  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.
