Intermittent Convex Integration in Mathematical Fluid Mechanics: Applications to the Euler Equations and the Chain Rule Problem

This thesis is concerned with intermittent convex integration and its applications to some equations in fluid mechanics. This technique originated from differential geometry with the proof of the famous Nash-Kuiper theorem on isometric embeddings. It was extended by Gromov and many other authors and is by now considered a powerful tool in the construction of (anomalous) solutions to certain PDEs. One of its most striking applications in the area of fluid mechanics is the proof of the flexible part of Onsager's conjecture. In this thesis, we consider two concrete examples of intermittent convex integration. The first one concerns the chain rule problem: given scalar functions β and ρ and a divergence-free vector field u, we ask the question whether one can express div(β(ρ)u) only in terms of β'(ρ) and div(ρu). This is clear if β is at least C^1 and u and ρ are Lipschitz continuous. We prove that in the class of Sobolev vector fields, the answer to that question is negative in general. We even show that for any distribution T which is the divergence of some L^1 function and a given β with suitable growth, there exists a divergence-free, Sobolev regular u and a ρ with div(β(ρ)u)=T and div(ρu) = 0. The second application deals with the two-dimensional Euler equations. We prove the existence of energy dissipating weak solutions with vorticity in a real Hardy space H^p with p<1. A novel difficulty here compared to previous works in convex integration is that working in real Hardy spaces requires control of higher order moments of the solutions. An additional difficulty is that we work on the full space rather than on a periodic domain.