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

We first derive the linearized system of (1.1) at the shear flow $(v_{*}=(U(x_{2}),0)^{T},\eta_{*}=0)$ given in (1.2) satisfied by the linearized solutions which we denote by $(v,\eta,p)$. Let $(S_{t}^{\epsilon},v^{\epsilon}(t,x),p^{\epsilon}(t,x))$ be a one-parameter family of solutions of (1.1) with $(S_{t}^{0},v^{0}(t,x),p^{0}(t,x))=(S_{*},v_{*},p_{*})$. Differentiating the Euler equation (1.1a) and (1.1b) with respect to $\epsilon$ and then evaluating it at $\epsilon=0$ yield

(1.3a)		$$\partial_{t}v+U(x_{2})\partial_{x_{1}}v+(U^{\prime}(x_{2})v_{2},0)^{T}+\nabla p=0,\quad\;\;\nabla\cdot v=0,\quad\;x_{2}\in(-h,0).$$
Taking its divergence and also evaluating the above linearized Euler equation at $x_{2}=-h$, we obtain
(1.3b)		$$-\triangle p=2U^{\prime}(x_{2})\partial_{x_{1}}v_{2},\quad x_{2}\in(-h,0),\;\text{ and }\;\partial_{x_{2}}p|_{x_{2}=-h}=0.$$
From the kinematic boundary condition (1.1c), we have
(1.3c)		$$\partial_{t}\eta=v_{2}|_{x_{2}=0}-U(0)\partial_{x_{1}}\eta.$$
Finally differentiating (1.1d), where the left side is $p^{\epsilon}(t,x_{1},\eta^{\epsilon}(t,x_{1}))$, and using $\partial_{x_{2}}p_{*}=-g$, we obtain
(1.3d)		$$p=g\eta-\sigma\partial_{x_{1}}^{2}\eta,\quad\text{ at }\ x_{2}=0.$$

The above (1.3a – 1.3d) form the linearization of the capillary gravity water wave problem (1.1) at the shear flow $(v_{*},S_{*},p_{*})$ with initial values $(v_{10}(x),v_{20}(x),\eta_{0}(x_{1}))$. In fact it can be reduced to an evolutionary problem of the unknowns $(v,\eta)$, while $p$ can be recovered by the boundary value problem of the elliptic system (1.3b) and (1.3d).

Due to its physical and mathematical significance there have been extensive studies of the Euler equation linearized at shear currents. Many of these works were on a fixed channel with slip boundary conditions

(1.4)

$$\text{\eqref{E:Euler-1}--\eqref{E:Euler-2} with }g=0,\;\;\;x_{2}\in(-h,0),\quad v_{2}(x_{1},0)=v_{2}(x_{1},-h)=0,$$

and some of the results have been extended to free boundary problems such as the gravity waves. The spectral analysis is naturally a crucial part of such linear systems. Eigenvalues yield linear solutions exponential in time, while the continuous spectra often lead to algebraic decay of solutions, the so-called inviscid damping due to the lack of a priori dissipation mechanism of the Euler equation.

$\bullet$ Eigenvalues. Since the variable coefficients in the linearized Euler system depend only on $x_{2}$, the subspace of the $k$-th Fourier mode is invariant under the linear evolution for any $k\in\mathbb{R}$. Hence it is a common practice to seek eigenvalues and eigenfunctions in the form of

(1.5)

$$v(t,x)=e^{ik(x_{1}-ct)}\big{(}v_{10}(x_{2}),v_{20}(x_{2})\big{)},\;\;\text{ also }\;\eta(t,x_{1})=e^{ik(x_{1}-ct)}\eta_{0}(x_{1})\;\text{in the free boundary case},$$

where apparently the eigenvalues take the form $\lambda=-ikc$ with the wave speed $c=c_{R}+ic_{I}\in\mathbb{C}$. The linear system is spectrally unstable if there exist such $c$ with $c_{I}>0$, which appear in conjugate pairs. Solutions in the above form with $c\in U([-h,0])$ are in a subtle situation and are referred to as singular modes (see Definition 2.1 and Remark 4.1 for singular and non-singular modes). In seeking solutions in the form of (1.5), the wave number $k\in\mathbb{R}$ is often treated as a parameter.

