Convex Integration

Under construction!

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:

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:

Key Players:

  • 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 – I’ve never met this person
  • 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.

What helped me was to read the following two papers, which discuss the positive part of Onsager’s Conjecture:

Finally, I think for intuition on the basic outline and strategy of Convex Integration for Eulers you could look at Philip Isett’s PhD Thesis:

For further reading, you may like to see the original paper on Convex Integration for fluids, by De Lellis and Shekelyhidi Jr. (insert link to paper later) 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: