A terrific buddy of mine recently asked me if I’ve ever gone back to study the original Nash and Embedding theorems on which Convex Integration for fluids is based on. The short answer is sadly no 😦 but I did have things to say about the original Nash and embedding theorem anyways ( LOL). I wasted no time in inundating my friend with my thoughts on this. Fortunately he’s a very patient guy.
But in writing that email I found a survey article that I think this great job explaining the connection so I wanted to share it here along with my own thoughts on this.
Happy New Year’s to you too! I hope your break is going well.
I have to admit that I have not looked at the original isometric embedding paper. I chose not to because those papers use a little differential geometry notation that I’m not comfortable with and I wanted to get into research quickly. However Phil did go back and work through that paper as a graduate student.
But since you asked, I did a quick search and I found this article by Klainerman on the Nash and bedding theorem and its connections to PDEs (see attached). I only skimmed the article but it includes references to the original papers and he attempts to explain things using the language of analysis.
Now I could be completely wrong about this, but I think the Nash embedding theorem deals with the question of can you find a smooth embedding of manifold into a small space. Think of trying to fit a sheet of paper into a unit ball in a way that doesn’t crumple the paper in a non-smooth fashion. The original idea was created by Nash but was later refined into what we call convex integration and an H-principle by Gromov. The idea they used is to take a guess of it solution / embedding, which doesn’t work, but instead has an error and then add a correction to your solution that will cancel some of the error. In order to decide how large (in a certain norm) the correction should be you will need to integrate the error. And if you iterate this process you notice that you are looking at approx. solutions which are points in a convex set of some function space. Hence the term convex integration!
Unfortunately, because I didn’t read the original papers my knowledge is limited to the context of fluid equations but if you have some differential geometry background and you are interested in reading the original Nash papers then you might gain some knowledge that I don’t have and you may be able to apply convex integration to problems that aren’t fluid PDEs. For example, last year Marta Lewicka (of Pitt) got a convex integration and H-principle result for the Monge-Ampere equations.
Sorry for the long message! But thanks for writing I always appreciate the interest and discussion – Happy New Year’s!