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

TL;DR

This paper analyzes the eigenvalues and inviscid damping of 2D capillary gravity water waves linearized at monotone shear flows, revealing eigenvalue branches, bifurcations, and decay rates of solutions under certain conditions.

Contribution

It provides a detailed spectral analysis of linearized capillary gravity water waves, including eigenvalue branches, bifurcation phenomena, and inviscid damping estimates, extending previous understanding of stability and decay.

Findings

Existence of two eigenvalue branches with asymptotics similar to irrotational waves

Bifurcation of unstable eigenvalues at inflection points of shear flow

Solutions exhibit inviscid damping with specific decay rates over time

Abstract

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 $x$.