2D Euler system for Hölder continuous vorticities

Abstract

This thesis investigates the two-dimensional incompressible Euler equations under smooth initial data, focusing on three formulations: 1. the classical Euler system, 2. the vorticity-stream forumlation, 3. the integrodifferential (particle-trajectory) formulation.

In Chapter 1, we first show that these three viewpoints are mathematically equivalent. We also develop core background material on vector fields, flows, and singular integral operators. These tools, particularly the singular integral operator theory, lay the groundwork for the vorticity-stream and integrodifferential formulations used in subsequent chapters.

Chapter 2 establishes the local-in-time existence and uniqueness of solutions when the initial vorticity lies in a Hölder space $C^\gamma$ (with $\gamma \in(0,1)$ ). By formulating the problem in terms of the integrodifferential formulation, we show that the relevant operator is locally Lipschitz in an appropriate Banach space. Applying a Picard-Lindelöf-type argument then yields the desired local well-posedness result. Key technical ingredients include singular integral estimates and careful composition bounds in Hölder spaces.

Finally, Chapter 3 addresses the global nature of these solutions. In two dimensions, the vorticity remains constant along particle trajectories and thus stays bounded for all time, preventing finite-time blow-up. Using a method inspired by the analysis of Beale, Kato, and Majda, we show that bounding the vorticity's supremum norm also bounds the velocity gradient, which in turn guarantees global existence. Thus, for sufficiently smooth initial vorticity in 2D, the solutions to the Euler equations extend indefinitely in time and remain regular.