Classical results on the spectra of the Euler equation (1.4) in a channel linearized at a shear flow include:

• •

  Unstable eigenvalues are isolated for any wave number $k\in\mathbb{R}$ and do not exist for $|k|\gg 1$.

• •

  Rayleigh’s necessary condition of instability [30]: unstable eigenvalues do not exist for any $k$ if $U^{\prime\prime}\neq 0$ on $[-h,0]$ (see also [8]).

• •

  Howard’s Semicircle Theorem [13]: for any $k\neq 0$, eigenvalues exist only with $c$ in the disk

  (1.6)

  $$\big{(}c_{R}-\tfrac{1}{2}(U_{max}+U_{min})\big{)}^{2}+c_{I}^{2}\leq\tfrac{1}{4}(U_{max}-U_{min})^{2}.$$

• •

  Unstable eigenvalues may exist with $c$ near inflection values of $U$ (Tollmien [34] formally, also [21]).

Many classical results can be found in books such as [7, 26] etc. For a class of shear flows, the rigorous bifurcation of unstable eigenvalues was proved, e.g., in [9, 22]. In particular, continuation of branches of unstable eigenvalues were obtained by Lin in the latter.

It has been extended to the linearized free boundary problem of gravity waves (i.e. $g>0$ and $\sigma=0$ in (1.1)) at shear flows (see [39, 14, 31] etc.) that: a.) assuming $U^{\prime}>0$ and $U^{\prime\prime}\neq 0$ on $[-h,0]$, there are no singular neutral modes in $U\big{(}(-h,0)\big{)}$ (i.e. solution in the form of (1.5) with $c\in U\big{(}(-h,0)\big{)}$); b.) the semicircle theorem still holds for unstable eigenvalues; and c.) for a class of shear flows, singular neutral modes may exist at inflection values of $U$ and the bifurcation and continuation of branches of unstable eigenvalues were also obtained. Compared to channel flows with fixed boundaries, new phenomena of the linearized gravity waves include: a.) in addition, critical values of $U$, where $U^{\prime}=0$, and $c=U(-h)$ may be limiting singular neutral modes; and b.) there are non-singular neutral modes, i.e. $c\in\mathbb{R}\setminus U([-h,0])$. Another related result is Miles’ critical layer theory [28, 5] on the instability of shear flows in two-phase fluid interface problem due to the resonance between the temporal frequency of the linear irrotational capillary gravity waves at the completely stationary water and the shear flow in the air in the above.

$\bullet$ Inviscid damping. The analysis of the inviscid damping phenomenon started with the Euler equation in a fixed periodic channel (1.4) linearized at the Couette flow $U(x_{2})=x_{2}$. In 1907, Orr [29] observed that the linearized vertical velocity $v_{2}(t,x)$ tends to zero as $t\to\infty$. Under the assumption $\int v_{10}(x_{1},x_{2})dx_{1}=0$, $\forall x_{2}\in(-h,0)$, which removes the shear flow component of the linear solutions through an invariant splitting, explicit calculations (see, e.g., [6, 24]) yield, as $t\to\infty$,

(1.7)

$$\omega_{0}\in L^{2}\implies|v|_{L^{2}}=o(1),\;\;\omega_{0}\in H^{1}\implies|v|_{L^{2}}=O(\tfrac{1}{|t|}),\;\;\omega_{0}\in H^{2}\implies|v_{2}|_{L^{2}}=O(\tfrac{1}{|t|^{2}}),$$

where $\omega_{0}$ denotes the initial vorticity. More general shear flows in a fixed channel have also been studied extensively. For a class of general stable shear flows, Bouchet and Morita [4] predicted similar decay estimates of the linearized velocity as well as the vorticity depletion phenomenon. For monotone shear flows without infection points, an $O(|t|^{-\nu})$ decay of the stream function was proved in [33] and then the (1.7) type decay in [40, 41] under a smallness assumption of $LU^{\prime\prime}$ (also $\omega_{0}|_{x_{2}=-h,0}=0$ in order for the $O(t^{-2})$ decay of $v_{2}$). A significant contribution is [35] by Wei-Zhang-Zhao where the (1.7) type estimates were obtained for general monotone shear flows without singular modes. In the follow-up works [36, 37, 38], vorticity depletion and velocity decay (as well as an $L_{t}^{2}$ decay if $\omega_{0}\in L^{2}$ only) were also obtained for a class of non-monotone shear flows. As the decay rates in (1.7) are basically optimal, some leading order effects from both the interior and the boundary were identified in [40, 20]. In the absence of boundary impact, for compactly supported initial vorticity, linear inviscid damping near a class of monotone shear flows was also obtained in Gevrey spaces [19]. In [12], a different approach using methods from the study of Schrödinger operators was successfully adopted to analyze inviscid damping. See also [2, 18] for important developments for the linear inviscid damping at circular flows in $\mathbb{R}^{2}$.

