1 Introduction

In this paper, we study a two-phase interface system which is modeled by the two dimensional Euler problem. Both fluids are considered immiscible and incompressible (see, for example, [9]). We use the superscript “+” to denote notations in the upper fluid and subscript “-” for the lower one. At time \(t\ge 0\), the upper fluid occupies \(\Omega _t^+\), and the lower fluid inhabits the region \(\Omega _t^-\). The fixed constants \(h_\pm >0\) are the locations of flat lid of the upper fluid (see, for example, [9]) and flat bed of the lower fluid. We denote \(S_t:=\partial \Omega _t^-\cap \partial \Omega _t^+=\{(t,x)| x_2=\eta (t,x_1)\}\) as the interface, which is considered as the graph of a smooth function \(\eta \). In addition, we assume that

$$\begin{aligned} \Omega _t^+\cup \Omega _t^-\cup S_t=\mathbb {T}\times (-h_-, h_+), \end{aligned}$$

where \(\mathbb {T}:=\mathbb {R}\setminus 2\pi \mathbb {Z}\). We suppose that the density of the upper fluid is a constant \(\rho ^+>0\) and that the lower fluid has constant density \(\rho ^->0\). Let \(v^\pm =v^\pm (t,x)\in \mathbb {R}^2\) be the velocity field and \(p^\pm =p^\pm (t,x)\in \mathbb {R}\) be the pressure. Then \(v^\pm \) and \(p^\pm \) satisfy the two dimensional incompressible Euler system as follows:

$$\begin{aligned}&\partial _t v^\pm +v^\pm \cdot \nabla v^\pm +\frac{1}{\rho ^\pm }\nabla p^\pm +g\vec {e}_2=0 \qquad \text {in}\quad \Omega _t^\pm , \end{aligned}$$
(1.1a)
$$\begin{aligned}&\nabla \cdot v^\pm =0 \qquad \text {in}\quad \Omega _t^\pm . \end{aligned}$$
(1.1b)

Here g is the gravitational acceleration and \(\vec {e}_2\) is the unit vector in \(x_2\)-axis. Since \(v^\pm \) restricted to \(S_t\) being a boundary velocity, the motion of the interface satisfies the kinematic boundary condition

$$\begin{aligned} \partial _t \eta =v^\pm \cdot (-\partial _{x_1}\eta , 1)^T \qquad x\in S_t. \end{aligned}$$
(1.1c)

Considering the surface tension, we also impose the dynamic boundary condition

$$\begin{aligned} p^+(t,x)-p^-(t,x)=\sigma \kappa \qquad x\in S_t, \end{aligned}$$
(1.1d)

where \(\sigma > 0\) and \(\kappa =\frac{\eta _{x_1x_1}}{(1+\eta _{x_1}^2)^{3/2}}\) is the mean curvature of \(S_t\) at x which corresponds to the surface tension. We assume that the upper fluid has finite altitude. It is justified that the behavior of the flow does not strongly affect the dynamics near the lower fluid if it evanesces at high altitude (e.g. [9]). The rigid flat lid of the upper fluid and bed of the lower fluid imply that

$$\begin{aligned} v^\pm \cdot \vec {e}_2=0 \qquad x_2=\pm h_\pm . \end{aligned}$$
(1.1e)

The local well-posedness theory for the interface problem (1.1) has been well studied. When the surface tension is considered, the full nonlinear problem is locally well-posed (cf., [3, 18]). For both the rotational and irrotational problems, the linearized system is ill-posed if the surface tension is ignored [1].

It is well known that shear flows are a fundamental class of stationary solutions in the form of

$$\begin{aligned} v_*^\pm (t,x)=(U^\pm (x_2), 0), \quad S_*=\mathbb {T}\times \{0\}. \end{aligned}$$
(1.2)

Our goal is to analyze this two-phase fluids interface system linearized at a pair of uniformly monotone shear flows

$$\begin{aligned} U^\pm \in C^{l_0}, \; l_0\ge 6, \; (U^+)'\ne 0, \;\ x_2\in [0, h_+], \;\ \text {and} \;\ (U^-)'\ne 0, \;\ x_2\in [-h_-, 0].\nonumber \\ \end{aligned}$$
(1.3)

1.1 Linearization

In this subsection, we linearize the Euler system (1.1) near shear flows (1.2). The linearization was previously obtained in [2], but we adopt a slightly more direct approach as in [11]. We consider an one-parameter family of solutions \(\big (S_t(\theta ), v^\pm (\theta ,t,x), p^\pm (\theta , t,x)\big )\) of system (1.1) with

$$\begin{aligned} \big (S_t(0), v^\pm (0,t,x), \nabla p^\pm (0,t,x)\big )=(S_*, v_*^\pm , -g\rho ^\pm \vec {e}_2). \end{aligned}$$

We abuse the notations and let \(v^\pm , p^\pm , \eta \) be the linearized solution in the following way:

$$\begin{aligned} v^\pm (t,x)=&\big (v_1^\pm (t,x), v_2^\pm (t,x)\big ):=\lim _{\theta \rightarrow 0}\partial _\theta v^\pm (t, x),\\ p^\pm (t,x):=&\lim _{\theta \rightarrow 0}\partial _\theta p^\pm (t, x), \;\ \eta (t, x_1):=\lim _{\theta \rightarrow 0 }\partial _\theta \eta (t, x_1). \end{aligned}$$

We differentiate (1.1a) with respect to \(\theta \) and then send \(\theta \) to 0 to have that

$$\begin{aligned} \partial _t v^\pm +v_*^\pm \cdot \nabla v^\pm +\nabla v_*^\pm \cdot v^\pm +\frac{1}{\rho ^\pm }\nabla p^\pm =0. \end{aligned}$$
(1.4)

Considering the second component of this equation, we have

$$\begin{aligned} \partial _t v_2^\pm +U^\pm (x_2)\partial _{x_1}v_2^\pm +\frac{1}{\rho ^\pm }\partial _{x_2}p^\pm =0, \qquad (x_1, \pm x_2)\in \mathbb {T}\times (0, h_\pm ). \nonumber \\ \end{aligned}$$
(1.5a)

We apply the divergence operator to (1.4) to obtain that

$$\begin{aligned} \nabla \cdot (v_*^\pm \cdot \nabla v^\pm +v^\pm \cdot \nabla v_*^\pm )+\frac{1}{\rho ^\pm }\Delta p^\pm =0. \end{aligned}$$

Sending \(\theta \) to 0 for the above equation, we obtain that

$$\begin{aligned} -\frac{1}{\rho ^\pm }\Delta p^\pm =2(U^\pm )'(x_2)\partial _{x_1}v_2^\pm ,\qquad (x_1, \pm x_2)\in \mathbb {T}\times (0, h_\pm ). \end{aligned}$$
(1.5b)

We differentiate (1.1c) with respect to \(\theta \) to obtain that

$$\begin{aligned} \partial _t \eta = v_2^\pm -U^\pm (x_2)\partial _{x_1}\eta , \qquad \qquad \{x_2=0\}. \end{aligned}$$
(1.5c)

Now we consider the variation of \(S_t\). We notice that (1.1d) means that

$$\begin{aligned} p^+\big (\theta , t, x_1, \eta (\theta , t, x_1)\big )-p^-\big (\theta , t, x_1, \eta (\theta , t, x_1)\big )=\sigma \kappa \big (\theta , t, x_1, \eta (\theta , t, x_1)\big ). \end{aligned}$$

Differentiating this equation with respect to \(\theta \) and sending \(\theta \) to 0, we obtain that

$$\begin{aligned} p^+(0)-p^-(0)=g\eta (\rho ^+-\rho ^-)+\sigma \partial _{x_1}^2\eta . \end{aligned}$$
(1.5d)

Linearizing (1.1e), we have

$$\begin{aligned} v_2^\pm (x_1, \pm h_\pm , t)=0. \end{aligned}$$
(1.5e)

We consider (1.5c) at \(\pm x_2=h_\pm \) and obtain that

$$\begin{aligned} \partial _{x_2} p^\pm =0, \qquad \pm x_2=h_\pm . \end{aligned}$$
(1.5f)

The system (1.5) forms a two-phase fluids interface problem linearized at a pair of shear flows (1.2). In the capillary gravity water wave problem (\(\rho ^+=0\)), the pressure p can be covered by a boundary value problem of elliptic system (1.5b), (1.5d), and (1.5f). In this two-phase fluids interface problem(\(\rho ^+>0\)), the pressure can be expressed in terms of velocity. We refer readers to Sect. 2 in [17]. Hence, the linearized system (1.5) can be viewed as an evolutionary problem of two unknowns \((v_2, \eta )\). The goal of this paper is to study the eigenvalue and the instability of the system (1.1) linearized at monotone shear flows (1.2).

1.2 Background

We notice that the variable coefficients in the linearized system (1.5) depend only on \(x_2\). Hence, each Fourier mode in \(x_1\) is decoupled from other modes. To study the linearized system, it is natural to seek eigenvalues and eigenfunctions in the form of

$$\begin{aligned} & \big (v_2^\pm (t, x_1, x_2), \eta (t, x_1), p^\pm (t, x_1, x_2)\big )=\big (v_2^\pm (x_2), \eta , p^\pm (x_2)\big )e^{ik(x_1-ct)},\qquad \nonumber \\ & \quad k\in \mathbb {Z}\setminus \{0\}, \end{aligned}$$
(1.6)

where the eigenvalues take the form \(\lambda =-ikc\) with the wave speed \(c=c_R+ic_I\in \mathbb {C}\) and wave number k. The linear solution is spectrally unstable if a wave speed c, which appears in conjugate pairs, has a positive imaginary part and wave number \(k>0\). Due to the symmetry of the spectrum, to find instability it suffices to consider the case of \(k>0\) and seek solution with \(c_I>0\).

Due to the significance in mathematics and physics, the two-phase fluids interface problem linearized at shear flows has been studied by some mathematicians. The simplest situation is the classical Kelvin-Helmholtz model where the upper layer fluid is air and the ocean stays below. In this model, the ocean is at rest and the velocity of the air is uniform, i.e., \(U^-\equiv 0\), \( U^+\equiv U_0\), for some constant \(U_0\). Without surface tension, it is (linearly) unstable if \(U_0\ne 0\). However with surface tension, the instability appears at very small wavelength when the wind speed becomes large. This is irrelevant with the practical phenomenon which was observed by Kelvin in 1871 (cf. [2, 4, 19]). Later on, in a series of papers [12,13,14] written by Miles, the wind speed is assumed to be a shear flow. Miles considered the wind generation process as the resonance phenomenon, which is the so-called quasi-laminar model. The atmosphere can be treated as a perturbation of vacuum when the density ratio is very small. If the water has infinite depth and the surface tension is ignored, the dispersion relation is \(c_k=\sqrt{g/k}\). Miles concluded that (linear) instability occurs if a critical layer exists where the unperturbed phase speed meets the wind speed at some \(x_2\) level, i.e. \(c_k = U^+(x_2)\), \( (U^+)'(x_2)\ne 0\), and \((U^+)''(x_2) < 0\). Bühler-Shatah-Walsh-Zeng [2] presented a rigorous derivation of the linearized evolution equations near a pair of shear flows. They also justified Miles’s conclusion [12] that the existence of critical layer may enable a resonance between the shear flow in the air and capillary gravity waves in the ocean, and showed linear instability of the shear flow in the air above the stationary water.

