Title: Capillary Gravity Water Wave Linearized at Monotone Shear Flows: Eigenvalues and Inviscid Damping
Xiao Liu
Mathematics Ph.D. Candidate
Georgia Institute of Technology
Email: xliu458@gatech.edu
Date: Friday, April 8, 2022
Time: 09:30 AM to 11:30 AM (EST)
Meeting Link: https://bluejeans.com/421317143/2787
Committee:
Dr. Chongchun Zeng(Advisor)—School of Mathematics, Georgia Tech
Dr. Ronghua Pan—School of Mathematics, Georgia Tech
Dr. Zhiwu Lin—School of Mathematics, Georgia Tech
Dr. Rafael De La Llave—School of Mathematics, Georgia Tech
Dr. Yao Yao—Department of Mathematics, National University of Singapore
Abstract:
This work is concerned with the two dimensional capillary gravity water waves of finite depth $x_2 \in (-h, 0)$ linearized at a uniformly monotonic shear flow $U(x_2)$. We focus on the eigenvalue distribution and linear inviscid damping. Unlike the linearized Euler equation in a fixed channel at a shear flow where eigenvalues exist only in low wave numbers $k$ of the horizontal variable $x_1$, we first prove that the linearized capillary gravity wave has two branches of eigenvalues $-ik c^\pm (k)$, where the wave speeds $c^\pm (k) = O(\sqrt{|k|})$ for $|k|\gg1$ have the same asymptotics as the those of the linear irrotational capillary gravity waves. Under the additional assumption of $U''\ne 0$, we obtain the complete continuation of these two branches, which are all the eigenvalues of the linearized capillary gravity waves in this (and some other) case(s). In particular, $-ik c^-(k)$ could bifurcate into unstable eigenvalues at $c^-(k)=U(-h)$. In general the bifurcation of unstable eigenvalues from inflection values of $U$ is also obtained. Assuming there are no singular modes, i.e. no embedded eigenvalues for any horizontal wave number $k$, linear solutions $(v(t, x), \eta(t, x_1))$ are considered in both periodic-in-$x_1$ and $x_1\in\R$ cases, where $v$ is the velocity and $\eta$ the surface profile. Each solution can be split into $(v^p, \eta^p)$ and $(v^c, \eta^c)$ whose $k$-th Fourier modes in $x_1$ correspond to the eigenvalues and the continuous spectra of the wave number $k$, respectively. The component $(v^p, \eta^p)$ is governed by a (possibly unstable) dispersion relation given by the eigenvalues, which is simply $k \to k c^\pm (k)$ in the case of $x_1 \in \R$ and is conjugate to the linear irrotational capillary gravity waves under certain conditions. The other component $(v^c, \eta^c)$ satisfies the linear inviscid damping as fast as $|v_1^c|_{L_x^2}, |\eta^c|_{L_2^x} = O(\frac 1{|t|})$ and $|v_2^c|_{L_x^2}=O(\frac 1{t^2})$ as $|t| \to \infty$. Furthermore, additional decay of $tv_1^c, t^2 v_2^c$ in $L_x^2 L_t^q$, $q\in (2, \infty]$, is obtained after leading asymptotic terms are singled out, which are in the forms of $t$-dependent translations in $x_1$ of certain functions of $x$. The proof is based on detailed analysis of the Rayleigh equation.
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