Abstract

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 focuses 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 the velocity and obtain faster decay for the remainders. This is a joint work with Xiao Liu. 

About the speaker

曾崇纯，美国佐治亚理工学院教授，美国数学会会士。主要从事微分方程与动力系统研究，在Invent. Math., Comm. Pure. Appl. Math.等顶级学术刊物发表论文多篇，曾获得美国Career奖和Sloan基金，并多次主持美国国家自然科学基金，现任J. Differential Equation和Discrete Contin. Dyn. Syst.等杂志编委。