A related problem is the one fluid problem, which is the so-called water wave problem. There are lots of classical results on the spectra of the two dimensional Euler equation linearized around shear flows \((U(x_2), 0)\) in a fixed channel (cf., [4]). Rayleigh [15] gave a necessary condition for the linear instability that the shear velocity profile should have an inflection point. Howard [6] showed that any unstable eigenvalue must have wave speed c lie in the complex plane within an upper semicircle,

$$\begin{aligned} \left( c_R-\frac{U_{\max }+U_{\min }}{2}\right) ^2+c_I^2\le \left( \frac{U_{\max }-U_{\min }}{2}\right) ^2, \;\ c_I>0. \end{aligned}$$

For a class of shear flows, the rigorous bifurcation of unstable eigenvalues was proved, e.g., in [5, 10]. Lin showed that for a certain class of shear flows, the neutral limiting wave speed must be an inflection value of the velocity profile and proved the global bifurcation of unstable modes from neutral modes rigorously [10]. There are also some results regarding the linearized free boundary problem of gravity waves at shear flows. Yih [20] showed that if the shear velocity profile U is monotone without inflection value, there are no singular neutral modes with c in the interior of the range of U. He also proved the existence of non-singular neutral modes with real wave speed outside the range of U and extended the Semicircle Theorem to the linearized gravity water wave near shear flows. For a certain class of shear flows, Hur and Lin [7, 8] proved that the linear instability may occur only when the wave speed c is near the value of U at bottom, critical values, or inflection values of U. They also gave an open set of wave numbers where linear instability exists. M. Renardy and Y. Renardy [16] found examples showing that instability can arise from these three locations through numerical methods. In the author’s recent work with Zeng [11], we considered capillary gravity water waves linearized at monotone shear flows and proved that in contrast to the fixed channel flow, there are exactly two non-singular modes (i.e. the corresponding eigenvalue problem has nontrivial solution when c is outside the range of U) for each wave number k under certain conditions which are real for all high wave numbers. We also obtained the bifurcation of unstable eigenvalues from inflection values of U and value of U at bottom. Moreover, we gave a complete picture of eigenvalue distribution for a certain class of monotone shear flows.

1.3 Main Results

The goal of this paper is to study the eigenvalue distribution of the linear system (1.5) and seek linear instability. In particular, we consider the case that the upper fluid is lighter than the lower fluid and the perturbed free surface \(\eta \ne 0\). If \(\eta =0\), system (2.2) is equivalent to the system which consists of two Rayleigh equations with fixed boundary without interface motion. In this case, for the Euler system linearized at monotone shear flows, the only possible locations of c where instability arises are inflection values of \(U^+\) or \(U^-\) (e.g. [10]). Throughout this paper, we let

$$\begin{aligned} & a:= \min \left\{ U^-\big ([-h_-, 0]\big ) \cup U^+\big ([0, h_+]\big )\right\} , \nonumber \\ & b:= \max \left\{ U^-\big ([-h_-, 0]\big )\cup U^+\big ([0, h_+]\big )\right\} . \end{aligned}$$
(1.7)

We first extend the Howard’s Semicircle theorem to the two-phase fluids interface problem under a mild condition. This shows that all the unstable wave speed must lie within a semi-circle in the upper half complex plane. The center and radius are determined by the a and b. The condition of the next theorem holds if the upper fluid is no heavier than the lower one, i.e. \(\rho ^+\le \rho ^-\), which is commonly seen in reality. It also holds for all large wave numbers.

Theorem 1.1

Assume that \(\eta \ne 0\) and that the following holds:

$$\begin{aligned} -g(\rho ^+-\rho ^-)+\sigma k^2\ge 0. \end{aligned}$$
(1.8)

Then any unstable wave speed \(c=c_R+ic_I\in \mathbb {C}\) with \(c_I>0\) must stay in the following upper semicircle:

$$\begin{aligned} \left( c_R-\frac{a+b}{2}\right) ^2+c_I^2\le \left( \frac{b-a}{2}\right) ^2, \qquad c_I>0. \end{aligned}$$
(1.9)

When seeking solutions in the form of (1.6), we treat the wave number \(k\in \mathbb {R}\) as a parameter. The next theorem shows that when |k| is large, there are exactly two eigenvalues in the whole complex plane with two branches of wave speed \(c^\pm (k)\in \mathbb {R}\). Under the assumption that the monotone shear flows have no inflection point and the gravitational acceleration g is large enough, we discuss two cases, depending on whether the intersection of the range of \(U^-\) and the one of \(U^+\) is empty. For each case, we prove that under a certain condition, there is no singular mode and both \(c^\pm (k)\) can be extended to be even and analytic functions for all \(k\in \mathbb {R}\). Moreover, for each k, \(-ikc^\pm (k)\) are the only eigenvalues of the linearized system (1.5). Basically, singular modes correspond to embedding eigenvalues with wave speed \(c\in U^-\big ([-h_-, 0]\big )\cup U^+\big ([0, h_+]\big )\). One can see the precise definition in Definition 2.1.

Theorem 1.2

Suppose \(U^\pm \in C^3\), \((U^\pm )'\ne 0\), \(\eta \ne 0\), and \(0< \rho ^+< \rho ^-\). Let a, b be defined in (1.7). Then the following hold:

  1. (1)

    There exists \(k_0>0\) such that for any \(|k|>k_0\), there are no singular modes and exist exactly two non-singular modes with \(c^+(k)\in (b, +\infty )\) and \(c^-(k)\in (-\infty , a)\).

  2. (2)

    \(\lim _{|k|\rightarrow \infty } c^\pm (k)/ \sqrt{\frac{\sigma |k|}{\rho ^-+\rho ^+}}=\pm 1\).

  3. (3)

    Assume that \((U^\pm )''\ne 0\) and

    $$\begin{aligned}&\underset{c\in \{U^+(0), U^+(h_+)\}}{\min }\int _{-h_-}^0 \frac{1}{\big (U^-(x_2)-c\big )^2}\textrm{d}x_2, \; \\&\quad \underset{c\in \{U^-(0), U^-(-h_-)\}}{\min }\frac{\rho ^-}{\rho ^+}\int _0^{h_+} \frac{1}{\big (U^+(x_2)-c\big )^2}\textrm{d}x_2>\frac{1}{g}. \end{aligned}$$

    If it that either holds

    1. (a)

      \(U^-\big ((-h_-, 0)\big )\cap U^+\big ((0, h_+)\big )=\emptyset \) and

      $$\begin{aligned}&\max _{c\in \{U^\pm (0), U^\pm (\pm h_\pm )\}} \Big \{\rho ^+\int _0^{h_+} \big (U^+(x_2)-c\big )^2\textrm{d}x_2\\&\quad +\rho ^-\int _{-h_-}^0\big (U^-(x_2)-c \big )^2\textrm{d}x_2\Big \}<\sigma , \end{aligned}$$

      or

    2. (b)

      \(U^-\big ((-h_-, 0)\big )\cap U^+\big ((0, h_+)\big )\ne \emptyset \), \((U^+)''(U^-)''>0\), and

      $$\begin{aligned} \max _{c\in \{a, b\}}\Big \{\rho ^+\int _0^{h_+} \big (U^+(x_2)-c\big )^2\textrm{d}x_2+\rho ^-\int _{-h_-}^0\big (U^-(x_2)-c \big )^2\textrm{d}x_2\Big \}<\sigma , \end{aligned}$$

    then the two branches \(c^\pm (k)\) can be extended to be even and analytic real-valued functions for all \(k\in \mathbb {R}\). \(c^+(k)>b\) and \(c^-(k)<a\) for all \(k\in \mathbb {R}\). Moreover, \(c^\pm (k)\) correspond to the only singular and non-singular modes of the linearized Euler system (1.5).

Remark 1.1

When \(\rho ^+=0\) and \((U^-)'\ne 0\), the results in Theorem 1.2(1) and (2) coincide the ones of capillary gravity water wave linearized at monotone shear flows which is obtained in [11].

We notice that \(c=a\) is the boundary of the domain where the bifurcation equation keeps the regularity. Based on Theorem 1.1, it might correspond to an isolated singular mode or a neutral limiting mode which is the limit of a sequence of unstable modes (see Definition 2.1). It is natural to study the possible bifurcation at \(c=a\) where the equation looses analyticity. In the following theorem, we consider the case where the upper fluid is lighter than the lower fluid. For the case of monotone shear flows which have no inflection value, we study the behavior of branch \(c^-(k)\) which is obtained in Theorem 1.2(1) as |k| tends to 0 from infinity. We show that there exists \(g_*\ge 0\) so that when \(g>g_*\), \(c^-(k)\) can be analytically extended to be an even and real-valued function for all \(k\in \mathbb {R}\). Moreover, \(c^-(k)\) stays on the left of a. Therefore, in this case, there is no singular mode and the shear flows are linearly stable.

Theorem 1.3

Assume that \(U^\pm \) satisfy (1.3), \((U^\pm )''\ne 0\), \(\eta \ne 0\), and \(0< \rho ^+< \rho ^-\). Let \(c^-(k)\) be obtained in Theorem 1.2(1). Then, there exists \(g_*> 0\) such that if \(g>g_*\), then \(c^-(k)\) can be extended as an even and analytic function such that for all \(k\in \mathbb {R}\), \(c^-(k)<a\). There is no singular mode at \(c=a\) for all \(k\in \mathbb {R}\).

We also study the case of \(g\le g_*\). Under some certain conditions, the value of g relative to \(g_*\) tells us when the instability happens near \(c=a\), i.e. the minimal value of \(U^+\big ([0, h_+]\big )\cup U^-\big ([-h_-, 0]\big )\). More precisely, if \(g= g_*\), there is no instability when c is near a. For the case of \(g<g_*\), whether the instability happens near \(c=a\) is related to the signs of \((U^\pm )''\). If \((U^\pm )''>0\), as |k| getting small from the infinity, the branch \(c^-(k)\) obtained in Theorem 1.2(1) bifurcates through \(c=a\) into a pair of branches symmetric about the horizontal axis in the complex plane. As |k| getting large from 0, there exists a branch \(\mathcal {C}(k)\) bifurcates through a and generates an unstable eigenvalue. In other words, the instability arises from \(c=a\). If \((U^\pm )''<0\), there is no instability near \(c=a\).

