Capillary gravity water waves linearized at monotone shear flows: eigenvalues and inviscid damping

This paper is concerned with the eigenvalues and linear inviscid damping of the 2D capillary gravity water waves of finite depth $x_2\in(-h,0)$ linearized at a monotone shear flow $U(x_2)$. Unlike the linearized Euler equation in a fixed channel where eigenvalues exist only in low horizontal wave number $k$, we first prove the linearized capillary gravity wave has two branches of eigenvalues $-ikc^\pm(k)$, 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 $U"\ne0$, we obtain the complete continuation of these two branches, which are all the eigenvalues in this (and some other) case(s). Particularly $-ikc^-(k)$ could bifurcate into unstable eigenvalues at $c^-(k)=U(-h)$. The bifurcation of unstable eigenvalues from inflection values of $U$ is also proved. Assuming no singular modes, i.e. no embedded eigenvalues for any wave number $k$, linear solutions $(v(t,x),\eta(t,x_1))$ are studieded in both periodic-in-$x_1$ and $x_1\in R$ cases, where $v$ is the velocity and $\eta$ the surface profile. Solutions can be split into $(v^p,\eta^p)$ and $(v^c,\eta^c)$ whose $k$-th Fourier mode in $x_1$ correspond to the eigenvalues and the continuous spectra of wave number $k$, respectively. The component $(v^p,\eta^p)$ is governed by a (possibly unstable) dispersion relation given by the eigenvalues, which are simply $k\to-ikc^\pm(k)$ in the case of $x_1\in R$. The other component $(v^c,\eta^c)$ satisfies the inviscid damping as fast as $|v_1^c|_{L_x^2},|\eta^c|_{L_x^2}=O(|t|^{-1})$ and $|v_2^c|_{L_x^2}=O(t^{-2})$ as $|t|\gg1$. Additional decay of $tv_1^c,t^2v_2^c$ in $L_x^2L_t^q$, $q\in(2,\infty]$, is obtained after leading asymptotic terms are removed, which are in the forms of $t$-dependent translations in $x_1$ of certain functions of $x$.