While this paper focuses on the linearized capillary gravity waves at shear flows, among the rich literatures on the related nonlinear dynamics of the 2-d Euler equation on fixed domains we refer the readers to [1] for nonlinear Lyapunov stability of steady states based on energy-Casimir functions by Arnold; the remarkable asymptotic stability of shear flows in Gevrey class [3, 16, 17, 27] based on the linear inviscid damping; and for nonlinear instability of steady states [10, 11, 23, 25], etc.

$\bullet$ Intuitions and goals on linearized capillary gravity waves. The goal of this paper is to study thoroughly the capillary gravity water waves linearized at a shear flow $U(x_{2})$ under the above monotonicity assumption (H), focusing on the spectral distribution, stability/instability, and, if the eigenvalues are properly separated from the continuous spectra, the spectral projections and the linear inviscid damping.

For an illustration, some explicit computations of the linearized capillary gravity wave system (1.3) at the Couette flow $U(x_{2})=x_{2}$ are given in Subsection 2.1. There it is easy to see that, on the one hand, the linear inviscid damping (1.7) holds for the rotational part of the solutions. On the other hand, there exist two branches of neutral modes $c^{\pm}(k)$ (see (2.3)) approaching infinity at the same rate as (2.4) of the irrotational capillary gravity waves linearized at zero. They form the two branches of the dispersion relation of the irrotational components in the linearized water wave system at the Couette flow, which is linearly stable.

Based on the above cited existing results on the channel flows with fixed boundaries, as well as those on the gravity water waves, and the explicit calculations of the capillary gravity waves linearized at the Couette flow, the analysis of the linearization (1.3) of the capillary gravity water waves at a general monotonic shear flow $U(x_{2})$ is expected to include the following.

a.) Eigenvalues for wave numbers $|k|\gg 1$. Like matrices, eigenvalues (or equivalently, singular and non-singular modes) of (1.3) correspond to roots of a “characteristic" function (the $\mathbf{F}(k,c)$ defined in (4.1)) analytic in $k\in\mathbb{R}$ and $c\in\mathbb{C}\setminus U([-h,0])$. In contrast to the linearized Euler equation on a fixed domain where no eigenvalues exist for any large wave number $k$, as seen in the linearized irrotational solutions of the capillary gravity waves at both the trivial (zero) solution and the Couette flow, likely there exist two non-singular neutral modes $c^{\pm}(k)$ for each wave number $|k|\gg 1$. These branches behave rather differently compared to the linearized gravity waves since the surface tension is dominant for $|k|\gg 1$. This part would be handled by an asymptotic analysis (Section 4).

b.) Analytic continuation and bifurcation of branches of eigenvalues and the spectral stability (Section 4). Each branch of non-singular modes could continue as long as they do not collide with each other or reach $U([-h,0])$, the boundary of the domain of analyticity of the characteristic function. The bifurcation analysis for $c$ near $U([-h,0])$, more specifically near inflection values of $U$ which possibly generates the instability (compare with the gravity wave case [14, 31, 15]) and $U(-h)$, would require very careful study of the dependence of the solutions to the classical Rayleigh equation (3.1) on the singular parameter $c$ (Section 3). Our main tool is a local transformation which isolates the singular part of the solutions.