Theorem 1.4

Assume that \(U^\pm \) satisfy (1.3), \((U^\pm )''\ne 0\), \(\eta \ne 0\), and \(0< \rho ^+< \rho ^-\). Let \(c^-(k)\) be obtained in Theorem 1.2(1) and \(g_*> 0\) be the one in Theorem 1.3. Assume that one of the following conditions is satisfied:

  1. (a)

    \(a=U^+(0)\notin U^-([-h_-, 0])\) and

    $$\begin{aligned} \frac{\rho ^-}{g(\rho ^--\rho ^+)}<\int _{-h_-}^0 \frac{1}{\big ( U^-(x_2)-U^+(0)\big )^2}\textrm{d}x_2; \end{aligned}$$
  2. (b)

    \(a=U^-(-h_-)\) or \(U^-(0)\);

  3. (c)

    \(a=U^+(h_+)\notin U^-([-h_-, 0])\) and

    $$\begin{aligned} \frac{\rho ^-}{g(\rho ^--\rho ^+)}<\int _{-h_-}^0 \frac{1}{\big ( U^-(x_2)-U^+(h_+)\big )^2}\textrm{d}x_2, \end{aligned}$$

Then, we have the following:

  1. (1)

    If \(g=g_*\), there exists a unique \(k_*>0\) such that \(c^-(k)\) can be extended to be an even and \(C^{1, \alpha }\) function for all \(k\in \mathbb {R}\) (for any \(\alpha \in [0, 1)\)) such that it is analytic everywhere except \(k=\pm k_*\) where \(c^-(\pm k_*)=a\) for all \(k\in \mathbb {R}\).

  2. (2)

    If \(g\in (0, g_*)\), there exist \(k_*^+>k_*^->0\) and \(\delta >0\) such that we have the following.

    1. (i)

      Assume that \((U^\pm )''>0\). Then \(c^-(k)\) can be extended to be an even and \(C^{1, \alpha }\) function for all \(|k|\ge k_*^+-\delta \) and analytic except at \(k=\pm k_*^+\) and it satisfies that

      $$\begin{aligned} & c^-(\pm k_*^+)=a,\quad c^-(k)<a, \; \forall |k|>k_*^+, \end{aligned}$$
      (1.10)
      $$\begin{aligned} & c_I^-(k)>0,\; c_R^-(k)>a, \;\forall |k|\in [k_*^+-\delta , k_*^+), \end{aligned}$$
      (1.11)

      and there exists a \(C^{1, \alpha }\) function \(\mathcal {C}(k)\) on \([0, k_*^-+\delta ]\) and analytic except at \(k=\pm k_*^-\) such that

      $$\begin{aligned} & \mathcal {C}(\pm k_*^-)=a, \quad \mathcal {C}(k)<a, \; \forall |k|\in [0, k_*^-), \end{aligned}$$
      (1.12)
      $$\begin{aligned} & \mathcal {C}_I(k)>0, \; \mathcal {C}_R(k)>a, \; \forall |k|\in (k_*^-, k_*^-+\delta ]. \end{aligned}$$
      (1.13)

      Moreover, there exists \(\gamma >0\) such that for \(c_R<a+\gamma \), all singular and non-singular modes with \(|k|\ge k_*^+-\delta \) are \(c^-(k)\) and the ones with \(|k|\in [0, k_*^-+\delta ]\) are \(\mathcal {C}(k)\).

    2. (ii)

      Assume that \((U^\pm )''<0\). Then \(c^-(k)\) can be extended to be an even and \(C^{1, \alpha }\) function for all \(|k|\ge k_*^+\), analytic except at \(k=\pm k_*^+\), and (1.10) holds. And there exists a \(C^{1, \alpha }\) function \(\tilde{C}(k)\) on \([0, k_*^-]\) and analytic except at \(k=\pm k_*^-\) such that (1.12) holds. In addition, for \(c\le a\), all singular and non-singular modes with \(|k|\ge k_*^+\) are \(c^-(k)\) and the ones with \(|k|\in [0, k_*^-]\) are \(\tilde{C}(k)\).

Remark 1.2

  1. 1.)

    In Theorem 1.4(2), \(c^-(k)\) may not be extended to be a \(C^{1, \alpha }\) function for all \(k\in \mathbb {R}\) since there might exist an isolated neutral limiting mode (see Definition 2.1) with \(c\ne a\) (see Lemma 3.4), which may break the continuation of the bifurcation curve.

  2. 2.)

    For the case of \(\rho ^+=0\), the system is the same as the capillary gravity water wave linearized at monotone shear flows. Theorem 1.1 in [11] shows that there exists \(g_\#>0\) which provides a sharp condition for linear instability arising from the value of velocity profile at the bottom. More precisely, when \(g\in (0, g_\#)\), an increasing convex shear flow is spectrally unstable and an increasing concave shear flow is spectrally stable. When \(g\ge g_\#\), monotone shear flow is always spectrally stable if the basic velocity profile has no inflection value. Our results in Theorem 1.3 and Theorem 1.4 for the case of \(a=U^-(-h_-)\) coincides this result with \(g_*=g_\#\).

When \(\rho ^+\ll \rho ^-\), we consider the two-phase fluid interface problem as a perturbation of capillary gravity water wave linearized at monotone shear flows. In particular, we discuss a special case where the velocity profiles in both shear flows are monotonically increasing and the speed of the lower fluid is strictly slower than the one in the upper fluid. We prove that if the one fluid free boundary problem has a neutral mode with wave speed \(c>U^-(0)\) in the range of \(U^+\), which means that there exists a critical layer location \(x_2\in [0, h_+]\), then small \(\epsilon \) leads to an unstable mode near such wave speed in the ocean-air system. This is due to the resonance between shear flows in the air and ocean. Such a situation can be viewed as an extension of the critical layer phenomenon studied by Miles [12,13,14] and Bühler-Shatah-Walsh-Zeng [2]. It is shown that in the one fluid problem, the right branch of wave speed \(c^+_0(k)\) is real and larger than \(U^-(0)\) for all \(k\in \mathbb {R}\) [11]. The eigenvalue distribution of the ocean-air system depends on the location of the neutral mode with zero wave number (i.e. \(c_0\) in Theorem 1.5) in the one fluid problem and the sign of \((U^+)''\). If \((U^+)''<0\), then the existence of critical layer within the range of \(U^+\) in the above sense leads to instability. We prove that if \((U^+)''>0\) and \(c_0\in \big (U^+(0), U^+(h_+)]\), then then for small \(\epsilon >0\), as |k| becomes small and tends to 0, the right branch \(c^+(k)\) of the two-phase interface problem disappears when it reaches \(U^+(h_+)\). If \((U^+)''>0\) and \(c_0\in \big (U^-(0), U^+(0)\big )\), then \(c^+(k)\) disappears when it reaches \(U^+(h_+)\) but reappears at \(U^+(0)\).

Theorem 1.5

Assume that \(U^\pm \in C^6\), \((U^\pm )'>0\), \((U^\pm )''\ne 0\), \(U^-(0)<U^+(0)\), \(\eta \ne 0\), and

$$\begin{aligned} \int _{-h_-}^0\big (U^-(x_2)-U^+(h_+)\big )^2\textrm{d}x_2<\frac{\sigma }{\rho ^-}. \end{aligned}$$
(1.14)

Let \(c^+(k)\) be obtained in Theorem 1.2(1) and \(c_0>U^-(0)\) be the unique solution to

$$\begin{aligned} \int _{-h_-}^0 \frac{1}{\big (U^-(x_2)-c\big )^2}\textrm{d}x_2 =\frac{1}{g}. \end{aligned}$$
(1.15)

Then, we have

  1. (1)

    If \(c_0>U^+(h_+)\), there exists \(\epsilon _0>0\) such that for any \(\frac{\rho ^+}{\rho ^-}\in (0, \epsilon _0]\), \(c^+(k)\) can be extended to be an even and analytic function for all \(k\in \mathbb {R}\) and \(c^+(k)>U^+(h_+)\). It is the only singular and non-singular mode for \(c_R>U^-(0)\).

  2. (2)

    If \(c_0\in (U^+(0), U^+(h_+)]\), there exists \(\epsilon _1>0\) such that for any \(\frac{\rho ^+}{\rho ^-}\in (0, \epsilon _1]\), there exists a unique \(k_*> 0\) such that \(c^+(\pm k_*)=U^+(h_+)\) and the following hold:

    1. (a)

      If \((U^+)''<0\), then \(c^+(k)\) can be extended to be an even and \(C^{1, \alpha }\) (for all \(\alpha \in [0, 1)\)) function for all \(k\in \mathbb {R}\) and analytic in k except at \(\pm k_*\). \(c^+(k)\) satisfies that

      $$\begin{aligned}&c^+(k)>U^+(h_+), \; \forall |k|\in (k_*, \infty ), \quad c_I^+(k)>0, c_R^+(k)\in \big (U^+(0), U^+(h_+)\big ), \; \\&\quad \forall |k|\in [0, k_*]. \end{aligned}$$
    2. (b)

      If \((U^+)''>0\), \(c^+(k)\) can be extended to be a real valued \(C^{1, \alpha }\) function (for all \(\alpha \in [0, 1)\)) for \(|k|\ge k_*\) and analytic in k if \(|k|>k_*\).

    Moreover, \(c^+(k)\) is the only singular and non-singular mode for \(c_R>U^+(0)\).

  3. (3)

    Suppose \(c_0\in (U^-(0), U^+(0)]\). Then there exists \(\epsilon _2>0\) such that for any \(\epsilon \in (0, \epsilon _2]\), there exist \(k_*^+>k_*^-> 0\) such that \( c^+(k_*^+)=U^+(h_+), c^+(k_*^-)=U^-(0)\).

    1. (i)

      If \((U^+)''<0\), then \(c^+(k)\) can be extended as an even \(C^{1, \alpha }\) function (\(\forall \alpha \in [0, 1)\)) for all \(k\in \mathbb {R}\) and analytic in k except at \(k=\pm k_*^\pm \) such that

      $$\begin{aligned}&c_I^+(k)>0, \quad c_R^+(k)\in \big (U^+(0), U^+(h_+)\big ), \; \forall |k|\in (k_*^-, k_*^+),\\&c^+(k)>U^+(h_+), \; \forall |k|\in (k_*^+, \infty ), \quad \\&c^+(k)\in \big ( U^-(0), U^+(0)\big ), \; \forall |k|\in [0, k_*^-). \end{aligned}$$
    2. (ii)

      If \((U^+)''>0\), then \(c^+(k)\) can be extended as an even \(C^{1, \alpha }\) real valued function (\(\forall \alpha \in [0, 1)\)) for \(|k|\ge k_*^+\) and analytic in k if \(k\ne \pm k_*^+\). And \(c^+(k)>U^+(h_+)\) for all \(|k|>k_*^+\). In addition, there exists an even real function \(\mathcal {C}(k)\) which is \(C^{1, \alpha }\) in k for all \(|k|\le k_*^-\) and analytic in k except at \(k=\pm k_*^-\) such that \(\mathcal {C}(\pm k_*^-)=U^+(0)\). And \(\mathcal {C}(k)\in \big ( U^-(0), U^+(0)\big )\) for all \(|k|<k_*^-\).

    Moreover, for \(c_R>U^-(0)\), all the singular and non-singular modes are \(c^+(k)\) and \(\mathcal {C}(k)\).

