Newton-Nash iterations in fluid equations

We will overview the Nash iteration schemes that are used to construct non-unique solutions to the incompressible Euler equations and explain the problem of unwanted high-high frequency interactions that present an obstacle to obtaining solutions with regularity just below the Onsager threshold. We then will explain the Newton iteration as a method to overcome this obstacle. We'll mention several results that use this technique to resolve Onsager-type conjectures in PDEs arising in incompressible fluids and conclude with potential future directions.

Based on joint works with Razvan-Octavian Radu and Mimi Dai.