c.) Spectral projections and the linear inviscid damping (Section 6). Assuming that the eigenvalues are properly separated from the continuous spectra, a decomposition of linear solutions $(v,\eta)=(v^{p},\eta^{p})+(v^{c},\eta^{c})$ into components corresponding to the eigenvalues and the continuous spectra, respectively, is expected. However, the boundedness of this spectral projection still needs to be obtained which is conceptually related to a lower bound of the angles both the two infinite dimensional components. This is a generalization of the Hodge decomposition of the free linear capillary gravity waves (linearized at zero) into the irrotational and the rotational parts. Both this intuition and the calculations of the capillary gravity waves linearized at the Couette flow suggest that $(v^{p},\eta^{p})$ is mostly related to the surface motion and dispersive (possibly with some unstable modes), while the other component $(v^{c},\eta^{c})$ is more determined by the internal rotations and thus by the vorticity. Whether the Euler equation is in a fixed domain or with free boundaries, the vorticity is transported in the same fashion by the fluid flow in the interior of the fluid domain. Hence it is natural to expect the linear inviscid damping (1.7) of $(v^{c},\eta^{c})$. One may further ask whether (1.7) is optimal in general. If so, a deeper question is whether it is possible to identify the leading order parts of $(v^{c},\eta^{c})$ for $|t|\gg 1$? These studies would be based on the careful analysis of the spectral contour integrals (Section 6) and the solutions to the homogeneous (Section 3) and non-homogeneous Rayleigh equation (Section 5).

We first give the main theorem on the eigenvalue distribution along with its implication on the linear stability. The results on the linear inviscid damping are somewhat more technical and only roughly outlined here. Their more precise statements are given in Theorems 2.1 and 2.2 in Subsection 2.2. See Definition 2.1, Lemma 4.1(5), and Remark 4.1 for what are referred to as singular and non-singular modes. Particularly, by slightly adjusting the same argument as in [13, 39], the Semi-circle Theorem still holds for the unstable modes of the linearized system (1.3) of the capillary gravity water waves at shear flows, namely, any unstable mode satisfies (1.6). We shall take this as granted in the rest of the paper.

The existence of the unbounded branches of non-singular neutral modes $c^{\pm}(k)$ are in contrast to the gravity waves or the Euler equation on fixed channels. The geometric multiplicity of $-ikc^{\pm}(k)$ occurs only among different $k$. These temporal frequencies $-kc^{\pm}(k)$ are asymptotic to those (see (2.4)) of the irrotational capillary gravity waves linearized at zero. Moreover, after normalizing the $L^{2}$ norm of the $v$ component of the eigenfunction to be $1$, the $L^{2}$ and $H^{1}$ differences in the $v$ and $\eta$ components, respectively, between the eigenfunctions of (1.3) and the linearized irrotational capillary gravity waves are of the order $O(|k|^{-\frac{3}{2}})$ as $|k|\to\infty$ (see Remark 6.1). In the case (happening only if (1.8) is not satisfied) where the branch $c^{-}(k)$ reaches $U(-h)$, where the bifurcation equation has the worst regularity, subtle bifurcations occur. This had been pointed out as a possibility in the linearized gravity waves [14, 31, 15], but not analyzed. In particular, it runs out that the sign of $U^{\prime\prime}$ determines whether $c^{-}(k)$ becomes unstable or disappears at $U(-h)$.

The spectral stability in the case $U^{\prime\prime}<0$ can also be obtained by directly modifying the usual proof of the Rayleigh theorem in the fixed channel flow case, as done in [39] for the gravity wave. Our proof provides a complete picture of the eigenvalue distribution as in the above theorem, however.

While $U(0)$ is never a singular mode, just like the Rayleigh’s theorem in the channel flow case the change of sign of $U^{\prime\prime}$ turns out to be necessary for the existence of interior singular modes, which is also sufficient if $\sigma\ll 1$. In the contrast this may not be sufficient if the stabilizing gravity $g$ and surface tension $\sigma$ are strong, see Remark 4.4.

In the following outline of the linear inviscid damping results, $k\in\mathbf{K}=\frac{2\pi}{L}\mathbb{N}$ if $x_{1}\in\mathbb{T}_{L}$, while $k\in\mathbf{K}=\mathbb{R}$ if $x_{1}\in\mathbb{R}$. The initial velocity $v_{0}=(v_{10},v_{20})$ is always assumed to satisfy $\partial_{x_{1}}^{n_{1}}\partial_{x_{2}}^{n_{2}}v_{0}\in L^{2}(\mathbb{T}_{L}\times[-h,0])$ or $L^{2}(\mathbb{R}\times[-h,0])$ for some $n_{1}$’s and $n_{2}$’s and similarly $\partial_{x_{1}}^{n}\eta_{0}\in L^{2}$. To avoid too much technicality, here we skip the detailed assumptions on their regularity in $x_{1}$, but focus on that of $x_{2}$ only. Precise statements are given in Theorems 2.1 and 2.2. The following $L_{t}^{q}$ is always for $t\in\mathbb{R}$.

