A Priori Estimates for Two-Dimensional Water Waves with Angled Crests

Abstract

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 $-\frac{\partial P}{\partial \mathbf{n}}\ge 0$ 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 $-\frac{\partial P}{\partial \mathbf{n}}$.

56 pages, 2 figures. Accepted by Cambridge Journal of Math

References in corpus (2)

• Two dimensional water waves in holomorphic coordinates
• On a class of self-similar 2D surface water waves

Cited by in corpus (6)

• Water waves problem with surface tension in a corner domain I: A priori estimates with constrained contact angle
• On the transition of the Rayleigh-Taylor instability in 2d water waves
• Justification of Peregrine soliton from full water waves
• Nonlinear modulational instabililty of the Stokes waves in 2d full water waves
• Water waves problem with surface tension in a corner domain II: the local well-posednes
• Elliptic estimates for Dirichlet-Neumann operator on a corner domain