Under construction! (Updated Dec 25 2017)
I’m making this page as a set of resources that I’ve been using in an attempt to learn convex integration. I’ll try to list out what little I have in order.
If you’ve never heard of convex integration before here are some cool Youtube videos containing the buzzwords and a general audience article from Quanta that even your parents will be able to understand:
- Convex Integration and Fluid Mechanics
- The H-Principle and Partial Differential Equations
- Mathematicians Find Wrinkle in Famed Fluid Equations
If you prefer to see a 1 hour talk on this here is Vlad Vicol’s presentation on Youtube of Convex integration for proving Onsager’s conjecture for the Euler Equations:
If you would like to know the key people using Convex Integration for fluid equations then here is my (crude and possibly incorrect) understanding as of 2017:
- Camillo De Lellis, Lazlo Schekelyhidi Jr. – First used Convex Integration, a tool from differential geometry, to construct a solution to a Euler’s equations
- Vlad Vicol – has collaborated with a lot of people and is very good at presenting this material
- Tristan Buckmaster – Has done some amazing work in his thesis and now applies convex integration to Navier – Stokes
- Philip Isett – Created a variety of tools to help the Convex Integration method and managed to get the first proof of Onsager’s Conjecture for 3D Euler. 2D is still open.
There are of course, more people than this interested in convex integration for fluids, but I think most of the papers related to Onsager’s conjecture are written by or with these people.
If you want to get started with reading these papers here are some resources that have helped me:
To start, you will need some basic understanding of Littlewood – Paley Theory. I certainly am no expert on this, so I use the following two references:
Also I used Hans Treibel’s Theory of Function Spaces Volume 2 (It looks like this) as a reference for understanding Besov spaces and interpolation of Holder Spaces. It’s honestly probably better as a reference than a textbook to learn from. (Upon second thought, this is probably more technical than one needs to start out).
What helped motivate me was to read the following two papers, which discuss the positive part of Onsager’s Conjecture:
- Constantin, Peter; E, Weinan; Titi, Edriss S.
- This paper has some typos in the calculations though
- A. CHESKIDOV, P. CONSTANTIN, S. FRIEDLANDER, AND R. SHVYDKOY
- I think it’s only the first 9 or so pages that deal with the energy conservation
Finally, I think for intuition on the basic outline and strategy of Convex Integration for Euler’s you could look at Philip Isett’s PhD Thesis:
Matt Novack of UT-Austin (If you’re a university looking to hire a talented fluids PDE guy, please check him out!) really liked the first three sections of this paper for intuition
I should comment that Phil uses a different notation than Camillo DeLellis, Vlad Vicol, Tristan Buckmaster, and Lazlo. You might find that the papers from Camillo DeLellis and the latter to be easier to read when starting out.
For further reading, you may like to see the original paper on Convex Integration for fluids, by De Lellis and Shekelyhidi Jr ( Dissipative continuous Euler flows ) however it is very geometric. Here are some notes that I jotted down on the basic idea of this paper (Insert notes later)
You could also look at some results by Isett, Vicol, Buckmaster etc. they apply convex integration to other fluid PDEs such as SQG or other Active Scalar problems (although I think currently, both of them are interested applying convex integration to the Navier-Stokes equations):
- Nonuniqueness of weak solutions to the SQG equation