<i>A priori</i> estimates for two-dimensional water waves with angled crests

We consider the two-dimensional water wave problem in the case where the free interface of the fluid meets a vertical wall at a possibly non-right angle and where the free interface can be non-$C^1$ with angled crests. We assume that the air has density zero, the fluid is inviscid, incompressible, irrotational, and subject to the gravitational force, and the surface tension is zero. In this regime, only a degenerate Taylor stability criterion $-\frac{\partial P}{\partial \mathbf{n}} \geqslant 0$ holds, with degeneracies at the singularities on the interface and at the point where it meets the wall if the angle is non-right. We construct a low-regularity energy functional and prove an a priori estimate. Our estimate differs from existing work in that it doesn’t require a positive lower bound for $-\frac{\partial P}{\partial \mathbf{n}}$.

Keywords

water wave equations, Riemann mapping coordinates, a priori estimates

2010 Mathematics Subject Classification

35B45, 35Q35

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R.H. Kinsey is supported in part by a University of Michigan Rackham Regents Fellowship and by NSF grants DMS-0800194 and DMS-1101434.

S. Wu is supported in part by NSF grant DMS-1101434.

Received 6 April 2017

Published 18 May 2018