Differential Equations Seminar

We consider the 2D capillary water wave equation in the presence of gravity. We assume that the fluid is inviscid, incompressible, irrotational and that the air density is zero. We construct an energy functional and prove an a priori estimate where the estimate is independent of the value of surface tension. We do not assume the Taylor sign condition in contrast to previous works. In addition our energy allows us to approximate singular solutions. When surface tension is zero, the energy reduces to the energy obtained by Kinsey and Wu for angled crest water waves. We show that in an appropriate scaling regime, the zero surface tension limit of our solutions is the one for the gravity water wave equation which includes waves with angled crests. Speaker(s): Siddhant Agrawal (Univ. of Michigan)