报告题目：The Existence of Three-Dimensional Multi-Hump Gravity-Capillary Surface Waves on Water of Finite Depth
报 告 人： 邓圣福， 华侨大学
腾讯会议ID：278 335 114 密码：123456
报告人简介：邓圣福，华侨大学教授，“闽江学者奖励计划”特聘教授，主要研究微分方程与动力系统理论及其在水波问题上的应用。先后主持国家自然科学基金面上项目3项、福建省和广东省自然科学基金多项，曾入选广东省“扬帆计划”引进紧缺拔尖人才、广东省高等学校“千百十人才培养工程”省级培养对象等。在Arch. Rational Mech. Anal.、SIAM J. Math. Anal.、Nonlinearity、J. Differential Equations、Physica D等国际重要学术期刊上发表论文40多篇。
报告摘要：This talk considers three-dimensional traveling surface waves on water of finite depth under the forces of gravity and surface tension using the exact governing equations, also called Euler equations. It was known that when two non-dimensional constants $b$ and $\lambda$, which are related to the surface-tension coefficient and the traveling wave speed, respectively, near a critical curve in the $(b, \lambda)$-plane, the Euler equations have a three-dimensional (3D) solution that has one hump at the center, approaches nonzero oscillations at infinity in the propagation direction, and is periodic in the transverse direction. We prove that in this parameter region, the Euler equations also have a 3D two-hump solution with similar properties. These two humps in the propagation direction are far apart and connected by
small oscillations in the middle. The result obtained here is the first rigorous proof on the existence of 3D multi-hump water waves. The main idea of the proof is to find appropriate free constants and derive the necessary estimates of the solutions for the Euler equations in terms of those free constants so that two 3D one-hump solutions that are far apart can be successfully matched in the middle to form a 3D two-hump solution if some values of those constants are specified from matching conditions. The idea may also be applied to study the existence of 3D $2^n$-hump water-waves.