Remark 1.3

If there exists \(\gamma >0\) such that \(\gamma <\theta \min \{U^+(h_+)-U^+(0), U^+(0)-U^-(0)\}\) for any \(\theta \in (0, 1)\) and \(c_0\in [U^-(0)+\gamma , U^+(0)]\cup \big (U^+(0)+\gamma , U^+(h_+)]\cup [U^+(h_+)+\gamma , \infty \big )\), then \(\epsilon _0\) in (1), \(\epsilon _1\) in (2), and \(\epsilon _2\) in (3) can be the same and independent of location of \(c_0\).

For this special case, we also study the behavior of the other branch \(c^-(k)\). It is related to the relative value of g and a gravitational strength \(g_\#\). If \(g>g_\#\), for small density ratio \(\frac{\rho ^+}{\rho ^-}\), \(c^-(k)\) always stays below the range of U. In the case of \(g<g_\#\), for each small density ratio \(\frac{\rho ^+}{\rho ^-}\), whether instability happens at \(U^-(-h_-)\) depends on the signs of \((U^-)''\). If \(U^-\) is strictly convex, as k getting small from infinity, \(c^-(k)\) bifurcates through \(U^-(-h_-)\) into a pair of branches which are symmetric about the real axis in the complex plane, and then leaves the semicircle and goes back to the real line through \(U^-(-h_-)\). If \(U^-\) is concave, there is no bifurcation at \(U^-(-h_-)\). Thus, no instability occurs near \(c=U^-(-h_-)\).

Theorem 1.6

Assume that \(U^\pm \) satisfies (1.3), \((U^\pm )'>0\), \(U^-(0)<U^+(0)\), \((U^\pm )''\ne 0\), \(\eta \ne 0\), and (1.14) holds. Then, there exists \(g_\#>0\) such that

  1. (1)

    If \(g>g_\#\), there exists \(\epsilon _3>0\) such that for any \(\frac{\rho ^+}{\rho ^-}\in (0, \epsilon _3]\), \(c^-(k)\) can be extended to be an even and analytic function for all \(k\in \mathbb {R}\) and \(c^-(k)<U^-(-h_-)\). Moreover, it is the only singular and non-singular mode for \(c_R\le U^-(0)\).

  2. (2)

    If \(g\in (0, g_\#)\), there exists \(\epsilon _4>0\) such that for any \(\epsilon \in (0, \epsilon _4]\), there exist \(k_\#^+>k_\#^->0\) such that \(c^-(\pm k_\#^\pm )=U^-(-h_-)\). In addition,

    1. (i)

      If \((U^-)''>0\), then \(c^-(k)\) can be extended as an even \(C^{1, \alpha }\) function (for any \(\alpha \in [0, 1)\)) for all \(k\in \mathbb {R}\) and analytic in k except at \(k=\pm k_\#^\pm \) such that

      $$\begin{aligned}&c^-(k)<U^-(-h_-), \; \forall |k|\in (k_\#^+, \infty )\cup [0, k_\#^-), \quad \\&\quad c_I^-(k)>0, \; c_R^-(k)>U^-(-h_-), \; \forall |k|\in (k_\#^-, k_\#^+). \end{aligned}$$
    2. (ii)

      If \((U^-)''<0\), then \(c^-(k)\) can be extended as an even \(C^{1, \alpha }\) real valued function (for any \(\alpha \in [0, 1)\)) for \(|k|\ge k_\#^+\), analytic in k except at \(k=\pm k_\#^+\) and \(c^-(k)<U^-(-h_-)\) for all \(|k|>k_\#^+\). There exists a \(C^{1, \alpha }\) function \(\tilde{C}(k)\) on \([0, k_\#^-]\) and analytic except at \(k=\pm k_\#^-\) such that \(\tilde{C}(k)<U^-(-h_-)\) for all \(|k|\in [0, k_\#^-)\).

    Moreover, for \(c_R\le U^-(0)\), all singular and non-singular modes are \(c^-(k)\) and \(\tilde{C}(k)\).

The hypothesis on the regularity of \(U^\pm \) in Theorem 1.2 is different than the ones in Theorems 1.4–1.6. In Theorem 1.2, \(c^\pm (k)\) correspond to non-singular modes and are not in the range of \(U^\pm \). In this case, the classical Rayleigh equation (3.6) has no singularity. Thus, \(c^\pm (k)\) can be extended without any further assumption on \(U^\pm \). However, since \(c^\pm (k)\) may enter the range of \(U^\pm \) in Theorems 1.4–1.6 and the bifurcation equation looses the regularity, Lemmas 2.5 and 2.6 imply that more regularity of \(U^\pm \) is needed to obtain the regularity of the crucial quantity \(Y^\pm \) in (2.13) which is essential to extend \(c^\pm (k)\).

1.4 Outline of the Proof

The proof of Theorem 1.1 is presented in Sect. 2. To prove Theorem 1.2, we first extend some results in [11] of the solution \(y^-(c, k, x_2)\) of the classical Rayleigh equation with boundary conditions (2.12b) to another solution \(y^+(c, k, x_2)\) with boundary conditions (2.12a). Correspondingly, we also extend the results of the crucial quantity \(Y^-(c, k)= \frac{(y^-)'(0)}{y^-(0)}\) in [11] to \(Y^+(c, k, x_2)=\frac{(y^+)'(0)}{y^+(0)}\). The free boundary condition on the interface (2.15) gives the relation between wave speed c and wave number k. It can be written in terms of \(Y^\pm (c, k)\) which are well defined if both \((U^+)''\ne 0\) and \((U^-)''\ne 0\) or |k| is large. This allows us to analyze the function \(F(c, k, \epsilon )\) in (3.1) based on the properties of \(Y^\pm (c, k)\). The zeros of \(F(c, k, \epsilon )\) correspond to the eigenvalues of the linearized system (1.5). To discuss the eigenvalue distribution, we treat the wave number k as a parameter and start with large |k|. We apply the Contraction Mapping Theorem on the region outside the range of \(U^\pm \) to prove statement (1) and (2)(See Lemma 3.2). Finally, after ruling out all the possible singular modes under certain conditions, we prove that without singular modes, the two branches wave speed \(c^\pm (k)\) can be extended for all \(k\in \mathbb {R}\) in statement (3). This follows directly from Lemma 3.5.

To prove Theorems 1.3 and 1.4, we study the properties of \(Y^\pm (c, k)\) and the explicit representation of \(y^+(c, 0)\)(resp. \(y^-(c, 0)\)) when \(c\notin U^+\big ((0, h_+)\big )\)(resp. \(c\notin U^-\big (-h_-, 0\big )\)). Thanks to the monotonicity of \(\partial _k Y^\pm \) and concavity of \(F(c, k, \epsilon )\) in \(k\ge 0\), we reduce the analysis to the case of \(k=0\). And then we apply a bifurcation analysis near \(c=a\) to study the behavior of \(c^-(k)\) locally. Finally, by applying a complex continuation argument of a holomorphic function, we extend \(c^-(k)\) as |k| getting small from infinity. The proofs of these two theorems follow from a series of Lemmas in Sect. 3.

To prove Theorems 1.5 and 1.6, we consider the interface problem as a perturbation of the one fluid free boundary problem. After overcoming the degeneracy at the bounds of \(U^\pm \), we prove the instability arising from these locations under a certain condition by applying the Implicit Function Theorem and Mean Value Theorem. Based on the eigenvalue distribution of the capillary gravity water wave linearized at monotone shear flows (see [11]), we apply a complex continuation argument of a holomorphic function and give a complete picture of eigenvalue distribution. The proofs of Theorem 1.5 and Theorem 1.6 are completed in Sect. 4.

2 Preliminary

To study the linear system (1.5), we consider the corresponding eigenvalue problem and seek eigenvalues and eigenfunctions in the form of (1.6). Using (1.6) in (1.5b), (1.5a) and (1.5d), we have

$$\begin{aligned} & \frac{1}{\rho ^\pm }(k^2-\partial _{x_2}^2)p^\pm (x_2)=2ik(U^\pm )'(x_2)v_2^\pm (x_2), \end{aligned}$$
(2.1a)
$$\begin{aligned} & ik\big (U^\pm (x_2)-c\big )v_2^\pm (x_2)+\frac{1}{\rho ^\pm }(p^\pm )'(x_2)=0, \end{aligned}$$
(2.1b)
$$\begin{aligned} & p^+(0)-p^-(0)=g\eta (\rho ^+-\rho ^-)-\sigma \eta k^2. \end{aligned}$$
(2.1c)

We combine (2.1a) acted by \(\partial _{x_2}\) and (2.1b) acted by \(k^2-\partial _{x_2}^2\) to obtain the following classical Rayleigh equations:

$$\begin{aligned} -(v_2^\pm )''(x_2)+\big (k^2+\frac{(U^\pm )''(x_2)}{U^\pm (x_2)-c}\big )v_2^\pm (x_2)=0, \qquad \pm x_2\in [0, h_\pm ]. \end{aligned}$$
(2.2a)

From (1.5e) and (1.5c), we obtain that

$$\begin{aligned} & v_2^\pm (\pm h_\pm )=0 \end{aligned}$$
(2.2b)
$$\begin{aligned} & ik\eta \big (U^\pm (0)-c\big )=v_2^\pm (0). \end{aligned}$$
(2.2c)

Adding (2.1b) acted on by \(\partial _{x_2}\) and (2.1a), then evaluating at \(x_2=0\) and combining with the last equation, we obtain the boundary condition at the interface:

$$\begin{aligned} ik\eta \big (g(\rho ^+-\rho ^-)-\sigma k^2\big )= & \rho ^-(U^-)'(0)v_2^-(0)-\rho ^+ (U^+)'(0)v^+_2(0)\nonumber \\ & +\rho ^+ (v_2^+)'(0)\big (U^+(0)-c\big )\nonumber \\ & -\rho ^-(v_2^-)'(0)\big (U^-(0)-c\big ). \end{aligned}$$
(2.2d)

Remark 2.1

To seek unstable solutions, we consider \(k\ne 0\) in (1.6) because zeroth mode corresponds to variation nearby shear flows, but we may still study the solution of system (2.2) with \(k=0\) since it is helpful for getting a picture of the eigenvalue distribution for all \(k\in \mathbb {Z}\).

We are interested in looking for linearized unstable solutions with \(\eta \ne 0\) and \(k\ne 0\). Without loss of generality, we normalize that by taking

$$\begin{aligned} ik\eta =1. \end{aligned}$$

Meanwhile, we let \(\epsilon :=\frac{\rho ^+}{\rho ^-}\) be the density ratio. Then we can rewrite (2.2c) and (2.2d) as

$$\begin{aligned} & v_2^\pm (0)=U^\pm (0)-c, \end{aligned}$$
(2.3a)
$$\begin{aligned} & g(\epsilon -1)-\frac{\sigma }{\rho ^-} k^2= (U^-)'(0)v_2^-(0)-\epsilon (U^+)'(0)v^+_2(0)+ \epsilon (v_2^+)'(0)\big (U^+(0)-c\big )\nonumber \\ & \quad -(v_2^-)'(0)\big (U^-(0)-c\big ). \end{aligned}$$
(2.3b)

We summarize the above calculation in the following:

Lemma 2.1

For \(k\in \mathbb {Z}\setminus \{0\}\), \(-ikc\) with \(c\notin U^-([-h_-, 0])\cup U^+([0, h_+])\) is an eigenvalue of the linearized Euler system (1.5) at the shear flow \(v_*^\pm =(U^\pm (x_2), 0)\) if there exists a non-trivial solution of (2.2a), (2.2b), and (2.3).

