Blow up for the 2D Euler Equation on Some Bounded Domains

TL;DR

This paper constructs a smooth solution to the 2D Euler equation on a bounded domain with cusps that remains locally well-posed but exhibits finite-time blow-up in vorticity continuity, highlighting boundary effects.

Contribution

It demonstrates finite-time vorticity blow-up for smooth solutions on domains with interior cusps, revealing boundary geometry's impact on solution regularity.

Findings

Existence and uniqueness of local smooth solutions

Finite-time vorticity discontinuity development

Boundary cusps influence on solution behavior

Abstract

We find a smooth solution of the 2D Euler equation on a bounded domain which exists and is unique in a natural class locally in time, but blows up in finite time in the sense of its vorticity losing continuity. The domain's boundary is smooth except at two points, which are interior cusps.