The Existence of Three-Dimensional Multi-Hump Gravity-Capillary Surface Waves on Water of Finite De

主讲人：邓圣福  华侨大学教授

时间：2021年10月11日9：00

地点：腾讯会议960 364 289  密码：123456

举办单位：数理学院

主讲人介绍：邓圣福,  华侨大学教授，从事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面上基金3项、教育部留学回国人员科研启动基金、中国博士后科学基金、福建省自然科学基金、广东省自然科学基金。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

内容介绍：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.