Remark 2.2

The solutions to system (2.2a), (2.2b), and (2.3) are even in k. In addition, the complex conjugate of solutions are still solutions and c is replaced by \(\bar{c}\).

Using the symmetry in Remark 2.2, we notice that the existence of a solution to system (2.2a), (2.2b), and (2.3) with \(c_I>0\) and \(k>0\) implies an exponential linear instability of Euler system. In the rest of this paper, k is assumed to be non-negative mostly. To study instability, we introduce some concepts as follows:

Definition 2.1

A pair (c, k) with real c, positive k is said to be a neutral mode if there exists a pair of non-trivial solutions to (2.2a), (2.2b), and (2.3). This is called an unstable mode if \(c_I>0\). We call it a neutral limiting mode if it is the limit of a sequence of unstable modes \(\{(c_n, k_n)\}\). Furthermore, if \(c\in U^+\big ([0, h_+]\big )\cup U^-(\big [-h_-, 0]\big )\), it is called a singular mode. Otherwise, it is a non-singular mode.

In the following, we prove Theorem 1.1 which shows that any unstable mode must have wave speed in an upper semicircle under a mild condition.

Proof of Theorem 1.1

Suppose \(c_I\ne 0\). Let \(\psi ^\pm (x_2)\) be defined by \(v_2^\pm (x_2)=\big (c-U^\pm (x_2)\big )\psi ^\pm (x_2)\). Then \(\psi ^\pm (x_2)\) satisfy

$$\begin{aligned} \Big (\big (U^\pm (x_2)-c\big )^2(\psi ^\pm ) '(x_2)\Big )'(x_2)-k^2\big (U^\pm (x_2)-c\big )^2\psi ^\pm (x_2)=0, \end{aligned}$$
(2.4)

along with

$$\begin{aligned} & \psi ^\pm (\pm h_\pm )=0, \qquad \psi ^\pm (0)=-1, \end{aligned}$$
(2.5)
$$\begin{aligned} & -g(\rho ^+-\rho ^-)+\sigma k^2=\rho ^+\big (U^+(0)-c\big )^2(\psi ^+)'(0)-\rho ^-\big (U^-(0)-c\big )^2(\psi ^-)'(0).\nonumber \\ \end{aligned}$$
(2.6)

We multiply (2.4) by \(\overline{\psi ^\pm }\) and integrate it from 0 to \(\pm h_\pm \) respectively to obtain that

$$\begin{aligned} & -\big (U^-(0)-c\big )^2(\psi ^-)'(0)=\int _{-h_-}^0 (U^--c)^2(k^2|\psi ^-|^2+|(\psi ^-)'|^2)\textrm{d}x_2,\qquad \quad \end{aligned}$$
(2.7a)
$$\begin{aligned} & \big (U^+(0)-c\big )^2(\psi ^+)'(0)=\int _0^{h_+}(U^+-c)^2 (k^2|\psi ^+|^2+|(\psi ^+)'|^2)\textrm{d}x_2. \end{aligned}$$
(2.7b)

Let \(Q^\pm := k^2|\psi ^\pm |^2+|(\psi ^\pm )'|^2\). Then \(Q^\pm >0\). We consider the imaginary parts of (2.7a) and (2.7b).

$$\begin{aligned} & {\text {Im}}\Big (-\big (U^-(0)-c\big )^2(\psi ^-)'(0)\Big )=-2c_I\int _{-h_-}^0 (U^--c_R)Q^-\textrm{d}x_2, \end{aligned}$$
(2.8a)
$$\begin{aligned} & {\text {Im}} \Big (\big (U^+(0)-c\big )^2(\psi ^+)'(0)\Big )=-2c_I\int _0^{h_+} (U^+-c_R)Q^+\textrm{d}x_2. \end{aligned}$$
(2.8b)

Combining (2.8a) multiplied by \(\rho ^-\) and (2.8b) multiplied by \(\rho ^+\), then using (2.6), we obtain that

$$\begin{aligned} & \int _0^{h_+}\rho ^+ U^+ Q^+\textrm{d}x_2+\int _{-h_-}^0 \rho ^- U^- Q^-\textrm{d}x_2\nonumber \\ & \qquad =c_R(\int _0^{h_+}\rho ^+ Q^+\textrm{d}x_2+\int _{-h_-}^0 \rho ^- Q^-\textrm{d}x_2), \end{aligned}$$
(2.9)

Similarly, we consider the real part of (2.7a) and (2.7b).

$$\begin{aligned} & {\text {Re}} \Big (-\big (U^-(0)-c\big )^2(\psi ^-)'(0)\Big )=\int _{-h_-}^0 \big ((U^--c_R)^2-c_I^2\big )Q^-\textrm{d}x_2,\qquad \quad \end{aligned}$$
(2.10a)
$$\begin{aligned} & {\text {Re}} \Big (\big (U^+(0)-c\big )^2(\psi ^+)'(0)\Big )=\int _0^{h_+} \big ((U^+-c_R)^2-c_I^2\big )Q^+\textrm{d}x_2. \end{aligned}$$
(2.10b)

Adding (2.10b) multiplied by \(\rho ^+\) and (2.10a) multiplied by \(\rho ^-\), then using (2.6), we obtain that

$$\begin{aligned}&\int _0^{h_+}U^{+2}\rho ^+Q^+\textrm{d}x_2+\int _{-h_-}^0 U^{-2}\rho ^-Q^-\textrm{d}x_2\nonumber \\&\quad =(c_R^2+c_I^2) (\int _0^{h_+}\rho ^+ Q^+\textrm{d}x_2+\int _{-h_-}^0 \rho ^- Q^-\textrm{d}x_2)\nonumber \\&\qquad -g(\rho ^+-\rho ^-)+\sigma k^2, \end{aligned}$$
(2.11)

where we used equation (2.9) and real part of (2.6). Now we consider the inequality

$$\begin{aligned} \int _0^{h_+}\rho ^+ Q^+(U^+-a)(U^+-b)\textrm{d}x_2+\int _{-h_-}^0 \rho ^- Q^-(U^--a)(U^--b)dx_2\le 0, \end{aligned}$$

where a and b are defined in (1.7). Substituting (2.9) and (2.11), we have

$$\begin{aligned}&\big (c_R^2+c_I^2-(a+b)c_R+ab\big ) \left( \int _0^{h_+}\rho ^+ Q^+dx_2+\int _{-h_-}^0 \rho ^- Q^-\textrm{d}x_2\right) \\&\quad \le -\big (g(\rho ^--\rho ^+)+\sigma k^2\big )\le 0. \end{aligned}$$

Since \(Q^\pm \ge 0\) and can not be both identically zero, \((c_R-\frac{a+b}{2})^2+c_I^2\le (\frac{a-b}{2})^2\). \(\square \)

To study the instability, we will address the possible locations of the limit of a sequence of unstable solutions to (2.2a), (2.2b), and (2.3), i.e. neutral limiting mode. In fact, not every neutral mode serves as a neutral limiting mode. First, we study the properties of the fundamental solutions of classical Rayleigh equations. Some results about the fundamental solutions were obtained in [11]. Let \(y^+(c, k, x_2)\) be the solution to (2.2a) on \([0, h_+]\) with the boundary conditions

$$\begin{aligned} y^+(h_+)=0, \quad (y^+)'(h_+)=1. \end{aligned}$$
(2.12a)

Suppose that \(y^-(c, k, x_2)\) be the solution to (2.2a) on \([-h_-, 0]\) satisfying

$$\begin{aligned} y^-(-h_-)=0, \quad (y^-)'(-h_-)=1. \end{aligned}$$
(2.12b)

When \(c\in U^+\big ([0, h^+]\big )\) (or \(U^-\big ([-h_-, 0]\big )\)), \(y^+(c, k, x_2)\) (resp. \(y^-(c, k, x_2)\)) is defined as \(\lim _{\theta \rightarrow 0^+}y^+(c+i\theta , k, x_2)\) (resp. \(\lim _{\theta \rightarrow 0^+} y^-(c+i\theta , k, x_2)\)). This satisfies the following condition for \( U^+(x_2)\ne c\) (resp. \(U^-(x_2)\ne c\)):

$$\begin{aligned} \lim _{\theta \rightarrow 0+}\big ( (y_R^+)'(c, k, x_c+\theta )-(y_R^+)'(c, k, x_c-\theta )\big ) =0, \\ \lim _{\theta \rightarrow 0+}(y_I^+)'(c, k, x_c-\theta )=-\pi \frac{(U^+)''(x_c)}{|(U^+)'(x_c)|}y^+(x_c). \end{aligned}$$

Here \( x_c=(U^+)^{-1}(c)\) is the preimage of \(c\in U^+([0, h_+])\) (resp. \(\lim _{\theta \rightarrow 0+}\big ((y_R^-)'(c, k, x_c+\theta )-(y_R^-)'(c, k, x_c-\theta )\big )=0,\; \lim _{\theta \rightarrow 0+}(y_I^-)'(c, k, x_c+\theta )=\pi \frac{(U^-)''(x_c)}{|(U^-)'(x_c)|}y^-(x_c)\), \(x_c=(U^-)^{-1}(c)\)). Moreover, because of the singularity of (2.2a), \((y_R^\pm )'(x_2)\) has a logarithmic singularity \(\log |x_2-x_c|\) near \(x_c\).

We also define some crucial quantities which are related to the Reynold stress. We let

$$\begin{aligned} Y^-(c, k)= & Y_R^-(c, k)+iY_I^-(c, k):=\frac{(y^-)'(0)}{y^-(0)}(c, k), \quad c\in \mathbb {C}\setminus U^-\big ([-h_-, 0]\big ), \nonumber \\ Y^-(c, k)= & \lim _{\beta \rightarrow 0+}Y^-(c+i\beta , k) \qquad c\in U^-\big ([-h_-, 0)\big ), \end{aligned}$$
(2.13a)
$$\begin{aligned} Y^+(c, k)= & Y_R^+(c, k)+iY_I^+(c, k):=\frac{(y^+)'(0)}{y^+(0)}(c, k), \quad c\in \mathbb {C}\setminus U^+\big ([0, h_+]\big ), \nonumber \\ Y^+(c, k)= & \lim _{\beta \rightarrow 0+}Y^+(c+i\beta , k) \qquad c\in U^+\big ((0, h_+]\big ), \end{aligned}$$
(2.13b)

One notice that \(Y^-\) (or \(Y^+\)) is not well defined at \(c=U^-(0)\) (resp. \(c=U^+(0)\)) because of the singularity of (2.2a) at \(x_2=0\). The domain of \(Y^\pm \) is given by

$$\begin{aligned} D(Y^\pm ):=\{(c, k)\in \mathbb {C}\times \mathbb {R} \mid c\ne U^\pm (0), y^\pm (c, k, 0)\ne 0\}. \end{aligned}$$
(2.14)

In \(D(Y^\pm )\), we can simplify the free boundary condition of our eigenvalue problem (2.3b) as

$$\begin{aligned} & g(1-\epsilon )+\frac{\sigma }{\rho ^-}k^2=\epsilon \Big ((U^+)'(0)\big (U^+(0)-c\big )-Y^+\big (U^+(0)-c)^2\Big )\nonumber \\ & \quad -(U^-)'(0)\big (U^-(0)-c\big )+Y^-\big (U^-(0)-c\big )^2. \end{aligned}$$
(2.15)

This allows us to focus on the system consisting of (2.2a), (2.12), and (2.15). We summarize the above analysis in the following lemma:

Lemma 2.2

The linearization of the interface problem (1.1) at the shear flow (1.2) has an eigenvalue \(-ikc\) if the solutions \(y^\pm (c, k, x_2)\) of (2.2a) satisfy (2.12) and the pair (c, k) satisfies (2.15).

