On the global well-posedness of the 2D Euler equations for a large class of Yudovich type data

TL;DR

This paper establishes the global well-posedness of the 2D Euler equations for a broad class of Yudovich-type initial data, including unbounded vorticities, ensuring existence, uniqueness, and regularity persistence.

Contribution

It introduces a new Banach space accommodating unbounded functions, proving global existence, uniqueness, and regularity persistence for the 2D Euler equations with such data.

Findings

Proved global well-posedness for a new class of unbounded vorticity data.

Established existence, uniqueness, and regularity persistence in the introduced Banach space.

Extended classical results to more general initial conditions.

Abstract

The study of the 2D Euler equation with non Lipschitzian velocity was initiated by Yudovich in [19] where a result of global well-posedness for essentially bounded vorticity is proved. A lot of works have been since dedicated to the extension of this result to more general spaces. To the best of our knowledge all these contributions lack the proof of at least one of the following three fundamental properties: global existence, uniqueness and regularity persistence. In this paper we introduce a Banach space containing unbounded functions for which all these properties are shown to be satisfied.