Wellposedness and singularity formation beyond the Yudovich class

TL;DR

This paper establishes a new local-in-time existence and uniqueness class for 2D Euler equations with unbounded vorticity, and demonstrates that solutions can develop finite-time singularities, providing insights into singular phenomena beyond the Yudovich class.

Contribution

It introduces a novel solution class for the 2D Euler equation with unbounded vorticity and analyzes finite-time blow-up within this framework.

Findings

Solutions can develop finite-time singularities.

Solutions can be continued as weak solutions after blow-up.

Provides a controlled setting to study singular phenomena.

Abstract

We introduce a local-in-time existence and uniqueness class for solutions to the 2d Euler equation with unbounded vorticity. Furthermore, we show that solutions belonging to this class can develop stronger singularities in finite time, meaning that they experience finite time blow up and exit the wellposedness class. Such solutions may be continued as weak solutions (potentially non-uniquely) after the singularity. While the general dynamics of 2d Euler solutions beyond the Yudovich class will certainly not be so tame, studying such solutions gives a way to study singular phenomena in a more controlled setting.