Singularity Formation in a Surface Wave Model

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 $\hat{I}{›}^{\hat{I}}\pm Hu$ for $0<\hat{I}\pm <1$, finite time singularity formation is also shown.

14 pages