I stumbled across the following video of the OG creator of convex integration Camillo De Lellis. He, his postdoc Maria Colombo, and one other collaborator have a new result posted to ArXiv (see here) this earlier month attempting to generalize the navier-stokes equations to non-local dissipation and prove a partial regularity result analogous to the Cafarelli-Kohn-Nirenberg.
This result builds up on some work done in 2002 by Kats-Pavlovic on a dyadic model of hyperdissipative (generalized) Navier-Stokes and also authors Tang-Yu ( who I am not familiar with).
Please see the 7 Minute mark for a neat table of most relevant weak solutions that have been studied for Navier-Stokes ( Energy Solutions, Leray-Hopf Solutions, Suitable Solutions, and classical Solutions).
You can also skip around to find nice summaries of the previous results he builds on.
The second video is a talk by Tristan Buckmaster on his recent result with Vlad Vicol showing a convex integration result for the navier-stokes equation (please see here). They call their new idea “intermittency” in which they use a Dirichlet Kernel in place of a single Beltrami wave.
- At the 29:25 mark: Tristan states a forthcoming result applying intermittent seat with convex integration to the 3D Euler equations in order to show non uniqueness of weak solutions in C_t^0H_x^\alpha, where alpha is less than 5 / 14. This is bigger than 1/3! (Previously 1/3 Holder regularity in time and space is the proven critical regularity for uniqueness)
I’m not sure why they are looking at this question but I’m wondering if it is to get practice before working on Loray-hopf weak solutions which live in a similar-looking Sobolov space for the x variable.
2. At the 51:40 mark: Tristan explains the importance of using the dirichlet kernel for the nice L1 and L2 norms, and why using a more typical C_0 norm on your corrections would not eork well for the linear term which has a Laplacian i.e. two derivatives, which would make the linearized term very bad.
3. At the 53 minute mark: Tristan explains his ambitious future work on proving non uniqueness of Leray-Hopf solutions to Navier-Stokes