We notice that when \(c\in U^+(0, h_+)\cup U^-(-h_-, 0)\), \(x_2\) which satisfies \(U^+(x_2)=c\) or \(U^-(x_2)=c\) plays an important role in the analysis since Rayleigh equation has singularity near such \(x_2\). For convenience, we introduce notations \(x_c^\pm \) to denote such \(x_2\).

Definition 2.2

If \(c\in U^-\big ([-h_-, 0] \big )\), let \(x_c^-\) be \(c=U^-(x_c^-)\). If \(c\in U^+\big ([0, h_+] \big )\), define \(x_c^+\) as \(c=U^+(x_c^+)\).

Remark 2.3

It is possible that \(c\in U^-\big ([-h_-, 0] \big )\cap U^+\big ([0, h_+] \big )\ne \emptyset \). In this case, \(c=U^+(x_c^+)=U^-(x_c^-)\), but \(x_c^+\) is not equal to \(x_c^-\).

To study the eigenvalue problem (2.2a) together with (2.11) and (2.15), it is essential to understand the behavior of the fundamental solutions to the classical Rayleigh equation. The asymptotic behavior of \(y^-\) and its regularity as \(x_2\) near \(x_c^-\) were studied in [11]. We cite the results as follows:

Lemma 2.3

( [11], Lemma 3.9, Lemma 3.19(1),(3)) Assume that \(U^-\in C^{l_0}, l_0\ge 6\) and \((U^-)'\ne 0\). Then the following hold:

  1. (1)

    For any \(\beta \in (0, \frac{1}{2})\), there exists \(C>0\) depending only on \(\beta , |(U^-)'|_{C^2}\), and \(|1/(U^-)'|_{C^0}\) such that, for any \(c\in \mathbb {C}\),

    $$\begin{aligned} & |y^-(x_2)\sqrt{k^2+1}-\sinh \big (\sqrt{k^2+1}(x_2+h_-)\big )|\nonumber \\ & \quad \le C(k^2+1)^{-\frac{\beta }{2}}\sinh \big (\sqrt{k^2+1}(x_2+h_-)\big ). \end{aligned}$$
    (2.16)
  2. (2)

    For any \(k\in \mathbb {R}\),

    $$\begin{aligned} & y^-(c, k, x_2)>0, \; \forall x_2\in (-h, 0], \; c\in \mathbb {R}\setminus U^-\big ((-h_-, 0) \big ),\quad \nonumber \\ & \quad y^-(x_c^-)> 0 \; \text {if}\; c\in U^-\big ((-h_-, 0]\big ) \end{aligned}$$
    (2.17)
  3. (3)

    There exists \(C>0\) depending only on \(U^-\) such that for any \(k\in \mathbb {R}\),

    $$\begin{aligned} & |y^-_I(c, k, 0)|\ge \frac{C}{k^2+1}|(U^-)''(x_c^-)|\sinh \big (\sqrt{k^2+1}(x_c^-+h_-)\big )\nonumber \\ & \quad \sinh \big (\sqrt{k^2+1}|x_c^-|\big ), \quad c\in U^-\big ([-h_-, 0]\big ), \end{aligned}$$
    (2.18)

One may follow the proof in [11] but consider the velocity profile \(U^+\) and boundary condition (2.12a) to get the following analogous properties for \(y^+(c, k, x_2)\):

Lemma 2.4

Assume that \(U^+\) satisfies (1.3). Then,

  1. (1)

    For any \(\beta \in (0, \frac{1}{2})\), there exists \(C>0\) depending only on \(\beta , |(U^+)'|_{C^2}\), and \(|1/(U^+)'|_{C^0}\) such that, for any \(c\in \mathbb {C}\setminus U^+([0, h_+])\),

    $$\begin{aligned} & |y^+(x_2)\sqrt{k^2+1}-\sinh \big (\sqrt{k^2+1}(x_2-h_+)\big )|\le C(k^2+1)^{-\frac{\beta }{2}}\nonumber \\ & \quad \sinh \big (\sqrt{k^2+1}(h_+-x_2)\big ). \end{aligned}$$
    (2.19)
  2. (2)

    For any \(k\in \mathbb {R}\),

    $$\begin{aligned} & y^+(c, k, x_2)<0, \; \forall x_2\in [0, h_+), \; c\in \mathbb {R}\setminus U^+\big ((0, h_+) \big ), \quad \nonumber \\ & \quad y^+(x_c^+)< 0 \; \text {if} \; c\in U^+\big ([0, h_+)\big ). \end{aligned}$$
    (2.20)
  3. (3)

    There exists \(C>0\) depending only on \(U^+\) such that for any \(k\in \mathbb {R}\),

    $$\begin{aligned} & |y^+_I(c, k, 0)|\ge \frac{C}{k^2+1}|(U^+)''(x_c^+)|\sinh \big (\sqrt{k^2+1}(h_--x_c)\big )\nonumber \\ & \quad \sinh \big (\sqrt{k^2+1}|x_c^+|\big ), \quad c\in U^+\big ([0, h_+]\big ). \end{aligned}$$
    (2.21)

From (2.15), one may notice that \(Y^\pm \) related to the Reynolds stress are crucial quantities for linearized problem, especially for the bifurcation analysis of eigenvalues. The basic properties of \(Y^-(c, k)\) were obtained in [11]. We cite some of them which will be used later in the next lemma.

Lemma 2.5

( [11], Lemma 3.20(3), Lemma 3.22, Lemma 3.24) Assume that \(U^-\in C^{l_0}, l_0\ge 6\) and \((U^-)'\ne 0\). Then \(Y^-\) is analytic in both \((c, k)\in D(Y^-){\setminus } \big (U^-([-h_-, 0]\times \mathbb {R}) \big )\), \(C^{l_0-3}\) in k, and locally \(C^\alpha \) in \((c, k)\in D(Y^-)\cap \{(c, k) \mid c_I\ge 0\}\) for any \(\alpha \in [0, 1)\). Then,

  1. (1)

    There exists \(C, \rho >0\) depending only on \(U^-\) such that

    $$\begin{aligned} & |Y^-(c, k)|\le C\big (\sqrt{1+k^2}+\big |\log \min \{1, |U^-(0)-c|\}\big | \big ), \quad \nonumber \\ & \quad \forall k\in \mathbb {R}, \; |c-U^-(0)|\le \rho . \end{aligned}$$
    (2.22)

    For any \(\beta \in (0, \frac{1}{2})\), there exist \(k_0>0\) and \(C>0\) depending only on \(\beta \), \(|(U^-)'|_{C^2}\), and \(|\frac{1}{(U^-)'}|_{C^0}\) such that,

    $$\begin{aligned} & |Y^-(c, k)- k\coth (kh_-)|\le C\big ((k^2+1)^{\frac{1-\beta }{2}}+|\log \min \{1, |U^-(0)-c|\}| \big ), \quad \nonumber \\ & \quad \forall |k|\ge k_0, c\ne U^-(0). \end{aligned}$$
    (2.23)
  2. (2)

    \(Y_I^-(c, k)=0\) for \(c\in \mathbb {R}{\setminus } U^-\big ((-h_-, 0] \big )\). If \(y^-(c, k, 0)\ne 0\),

    $$\begin{aligned} & Y_I^-(c, k)=\frac{\pi (U^-)''(x_c^-)y^-(c, k, x_c^-)^2}{|(U^-)'(x_c^-)||y^-(c, k, 0)|^2}, \quad c\in U^-\big ((-h_-, 0) \big ). \end{aligned}$$
    (2.24)
    $$\begin{aligned} & Y^-(c, k)\nonumber \\ & \quad = {\left\{ \begin{array}{ll} \frac{1}{\pi }\int _{U^-([-h_-, 0])}\frac{Y^-_I(c', k)}{c-c'}dc'+k\coth (kh_-), \quad c\notin U^-\big ([-h_-, 0]\big ), \\ -\mathcal {H}\big (Y^-_I(\cdot , k) \big )(c)+iY^-_I(c, k)+k\coth (kh_-), \quad c\in U^-\big ([-h_-, 0]\big ). \end{array}\right. }\nonumber \\ \end{aligned}$$
    (2.25)

    here \(\mathcal {H}\) means Hilbert transform in \(c\in \mathbb {R}\).

    $$\begin{aligned} \mathcal {H}(f)(c)=\frac{1}{\pi }P.V.\int _{\mathbb {R}}\frac{f'(c')}{c-c'}\textrm{d}c', \end{aligned}$$

    where \(P.V.\int \) denotes the principle value of the singular integral.

Similarly, we follow the proof in [11] to obtain the standard properties of \(Y^+(c, k)\) as following. These properties of \(Y^\pm (c, k)\) will be used very often in the asymptotic analysis and bifurcation analysis in the next two sections.

Lemma 2.6

Assume that \(U^+\) satisfies (1.3). Then \(Y^+\) is \(C^{l_0-3}\) in k and locally \(C^\alpha \) in \((c, k)\in D(Y^+)\cap \{(c, k) \mid c_I\ge 0\}\) for any \(\alpha \in [0, 1)\) and analytic in \((c, k)\in D(Y^+){\setminus } \big (U^+([0, h_+])\times \mathbb {R} \big )\). The following hold:

  1. (1)

    There exists \(C, \rho >0\) depending only on \(U^+\) such that

    $$\begin{aligned} & |Y^+(c, k)|\le C\big (\sqrt{1+k^2}+\big |\log \min \{1, |U^+(0)-c|\}\big | \big ), \quad \nonumber \\ & \quad \forall k\in \mathbb {R}, \; |c-U^+(0)|\le \rho . \end{aligned}$$
    (2.26)

    For any \(\beta \in (0, \frac{1}{2})\), there exist \(k_0>0\) and \(C>0\) depending only on \(\beta \), \(|(U^+)'|_{C^2}\), and \(|\frac{1}{(U^+)'}|_{C^0}\) such that,

    $$\begin{aligned} & |Y^+(c, k)+ k\coth (kh_+)|\le C\big ((k^2+1)^{\frac{1-\beta }{2}}+|\log \min \{1, |U^+(0)-c|\}| \big ), \quad \nonumber \\ & \quad \forall |k|\ge k_0, c\ne U^+(0). \end{aligned}$$
    (2.27)
  2. (2)

    For \(c\in \mathbb {R}{\setminus } U^+\big ([0, h_+) \big )\), \(Y_I^+(c, k)=0\). If \(y^+(c, k, 0)\ne 0\),

    $$\begin{aligned} & Y_I^+(c, k)=-\frac{\pi (U^+)''(x_c^+)y^+(c, k, x_c^+)^2}{|(U^+)'(x_c^+)||y^+(c, k, 0)|^2}, \quad c\in U^+\big ((0, h_+) \big ). \end{aligned}$$
    (2.28)
    $$\begin{aligned} & Y^+(c, k)\nonumber \\ & \quad = {\left\{ \begin{array}{ll} \frac{1}{\pi }\int _{U^+([0, h_+])}\frac{Y^+_I(c', k)}{c'-c}dc'-k\coth (kh_+), \quad c\notin U^+\big ([0, h_+]\big ), \\ -\mathcal {H}\big (Y^+_I(\cdot , k) \big )(c)+iY^+_I(c, k)-k\coth (kh_+), \quad c\in U^+\big ([0, h_+]\big ). \end{array}\right. }\nonumber \\ \end{aligned}$$
    (2.29)

Remark 2.4

  1. 1).