Main results on inviscid damping. Assume (H) and there are no singular modes for all $k\in\mathbf{K}$, then any solution $(v(t,x),\eta(t,x_{1}))$ to (1.3) can be decomposed into $v=v^{p}+v^{c}$ and $\eta=\eta^{p}+\eta^{c}$, where $(v^{p},\eta^{p})$ belongs to the invariant subspace generated by the eigenfunction of the non-singular modes for all $k\in\mathbf{K}$, while $(v^{c},\eta^{c})$ and its vorticity $\omega^{c}$ satisfy the following estimates.

1. (1)

   If the initial vorticity $\omega_{0}\in L_{x_{2}}^{2}$, then for any $q\in[2,\infty]$, $v^{c}\in L_{x}^{2}L_{t}^{q}$ and $\eta^{c}\in L_{t}^{q}$.

2. (2)

   If $\omega_{0}\in H_{x_{2}}^{1}$, then $tv_{2}^{c}$, $t\eta^{c}\in L_{x}^{2}L_{t}^{q_{1}}$ for any $q_{1}\in[2,\infty]$. Moreover there exists $\Omega^{c}(x)\in H_{x_{2}}^{1}$ such that for any $q_{2}\in(2,\infty]$,

   $$tv_{1}^{c}-U^{\prime}(x_{2})^{-1}\partial_{x_{1}}^{-1}\Omega^{c}(x_{1}-U(x_{2})t,x_{2}),\,\omega^{c}-\Omega^{c}(x_{1}-U(x_{2})t,x_{2}),\,\partial_{x_{2}}^{2}v_{2}^{c}-\partial_{x_{1}}\Omega^{c}(x_{1}-U(x_{2})t,x_{2})\in L_{x}^{2}L_{t}^{q_{2}}.$$

3. (3)

   If $\omega_{0}\in H_{x_{2}}^{2}$, then there exist $\Lambda_{B}(x),\Lambda_{T}(x)\in H_{x_{2}}^{1}$ such that for any $q_{2}\in(2,\infty]$,

   $$t^{2}v_{2}^{c}-U^{\prime}(x_{2})^{-2}\partial_{x_{1}}^{-1}\Omega^{c}(x_{1}-U(x_{2})t,x_{2})-\Lambda_{B}(x_{1}-U(-h)t,x_{2})-\Lambda_{T}(x_{1}-U(0)t,x_{2})\in L_{x}^{2}L_{t}^{q_{2}}.$$

In the above results, the assumption of the non-existence of singular modes, which is equivalent to the absence of embedded eigenvalues of (1.3) for each wave number $k$, turns out to yield the spectral decomposition of the phase space of (1.3) into the invariant subspaces corresponding to the non-singular modes/point spectra and the continuous spectra $-ikU([-h,0])$ for each $k\in\mathbb{R}$.

The component $(v^{c},\eta^{c})$ corresponds to the continuous spectra and enjoys temporal algebraic decay as in the case (1.4) of the Euler equation in a fixed channel. For the case of $x_{1}\in\mathbb{R}$, certain stronger decay in $|k|\ll 1$ (for long waves) is also assumed on the initial values, see Theorem 2.2 and Remark 2.2. Additional to the above $L_{t}^{q}$ bounds, derivatives-in-$t$ estimates are given in Theorems 2.1 and 2.2 as well which also imply pointwise-in-$t$ decay. Compared with (1.7), these additional $L_{t}^{q}$ estimates represent an improvement of decay of roughly an order of $O(t^{-\frac{1}{q}})$ (after appropriate $t$-dependent translations in $x_{1}$ of some asymptotic leading terms are identified and singled out in the cases of $tv_{1}^{c}$, $t^{2}v_{2}^{c}$, etc.). For the Euler equation in a fixed channel (1.4), a.) when $\omega_{0}\in L^{2}$, the $|v|_{L_{t}^{2}}$ estimate was also obtained in [36, 38]; b.) comparable asymptotic leading terms were identified in Lemma 3 of [40] and in Lemma 5.1 of [20]. The Fourier transforms (in $x_{1}$) of these leading terms $\Omega^{c}$, $\Lambda_{T}$, and $\Lambda_{B}$ are given explicitly in (6.9), (6.16), and (6.15), which represent the impact of the interior flow and the top and bottom boundaries, respectively. See also (2.5) for singular elliptic boundary value problems satisfied by $\Lambda_{T}$ and $\Lambda_{B}$. In particular, the free boundary effect is explicitly reflected in the boundary conditions (2.11c) of the corresponding Rayleigh equation (2.11) and the form (6.15) of $\Lambda_{T}$. The error estimates in addition to these leading asymptotic terms also justify that the estimates of $tv_{1}$ and $t^{2}v_{2}$ in (1.7) are optimal. The precise asymptotic leading terms could be useful for further analysis.

