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Capillary Gravity Water Waves Linearized at Monotone Shear Flows: Eigenvalues and Inviscid Damping

发布时间:2022-04-19 浏览量:57

时   间:  2022-04-19 10:00 — 11:00

地   点:  腾讯会议 APP()
报告人:  Chongchun Zeng
单   位:  Georgia Institute of Technology
邀请人:  谢春景
备   注:  Tencent meeting: ID: 232-703-653; Password: 220419
报告摘要:  

We consider the 2-dim capillary gravity water wave problem -- the free boundary problem of the Euler equation with gravity and surface tension -- of finite depth $x_2 \in (-h,0)$ linearized at a uniformly monotonic shear flow $U(x_2)$. Our main focus are eigenvalue distribution and inviscid damping. We first prove that in contrast to finite channel flow and gravity waves, the linearized capillary gravity wave has two unbounded branches of eigenvalues for high wave numbers. They may bifurcate into unstable eigenvalues through a rather degenerate bifurcation. Under certain conditions, we provide a complete picture of the eigenvalue distribution. Assuming there are no singular modes (i.e. embedded eigenvalues), we obtain the linear inviscid damping. We also identify the leading asymptotic terms of velocity and obtain stronger decay for the remainders. This is a joint work with Xiao Liu.