Eugen Varvaruca (Reading)
Existence of steady free-surface water waves with corners of 120
degrees at their crests in the presence of vorticity
Abstract: We present some recent results on singular solutions of the problem of
travelling gravity surface water waves on flows with vorticity. It has
been known since the work of Constantin and Strauss in 2004 that there
exist spatially periodic waves of large amplitude for any vorticity
distribution. We show that, for any nonpositive vorticity distribution
and for any value of the period of the waves, a sequence of
large-amplitude regular waves converges in a weak sense to an extreme
wave with stagnation points at its crests. A similar results holds for
nonnegative vorticity distributions, under some smallness assumptions,
and provided that the period of the waves is sufficiently large. The
proof is based on new a priori estimates, obtained by means of the
maximum principle, for the fluid velocity and the wave height along the
family of regular waves whose existence was proved by Constantin and
Strauss. We also show, by new geometric methods, that this extreme wave
has corners of 120 degrees at its crests, as conjectured by Stokes in
1880. The results presented were obtained in collaboration with Ovidiu
Savin and Georg Weiss.