## CAPILLARY GRAVITY WATER WAVE LINEARIZED AT MONOTONE SHEAR FLOWS: EIGENVALUE AND INVISCID DAMPING

##### 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.