Singularity Formation in a Surface Wave Model

TL;DR

This paper investigates finite-time singularity formation in a nonlocal Burgers equation with Hilbert transform and fractional generalizations, demonstrating blow-up for broad initial conditions relevant to vortex patch dynamics.

Contribution

It proves finite-time blow-up for a class of initial data in a nonlocal Burgers equation and extends results to fractional nonlocal terms, advancing understanding of singularity formation.

Findings

Finite-time blow-up for the Burgers-Hilbert system.

Extension of blow-up results to fractional nonlocal operators.

Relevance to vortex patch boundary dynamics.

Abstract

In this paper we study the Burgers equation with a nonlocal term of the form $Hu$ where $H$ is the Hilbert transform. This system has been considered as a quadratic approximation for the dynamics of a free boundary of a vortex patch. We prove blow up in finite time for a large class of initial data with finite energy. Considering a more general nonlocal term, of the form $\Lambda^\alpha Hu$ for $0<\alpha< 1$, finite time singularity formation is also shown.