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

摘要

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-trivial angle; our problem also covers interfaces with angled crests. We assume that the fluid is inviscid, incompressible, and irrotational, with no surface tension and with air density zero. We construct a low-regularity energy and prove a closed energy estimate for this problem. Our work differs from earlier work in that, in our case, only a degenerate Taylor stability criterion holds, with the inward-facing normal derivative of the pressure non-negative, instead of the strong Taylor stability criterion, which requires that the inward-facing normal derivative of the pressure be bounded below by a strictly positive constant.