of Infinite Dimensional Dynamical Systems (CIDDS)

Part I: Nonlinear localization and regularity of Brinkman-Forchheimer equations. Autumn Semester 2022:

Lecture 1: Introduction

Lecture 2: Steady Stokes problem- H^{-1} to H^1 regularity

Lecture 3: Steady Stokes problem: L^2 to H^2 regularity

Lecture 4: Steady Stokes problem: non-linear localization

Lecture 5: Steady-Stokes problem: nonlinear localization ( continued) Slides

Lecture 6: Stokes operator revisited Slides

Lecture 7: Non-steady Navier-Stokes-BF problem: localization Slides

Lecture 8. Non-steady NS-BF equation (continued) Slides

Part II. Vishik vortices and non-uniqueness for Navier-Stokes equations. Spring Semester 2023:

Lecture 1: Introduction Slides

Lecture 2. Linear stability analysis for 2D Euler equations Slides

Lecture 3: Linear stability, 2D case (continued) Slides

Lecture 4 (part 1) Introduction to the operator theory Slides

Lecture 4(part 2) Applications to hydrodynamics Slides

Lecture 5 Applications to hydrodynamics (continued) – slides are contained in the previous lecture

Lecture 6: Instability in the Rayleigh equation (completion) Slides

Lecture 7: Instability in 3D case via big vortex rings Slides

Lecture 8. Essentially unstable manifolds Slides

Lecture 9: Completion of the non-uniqueness proof Slides

Part III. PDEs in unbounded domains (including Navier-Stokes type equations): Autumn semester 2023:

https://us06web.zoom.us/j/81988585186?pwd=T1lJp14pUiHCzYCU1BOQLffJtiqsw2.1

Meeting ID: 819 8858 5186

Passcode: 829887

Lecture 1. Introduction and examples of weighted estimates

Lecture 2. Weighted and uniformly local Sobolev spaces. Slides

Lecture 3. Attractors in unbounded domains.

Lecture 4. Structure assumptions and compactness in uniform topology

Lecture 5. Damped Burger’s equation in uniformly local spaces

Lecture 6. Damped Burger’s equation in uniformly local spaces: 2D case

Lecture 7. Navier-Stokes equation in uniformly local spaces

Lecture 8. Multi-vortices and lower bounds for the dimension of attractors: Introduction

Lecture 9. Multi-vortices and instability: Lyapunov-Schmidt reduction