Seminar on the Navier-Stokes Equations

The seminar will take place on a weekly basis in the summer semester for masters and PhD students. One will need to have taken the Functional analysis course and have a strong interest in PDE's and analysis. This course can count for masters students to your seminar certificate. Once you have seen your section of the paper you will have plenty of time to prepare and decide if it is appropriate for you.   

Registration

To register please everyone email Dr. Jack Skipper and Prof. Emil Wiedemann and further come to the organisational meeting on February 14th. 

If you cannot make or missed this meeting then still email us and we can meet you separately and add you to the seminar. We are just starting the topics early so that there is plenty of time for people to prepare. 

Outline

In this seminar we will discuss the famous paper of Caffarelli-Kohn-Nirenberg, titled "Partial Regularity of Suitable Weak Solutions of the Navier-Stokes Equations." The results of this paper are strongly related to the prestigious Clay Mathematics Institute Millennium Problem for the Navier-Stokes Equations. For extra details one can consider an interesting description of the problem and current progress in the area given by Prof. Charles Fefferman.  

For help it would be useful to look at the modern book of "The Three-Dimensional Navier-Stokes Equations" (Link to E-Book at bottom of page) by James C. Robinson, Jose L. Rodrigo and Witold Sadowski (especially Chapters 15 and 16).  Some more copies I believe have been ordered. Further, the paper by Fanghua Lin gives a more modern approach and new proof of the CKN theorem and would be useful to look at. 

Other related topics can be discussed depending on numbers and interests of the attendees of the seminar. 

Talks

The outline of talks is here and will take place in E.60, Helmholtzstr. 18 on Tuesdays at 16:15 to 18:00. The dates are approximate and may change if important people can't make it or people are ill. There are two Tuesdays left in the term for talks. 

  1. Introduction of the CKN paper (7.5 sides) Jack Skipper (23.04.2019)
  2. Def. "suitable weak solution" (in section 2) then appendix proof.  David Berger+Farid Mohamed (30.04.2019)
  3. Finishing off the appendix (11.5 sides combined) David Berger+Farid Mohamed (07.05.2019)
  4. Section 2 "Background and Definitions" (6 sides) Fabian Rupp (14.05.2019)
  5. Section 3 "Some Dimensionless Estimates" (4 Sides) Ibrokhimbek Akramov (21.05.2019)  
  6. Section 4 "The Inductive Argument"  (7 sides) Marcus Mueller (04.06.2019)
  7. Section 5 "The Blow-up Estimate" Marius Mueller+Lukas Niebel  (18.06.2019)
  8. Section 5 continued (11 sides combined) Marius Mueller+Lukas Niebel  (25.06.2019)
  9. Sections 6 and 7 "Estimating the Singular Set" and "Estimates for u and p in Weighted Norms" (7 pages) Dennis Gallenmueller (02.07.2019)
  10. Section 8 "Theorems C and D" (8 sides)  Emil Wiedemann (or volunteer)  (Jack Skipper) (09.07.2019)
  11. Similar results on the Navier-Stokes equations with fractional Laplacian. Frederic Weber+Edward Sockin (16.07.2019)
  12. Similar results on the Navier-Stokes equations with fractional Laplacian. Frederic Weber+Edward Sockin (23.07.2019)

The level of the talks should be accessible to a fellow Masters student and so one should research the methods used in your section and expand on the details where appropriate. If you have any questions, or want to discuss your section for help or extra understanding please feel free to email and/or talk to Jack Skipper, Emil Wiedermann or another attendee of the seminar. Make sure you do this in plenty of time before your talk so that it is useful. 

To help prepare your presentation and to further your written ability I would like each participant to write up, in Latex, their section, adding in all the explanation and detail they require for their audience. This will benefit everyone and we should finish with a nice document containing our work. To aide in this I will give a template here and I will keep a main pdf updated of the work as and when people send me their notes. Furthermore, everyone should regularly attend the seminar. 

Supervision

The seminar will be run by Dr. Jack Skipper and Prof. Emil Wiedemann.

Please E-mail Dr. Jack Skipper or Prof. Emil Wiedemann.

Organisational meeting at 11:30 on 14th February in room E.60, Helmholtzstr. 18 (After the final lecture of Advanced Topics in Partial Differential Equations).

Language and Presentation

The organisation will be happening in English yet your talk can be done in either German or English. Just notify Jack Skipper or Emil Wiedemann in good time before your talk what language you would like to present in. 

You can present in any fashion yet we recommend chalk and board, if you would like to present though other medium then please email us in good time beforehand so that we can prepare.