    (2.17) and (2.20) still hold without assuming \((U^\pm )'\ne 0\).

  2. 2).

    For \(c\in U^-\big ((-h_-, 0)\big )\), if \(Y^-(c, k)\) is defined as \(\lim _{\beta \rightarrow 0+}Y^-(c-i\beta , k)\), then \({{\,\textrm{sgn}\,}}(Y^-_I)\) is different, i.e. \(-Y^-_I\) satisfies (2.24). \(Y_I^+\) has an analogous property. In this paper, we shall only use the properties of \(Y^\pm _I\) for \(c_I\ge 0\). Based on (2.18), \(Y^-\) is well defined for \(c\in U^-\big ((-h_-, 0)\big )\) under the assumption that \((U^-)''\ne 0\). According to (2.21), \(Y^+\) is well defined for \(c\in U^+\big ((0, h_+)\big )\) if \((U^+)''\ne 0\).

3 Distribution of Eigenvalues

In this section, we shall consider the situation where the upper layer fluid is lighter than the lower layer fluid. In other words, we shall discuss the linear instability of shear flows (1.2) which satisfy (1.3) for \(0\le \rho ^+\le \rho ^-\). Some of the results are also true for large density ratio \(\epsilon =\frac{\rho ^+}{\rho ^-}\). Since the linear system (1.5) preserves Fourier mode \(e^{ikx_1}\) for any \(k\in \mathbb {R}\), we will treat the wave number \(k\in \mathbb {R}\) as a parameter. According to Lemma 2.1, \(-ikc\) with \(c\in \mathbb {C}{\setminus } \Big (U^-\big ([-h_-, 0] \big )\cup U^+\big ([0, h_+] \big )\Big )\) is an eigenvalue of the linear system (1.5) with parameter k if

$$\begin{aligned} & F(c, k, \epsilon ):= \epsilon \Big ((U^+)'(0)\big (U^+(0)-c\big ) -Y^+\big (U^+(0)-c\big )^2\Big )\nonumber \\ & \quad -(U^-)'(0)\big (U^-(0)-c\big )\nonumber \\ & \quad +Y^-\big (U^-(0)-c\big )^2 -\big (g(1-\epsilon )+\frac{\sigma }{\rho ^-}k^2\big ), \end{aligned}$$
(3.1a)

equals zero. Here, \(Y^\pm \) are defined in (2.13). According to (2.22), (2.24), (2.26), and (2.28), we also define that

$$\begin{aligned} & F(c, k, \epsilon ):=\lim _{\beta \rightarrow 0+}F(c+i\beta , k, \epsilon )=\overline{\lim _{\beta \rightarrow 0+}F(c-i\beta , k, \epsilon )}, \quad \nonumber \\ & \quad \forall c\in U^-\big ([-h_-, 0)\big )\cup U^+\big ((0, h_+]\big ). \end{aligned}$$
(3.1b)

Then, the zeros of \(F(c, k, \epsilon )\) correspond to singular or non-singular mode (c, k) which is defined in Definition 2.1. Based on Lemma 4.1 in [11], Lemma 2.5, and Lemma 2.6, we have some basic properties of \(F(c, k, \epsilon )\) for each \(\epsilon \ge 0\).

Lemma 3.1

Assume \(U^\pm \) satisfy (1.3). Then the following hold:

  1. (1)

    \(F(c, k, \epsilon )\) is well defined for all \(k\in \mathbb {R}\), \(c\in \mathbb {C}\), and \(\epsilon \ge 0\).

  2. (2)

    When restricted to \(c_I\ge 0\), \(F(c, k, \epsilon )\) is \(C^{l_0-3}\) in k, \(C^{1, \alpha }\) in c, for any \(\alpha \in [0, 1)\), and \(C^\infty \) in \(\epsilon \ge 0\).

To study the eigenvalue distribution, we consider wave number k as a parameter. We first address the location of wave speed \(c\in \mathbb {C}\) for large |k|. This is also helpful to understand the location of c when |k| is small and thus get a picture of eigenvalue distribution.

Lemma 3.2

Assume that \(U^\pm \) satisfy (1.3). Then there exist some \(k_0>0\) and \(C>0\) depending only on \(|(U^\pm )'|_{C^2}\), and \(|1/(U^\pm )'|_{C^0}\), such that for each \(\epsilon \in [0, 1]\) and all \(|k|>k_0\), \(F(c, k, \epsilon )\) defined by (3.1) has exactly two solutions \(c^\pm (k)\) depending on k analytically. Moreover,

$$\begin{aligned} & c^\pm (k)\in \mathbb {R}, \; c^\pm (k)=c^\pm (-k), \end{aligned}$$
(3.2)
$$\begin{aligned} & \Big |c^\pm (k)-\frac{U^+(0)\epsilon +U^-(0)}{1+\epsilon }{\mp }\sqrt{-\frac{\epsilon \big (U^+(0)-U^-(0)\big )^2}{(1+\epsilon )^2}+\frac{\sigma |k|}{\rho ^-(1+\epsilon )}}\Big |\le C, \end{aligned}$$
(3.3)
$$\begin{aligned} & \big |\partial _c F\big (c^\pm (k), k, \epsilon \big ){\mp } 2|k|^{\frac{3}{2}}\sqrt{\sigma (1+\epsilon )/\rho ^-}\big |\le C|k|. \end{aligned}$$
(3.4)

Proof

Since \(\epsilon \in [0, 1]\), (1.8) holds for all \(k\in \mathbb {R}\). According to (2.23), and (2.27), \(Y^\pm (c, k)\) are comparable to \({\mp } |k|\) as |k| tends to \(\infty \). Then, for large |k|, \(F(c, k, \epsilon )\) behaves more or less like a quadratic function in k with a negative leading coefficient. Hence, there exist \(N_1, \gamma >0\) such that for all \(|k|>N_1\), if \( y^\pm (c, k, x_2)\) with \((c_R-\frac{a+b}{2})^2+c_I^2\le b+2\gamma -a\), solves the system (2.12a) and (2.12b), then \(F<-\frac{\sigma }{2\rho ^-}k^2<0\). Theorem 1.1 implies that if \(-ikc\) is an eigenvalue with c lying outside the semicircle (1.9), then \(c\in \mathbb {R}{\setminus } [a, b]\). Therefore, if \(\big (c, k, y^\pm (c, k, x_2)\big )\) with \(|k|>N_1\) solves the system (2.12a), (2.12b), and (3.1a), then \(c_R\in [a-\gamma , b+\gamma ]\), where a and b are defined in (1.7). Let

$$\begin{aligned} S_+:=\{c>b+\gamma \}, \quad S_-:=\{c<a-\gamma \}. \end{aligned}$$
(3.5)

By the definition of \(F(c, k, \epsilon )\), F is \(C^\infty \) in \((c, k)\in S_\pm \times \mathbb {R}\). For each \(\epsilon \in [0, 1]\), we consider \(F(c, k, \epsilon )=0\) as a quadratic equation of c; its roots satisfy

$$\begin{aligned} c=f^\pm (c, k)= \frac{-B\pm \sqrt{B^2-4\big (Y^-(c, k)-\epsilon Y^+(c, k)\big )A}}{2\big (Y^-(c, k)-\epsilon Y^+(c, k)\big )}, \end{aligned}$$

where

$$\begin{aligned} A:=&\epsilon \big ((U^+)'(0)U^+(0)-Y^+U^+(0)^2\big )-(U^-)'(0)U^-(0)+Y^-U^-(0)^2\\&-g(1-\epsilon )-\frac{\sigma }{\rho ^-}k^2,\\ B:=&(U^-)'(0)-2U^-(0)Y^--(U^+)'(0)\epsilon +2Y^+U^+(0)\epsilon . \end{aligned}$$

Using formulas (2.25) and (2.24) with estimate (2.16), we obtain that there exist \(C>0\) and large \(N_1>0\) such that for all \(c\in S_+\cup S_-\) and \(|k|>N_1\),

$$\begin{aligned}&|Y^-(c, k)-k\coth (kh_-)|=\frac{1}{\pi }\Big |\int _{U^-([-h_-, 0])} \frac{Y_I^-(c', k)}{c-c'}dc'\Big | \\&\quad \le C \int _{U^-[-h_-, 0]}\frac{y^-(c, k, x_c)^2}{|y^-(c, k, 0)|^2}dc \\&\quad \le C \int _{-h_-}^0 e^{\sqrt{k^2+1}x_c}dc\le C(|k|+1)^{-1}. \end{aligned}$$

A similar computation based on (2.29), (2.19), and (2.28) leads to

$$\begin{aligned} |Y^+(c, k)+k\coth (kh_+)|\le C(|k|+1)^{-1}. \end{aligned}$$

According to these two estimates and the fact that \(\coth (x)=1+2(e^{2x}-1)^{-1}\), there exists \(k_0>N_1\) and \(C>0\) depending only on \(|(U^\pm )'|_{C^2}\), and \(|1/(U^\pm )'|_{C^0}\), such that, for all \(|k|>k_0\),

$$\begin{aligned} \Big |f^\pm (c, k)-\frac{U^+(0)\epsilon +U^-(0)}{1+\epsilon }{\mp }\sqrt{-\frac{\epsilon \big (U^+(0)-U^-(0)\big )^2}{(1+\epsilon )^2}+\frac{\sigma |k|}{\rho ^-(1+\epsilon )}}\Big |\le C. \end{aligned}$$

Then, \(f^\pm (c, k):S_\pm \rightarrow S_\pm \) for \(|k|>k_0\). It remains to evaluate \(\partial _c f^\pm (c, k)\) and show \(f^\pm (c, k)\) is a contraction acting on \(S_\pm \). We first compute \(\partial _c Y^\pm \). While \(\partial _c Y^-\) was obtained and evaluated in [11], we compute \(\partial _c Y^\pm \) here for self-completeness.

