Singularity formations for 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 (see Biello and Hunter 2010 Commun. Pure Appl. Math. LXIII 0303–36; Marsden and Weinstein 1983 Physica D 7 305–23). We prove blowup in finite time for a large class of initial data with finite energy. Considering a more general nonlocal term, of the form ΛαHu for 0 < α < 1, finite time singularity formation is also shown.