地点：腾讯会议960 364 289 密码：123456
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.