The component $(v^{p},\eta^{p})$ are given by superpositions of the eigenfunctions of those non-singular modes, which is governed by a (possibly unstable) multi-branched dispersion relation given by $k\to-kc$ for all non-singular modes $c$ of the $k$-th Fourier modes in $x_{1}$. According to the above spectral analysis, this dispersion relation is asymptotic to that of the linear irrotational capillary gravity wave for $|k|\gg 1$. In the case of $x_{1}\in\mathbb{R}$, in the absence of singular modes, all non-singular modes are given by $c^{\pm}(k)$ which are neutral/stable. The dispersion of $(v^{p},\eta^{p})$ implies that it should decay if $x_{1}\in\mathbb{R}$, but at a slower rate. Hence the dynamics of (1.3) has two layers: faster inviscid decay of $(v^{c},\eta^{c})$ leaves the remaining $(v^{p},\eta^{p})$ decaying at a slower rate due to the dispersion like a linear irrotational capillary gravity wave.

In the periodic-in-$x_{1}$ case, as the non-existence of singular modes is assumed only for $k\in\frac{2\pi}{L}\mathbb{N}$, there can still be other non-singular modes besides $c^{\pm}(k)$ which may have bifurcated from inflection values of $U([-h,0])$ (as well as unstable modes from $U(-h)$) at some $k\notin\frac{2\pi}{L}\mathbb{N}$. In particular instability may appear in finitely many dimensions in low wave numbers.

In the preliminary analysis in Subsection 2.3, we first apply the Fourier transform in $x_{1}$ to (1.3), resulting in decoupled systems for each wave number $k$. The problem can be further reduced to the evolution of $\hat{v}_{2}(t,k,x_{2})$, the Fourier transform of $v_{2}$. The Laplacian transform¹¹1Working on the Laplace transform of the unknowns is essentially equivalent to analyzing the resolvent of the linear operator in (1.3). $V_{2}(k,c,x_{2})$ of $\hat{v}_{2}(t,k,x_{2})$, where $s=-ikc$ is the Laplace transform variable, satisfies a non-homogeneous boundary value problem (2.11) of the Rayleigh equation, solutions to the associated homogeneous problem of which correspond to eigenvalues and eigenfunctions.

A detailed analysis of the homogeneous Rayleigh equation (3.1), carried out in Section 3, lays the foundation of the study of both the eigenvalue distribution and the inviscid damping. The dependence of the estimates of the solutions on the wave number $k$ is also carefully tracked.

We first study the Rayleigh equation away from the singularity where $|U(x_{2})-c|\geq O(\mu)$, $\mu=\frac{1}{\langle k\rangle}=(1+k^{2})^{-\frac{1}{2}}$. Near the singularity where $|U(x_{2})-c|\leq O(\mu)$, different from those in, e.g., [35, 20], our approach is an improved version of the technique in [5] based on the ODE blow-up and invariant manifold method. Through a transformation, solutions to the homogeneous Rayleigh equation near $U(x_{2})-c=0$ are expressed in a form involving the explicit $\log(U-c)$ and the heaviside function with coefficients smooth in $(k,c_{R},x_{2})$.

