Ill-Posedness and Loss of Regularity in Models of Incompressible Fluids

Abstract

The main goal of this thesis is to explore questions related to well-posedness/ill-posedness, propagation, and loss of regularity in models of incompressible fluids. Specifically, we focus on these issues in the context of the Euler equations and related models. The Euler equations are nonlinear, nonlocal partial differential equations that describe the motion of ideal incompressible fluids. The complexity of the Euler system leads to rich and interesting dynamics. In the first part of the thesis, Chapter 2, we study the Euler equations with Riesz forcing, which arises as a model when physical quantities such as temperature or magnetic fields are coupled with Euler dynamics. In this Chapter 2, we prove the strong ill-posedness of bounded solutions to the Euler with Riesz forcing system. In the second part of the thesis, Chapter 3, we investigate the Euler equations with rough but continuous initial data. We address the question of whether the modulus of continuity can always be propagated for the Euler equations. Our answer to this question is negative, as we construct a family of moduli of continuity for the two dimensional Euler equations that are not propagated.