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:

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