We focus on a pair of fundamental solutions $y_{\pm}(k,c,x_{2})$ to the homogeneous Rayleigh equation which satisfy the corresponding homogeneous boundary conditions (2.11b)-(2.11c) in (2.11) at $x_{2}=0,-h$, respectively (boundary condition (2.11c) reflects the free boundary setting). For $y_{\pm}$, we establish a.) their a priori bounds; b.) the convergence to their limits $y_{0\pm}(k,c_{R},x_{2})$ as $c_{I}\to 0+$; and c.) the smoothness of $y_{0\pm}$, particularly, in $c_{R}$. Recall $U\in C^{l_{0}}$, we prove $y_{0\pm}$ is $C^{l_{0}-2}$ in $c_{R}$ except at $c_{R}=U(-h),U(0)$. Due to the analyticity of $y_{\pm}$ in $c$ with $c_{I}>0$, the estimates of $y_{0\pm}$ also yield those of $y_{\pm}$ for $c_{I}>0$. Eventually general solutions to the non-homogeneous boundary value problem (2.11) of the Rayleigh equation are expressed using $y_{\pm}$. Finally, the quantity $Y(k,c)=\partial_{x_{2}}y_{-}(k,c,0)/y_{-}(k,c,0)$ related to the Reynolds stress is carefully studied, which plays an important role in the analysis of the Rayleigh equation.

The analysis of the Rayleigh equation near the singularity based on a canonical form presented in Section 3 gives the most detailed information of the solutions to this singular ODE. In the representation formula (3.74), additional to the explicit form of the singular part of the solutions, the Taylor expansion of the smooth transformation $B(\cdot)_{2\times 2}$ can be carried out to an arbitrary order if needed. The section is a lengthy, but we believe this technique applied to the Rayleigh equation is widely useful for various purposes. In a forthcoming work we are studying the linearized Euler equation at a non-monotonic shear flow with a similar approach.

In Section 4 we prove the results on the eigenvalue distribution based on the detailed analysis in Section 3. We first obtain $c^{\pm}(k)$ for $|k|\gg 1$, followed by an argument based on analytic continuation and index calculation. Bifurcations may occur at inflection values of $U$ and particularly subtle at $c=U(-h)$, which are on the boundary of the analyticity of the bifurcation equation $F(k,c)=0$. The regularity obtained in Section 3 implies, when restricted to $c_{I}\geq 0$, $F\in C^{l_{0}-2}$ near $c\in U\big{(}(-h,0)\big{)}$ and $F\in C^{1,\alpha}$ near $c=U(-h)$. This makes the bifurcation analysis possible near $c=U(-h)$ and much easier even in the relatively classical case near inflection values of $U$.

Among the results in Theorem 1.1, in statement (1), $c^{\pm}(k)$ are obtained for large $|k|$ in Lemma 4.2(3) with more detailed estimates, the extension of $c^{\pm}(k)$ in Corollary 4.3.1, and the semi-simplicity of the eigenvalues $-ikc^{\pm}(k)$ in Lemma 4.2(3), Proposition 4.4, and Corollaries 4.3.1 and 6.2.1. Under the additional assumption of non-existence of singular modes, the non-existence of other non-singular modes is proved in Proposition 4.4. The analysis of the critical points of $c^{\pm}(k)$ is given in Lemmas 4.7. The conjugacy to the linearized irrotational waves is proved in Proposition 6.12. See also Remark 6.5. The existence of $g_{\#}$ is proved in Lemma 4.6, along with the existence of $k_{\#}$ and/or $k_{\#}^{\pm}$ in statement (3). The rest of statement (3) is proved at the end of Subsection 4.2 after a series of lemmas. Statement (4) is proved in Subsection 4.3 with more details.

Under the assumption of the absence of singular modes, general solutions $y_{B}(k,c,x_{2})$ to the non-homogeneous boundary value problem (2.11) of the Rayleigh equation are studied in Section 5, which are expressed in the variation of parameter formula using $y_{\pm}$ obtained in Section 3. We establish the basic a priori and convergence (as $c_{I}\to 0+$) estimates in Subsection 5.1. The latter is often referred to as the limiting absorption principle (e.g. [36, 20]). For the inviscid damping estimates, it is crucial to obtain the smoothness of $y_{B}$ in $c$ (in Subsection 5.2). Since singularity occurs in the Rayleigh equation along $c=U(x_{2})$, $\partial_{c}^{j}y_{B}$, $j=1,2$, behaves badly there. Instead we apply a differential operator $D_{c}$ to the Rayleigh system (2.11) which differentiates along the direction of $c_{R}=U(x_{2})$, hence $D_{c}^{j}y_{B}$ satisfies another boundary value problem of the Rayleigh equation in the form of (2.11) and enjoys reasonable estimates. Essentially this approach is similar to those used in [35, 19, 38] for the Euler equation on fixed channels. The main results of Subsection 5.2 are the estimates of $\partial_{c}^{j_{1}}\partial_{x_{2}}y_{B}$, $j_{1}=1,2$ and $j_{2}=0,1$, with the most singular terms identified.

The splitting and the linear inviscid damping estimates of solutions $(v,\eta)$ to the linearized capillary gravity waves (1.3) are obtained in Section 6. While the vorticity $\omega$ is not sufficient to recover the whole solution (which is different from the fixed channel case as in e.g. [41, 35]), the solutions are expressed in terms of the inverse Laplace transform of $V_{2}(k,c,x_{2})$, the Laplace transform of $\hat{v}_{2}(t,k,x_{2})$, which is estimated in Section 5. Unlike e.g. [35, 20], technically we do not immediately push the contour integral (in $c$) of the inverse Laplace transform to the limit spectra set $U([-h,0])$, but first keep it along the boundary of a small neighborhood of it in the complex plane. This allows easy integration by parts in $c$ to establish the decay estimates in $t$ after the leading asymptotic terms are obtained by applying the Cauchy integral theorem to the most singular terms of $\partial_{c}^{j}V_{2}$, $j=1,2$. In fact, in deriving the decay estimates of $v$, $\eta$, $tv_{2}$, and $t\eta$ where the leading asymptotic terms were not involved, a priori estimates, but not the limiting absorption principle, is sufficient.

The above approach to obtain the inviscid decay also applies to the Euler equation in a fixed channel linearized at a shear flow $U(x_{2})$. Similarly, while the asymptotic leading order terms of $tv_{1}$, $\omega$, and $\partial_{x_{2}}^{2}v_{2}$ are all generated by the asymptotic vorticity $\Omega^{c}$, that of $t^{2}v_{2}$ involves two additional functions $\Omega_{T}$ and $\Omega_{B}$ due to the contributions from the top and bottom boundaries. We give a brief summary of the results for the channel flow in Subsection 6.4 and see also Remark 6.6.

Notation: Throughout the paper, $C>0$ denotes a generic constant which might change from line to line, but always independent of $k$, $c$, and $x_{2}$; $\delta(x)$ the delta function; $P.V.$ (or $(P.V.)_{c}$) the principle value (or the principle value with respect to variable $c$ etc.). The Japanese bracket $\langle k\rangle=\sqrt{k^{2}+1}$ is adopted. For $c=c_{R}+ic_{I}$ close to $U([-h,0])$, $x_{2}^{c}$ denotes $U^{-1}(c_{R})$ after some extension of $U$. We always denote $\mu=\langle k\rangle^{-1}$ and ${}^{\prime}=\partial_{x_{2}}$.

We first describe two main relevant properties using the Couette flow as an illustration. The linearized velocity can be decomposed uniquely into the rotational and irrotational/potential parts (see e.g. [32])

$$v=v^{ir}+v^{rot},\ \text{ where }\,\nabla\cdot v^{ir,rot}=0,\;$$

where

$$v^{ir}=\nabla\varphi,\quad\Delta\varphi=0,\;\;x_{2}\in(-h,0),\,\text{ and }\,\partial_{x_{2}}\varphi|_{x_{2}=-h}=0,$$

and $v^{rot}$ satisfies

$$\nabla\cdot v^{rot}=0,\quad v_{2}^{rot}|_{x_{2}=-h,0}=0.$$

In particular, the rotational part can almost be determined by the vorticity $\omega$ in the same way as in the Euler equation (1.4) in the fixed channel $x_{2}\in(-h,0)$ with slip boundary condition

(2.1)

$$v^{rot}=(-\partial_{x_{2}},\partial_{x_{1}})^{T}\Delta^{-1}\omega+(a,0)^{T},\,\text{ and }\,\omega=\nabla\times v=\partial_{x_{1}}v_{2}-\partial_{x_{2}}v_{1},$$

where $a$ is a constant and $\Delta^{-1}$ is the inverse Laplacian in the 2-d region $x_{2}\in(-h,0)$ ($L$-periodic in $x_{1}$ or $x_{1}\in\mathbb{R}$) under the zero Dirichlet boundary condition along $x_{2}=-h,0$. In the periodic-in-$x_{1}$ case, the constant $a$ may be non-zero and is determined by the physical quantity circulation.