We differentiate (2.2a) in \(c\in \mathbb {R}{\setminus } U^-([-h_-, 0])\) and \(c\in \mathbb {R}{\setminus } U^+([0, h_+])\), respectively,

$$\begin{aligned} {\left\{ \begin{array}{ll} -\partial _c (y^\pm )''(x_2)+\big (k^2+\frac{(U^\pm )''}{U^\pm -c}\big )\partial _c y^\pm (x_2)=-\frac{(U^\pm )''}{(U^\pm -c)^2}y^\pm (x_2), \quad \! \pm x_2\in (0, h_\pm ),\\ \partial _c y^\pm (\pm h_\pm )=\partial _c (y^\pm )'(\pm h_\pm )=0. \end{array}\right. }\nonumber \\ \end{aligned}$$
(3.6)

We combine (2.2a) multiplied by \(\partial _c y^\pm (x_2)\) and (3.6) multiplied by \(y^\pm (x_2)\) to obtain that

$$\begin{aligned} -(y^\pm )'' \partial _c y^\pm +\partial _c (y^\pm )'' y^\pm =\frac{(U^\pm )''}{(U^\pm -c)^2}(y^\pm )^2, \qquad \pm x_2\in (0, h_\pm ). \end{aligned}$$

Using the boundary conditions in (3.6), we integrate the above equation on \([-h_-, 0]\) and \([0, h_+]\), respectively, and obtain that

$$\begin{aligned} -(y^+)'(0)\partial _c y^+(0)+\partial _c (y^+)'(0)y^+(0)=-\int _0^{h_+} \frac{(U^+)''}{(U^+-c)^2}(y^+)^2 \textrm{d}x_2,\\ -(y^-)'(0)\partial _c y^-(0)+\partial _c (y^-)'(0)y^-(0)=\int _{-h_-}^0 \frac{(U^-)''}{(U^--c)^2}(y^-)^2 \textrm{d}x_2. \end{aligned}$$

This implies that

$$\begin{aligned} & \partial _c Y^+(c, k)=-\frac{1}{y^+(0)^2}\int _0^{h_+}\frac{(U^+)''}{(U^+-c)^2}(y^+)^2 \textrm{d}x_2, \end{aligned}$$
(3.7a)
$$\begin{aligned} & \partial _c Y^-(c, k)=\frac{1}{y^-(0)^2}\int _{-h_-}^0\frac{(U^-)''}{(U^--c)^2}(y^-)^2 \textrm{d}x_2. \end{aligned}$$
(3.7b)

By Lemmas 2.3 and 2.4, we obtain that there exist \(C_1>0\) such that for all \(|k|>k_0\) and \(c\in S_+\cup S_-\),

$$\begin{aligned} |\partial _c Y^\pm |\le \frac{C_1}{|k|}. \end{aligned}$$

We compute \(\frac{df^\pm (\cdot , k)}{dc}\). Based on (2.23), (2.27), and the estimate of \(|\partial _c Y^\pm |\), we can choose \(k_0\) large such that the term involving \(\frac{dA}{dc}\) dominates in \(\frac{df^\pm (\cdot , k)}{dc}\). Then one may check that

$$\begin{aligned} \left| \frac{df^\pm (\cdot , k)}{dc}\right| \le \frac{C}{\sqrt{|k|}}<1, \quad \forall |k|\ge k_0, \end{aligned}$$

where C is independent of k, c, and \(\epsilon \). Therefore, \(f^\pm (c, k)\) are contractions acting on \(S^\pm \) respectively. Then their fixed points \(c^\pm (k)\) are the only solutions to system (2.2a), (2.2b), and (2.3) on \(S_\pm \). Moreover, they are analytic in k. Finally, one may compute

$$\begin{aligned} \partial _c F= & \epsilon \Big (-(U^+)'(0)+2\big (U^+(0)-c\big )Y^+-\partial _c Y^+\big (U^+(0)-c)^2\Big )\nonumber \\ & +\partial _cY^-\big (U^-(0)-c)^2-2\big (U^-(0)-c\big )Y^-+(U^-)'(0). \end{aligned}$$
(3.8)

Then by using the above estimates of \(|Y^\pm \pm k\coth (kh_\pm )|\), \(\partial _c Y^\pm \), and \(c^\pm (k)\), the last desired estimate of \(\partial _c F\big (c^\pm (k), k, \epsilon \big )\) can be obtained. This completes the proof of the lemma. \(\square \)

Remark 3.1

  1. (1).

    When \(\epsilon =0\), \(c^\pm (k)\) obtained in Lemma 3.2 coincide with those obtained in [11]. Later in Sect. 4, this fact will be used.

  2. (2).

    Recall that a and b are defined in (1.7). Suppose that \(k_+\ge 0\) has the biggest |k| such that \(F(b, k_+, \epsilon )=0\), and \(k_-\ge 0\) has the biggest |k| such that \(F(a, k_-, \epsilon )=0\). By Lemma 4.3 in [11] and Theorem 1.1, for each \(\epsilon \in [0, 1]\), \(c^+(k)\) (resp. \(c^-(k)\)) obtained in Lemma 3.2 can be extended to be an even and analytic function of k for all \(|k|\in [k_+, \infty )\) (resp. \(|k|\in [k_-, \infty )\)) such that

    $$\begin{aligned} F\big (c^+(k), k, \epsilon \big )=0, \; \forall |k|\in [k_+, \infty ), \quad c^+(k)>b, \; \forall |k|\in (k_+, \infty ), \nonumber \\ F\big (c^-(k), k, \epsilon \big )=0,\; \forall |k|\in [k_-, \infty ), \quad c^-(k)<a,\; \forall |k|\in (k_-, \infty ). \end{aligned}$$
    (3.9)

    Moreover, \(c^+(k)\) (resp. \(c^-(k)\)) can be extended to be an analytic function for all \(|k|\in [0, \infty )\) if for any \(k\in \mathbb {R}\), (b, k) (resp. (a, k)) is not a neutral mode. In addition, for \(c_R>b\) and \(c_I\in \mathbb {R}\), \(c^+(k)\) is the unique root of \(F(\cdot , k, \epsilon )=0\), (resp. for \(c_R<a\) and \(c_I\in \mathbb {R}\), \(c^-(k)\) is the unique root of \(F(\cdot , k, \epsilon )=0\).)

Since \(F(c, k, \epsilon )\in \mathbb {R}\) for all \(c\in \mathbb {R}{\setminus } [a, b]\), \(\partial _c F(c^\pm (k), k, \epsilon )\) does not change sign along these simple roots. Hence, the signs of \(\partial _c F\) in (3.4), Lemmas 2.5, 2.6, and 3.1 imply that

$$\begin{aligned} \partial _c F\big (c^+(k), k, \epsilon \big )>0, \; \forall |k|\in [k_+, \infty ), \quad \partial _c F\big (c^-(k), k, \epsilon \big )<0, \; \forall |k|\in [k_-, \infty ).\nonumber \\ \end{aligned}$$
(3.10)

Particularly, since \(Y^-(c, k)\) has logarithmic singularity \(\log |U^-(0)-c|\) near \(c=U^-(0)\),

$$\begin{aligned} F\big (U^-(0), k, 0\big )=-g-\frac{\sigma }{\rho ^-}k^2<0, \quad \partial _c F\big (c^+(k), k, 0\big )>0, \; \forall k\in \mathbb {R}.\qquad \end{aligned}$$
(3.11)

According to Remark 3.1(2), \(c^\pm (k)\) will keep continuity in k as simple roots of the analytic function \(F(\cdot , k, \epsilon )\) when |k| getting small. We shall track two roots \(c^\pm (k)\) of function \(F(c, k, \epsilon )\) as |k| decreasing to study possible bifurcation and seek instability. Before this, we need more properties of \(F(c, k, \epsilon )\). In the following lemma, we summarize the properties of the second derivative of \(F(c, k, \epsilon )\). In the proof, even though some properties of \(Y^-(c, k)\) other than Lemma 2.5 were obtained in [11], we present the proof for self-completeness.

Lemma 3.3

Assume that \(U^\pm \) satisfy (1.3). Suppose that \(K=k^2\) and \(\epsilon \ge 0\). Let a and b be defined in (1.7). Then, for \(c\in \mathbb {R}{\setminus } \Big ( U^-\big ((-h_-, 0]\big )\cup U^+\big ([0, h_+)\big )\Big )\), the following hold:

  1. (1)
    $$\begin{aligned} \partial _{KK}F(c, k, \epsilon )<0. \end{aligned}$$
    (3.12)
  2. (2)

    If both \((U^+)''>0\) and \((U^-)''>0\), then \(\partial _{Kc}F(c, k, \epsilon )<0\) for all \(c\le a\) and \(k\in \mathbb {R}\).

  3. (3)

    If both \((U^+)''<0\) and \((U^-)''<0\), then \(\partial _{Kc}F(c, k, \epsilon )>0\) for all \(c\ge b\) and \(k\in \mathbb {R}\).

Proof

Let \(K:=k^2\). We notice that for \(c\in \mathbb {C}{\setminus } U^-([-h_-, 0])\), \(\partial _K y^-\) satisfies the following equations.

$$\begin{aligned} {\left\{ \begin{array}{ll} -\partial _K (y^-)''(x_2)+\big (k^2+\frac{(U^-)''}{U^--c}\big )\partial _K y^-(x_2)=-y^-(x_2), \qquad x_2\in (-h_-, 0),\\ \partial _K y^-(- h_-)=\partial _K (y^-)'(- h_-)=0. \end{array}\right. } \end{aligned}$$
(3.13a)

For \(c\in \mathbb {C}\setminus U^+([0, h_+])\), \(\partial _K y^+\) satisfies the following system.

$$\begin{aligned} {\left\{ \begin{array}{ll} -\partial _K (y^+)''(x_2)+\big (k^2+\frac{(U^+)''}{U^+-c}\big )\partial _K y^+(x_2)=-y^+(x_2), \qquad x_2\in (0, h_+),\\ \partial _K y^+( h_+)=\partial _K (y^+)'( h_+)=0. \end{array}\right. } \end{aligned}$$
(3.13b)

For \(c\in \mathbb {R}{\setminus } \big (U^-([-h_-, 0])\cup U^+([0, h_+])\big )\), we combine (2.2a) multiplied by \(\partial _K y^\pm (x_2)\) and (3.13) multiplied by \(y^\pm (x_2)\), respectively, to obtain that

$$\begin{aligned} & -(y^-)''\partial _K y^-+\partial _K(y^-)'' y^-=(y^-)^2, \quad x_2\in (-h_-, 0). \end{aligned}$$
(3.14a)
$$\begin{aligned} & -(y^+)''\partial _K y^++\partial _K(y^+)'' y^-=(y^+)^2, \quad x_2\in (0, h_+). \end{aligned}$$
(3.14b)

Using the boundary conditions in (3.13), we integrate (3.14a) on \([-h_-, 0]\) and integrate (3.14b) on \([0, h_+]\), and then divide it by \(y^\pm (0)^2\), respectively,

$$\begin{aligned} \partial _K Y^-(c, k)=\frac{1}{y^-(0)^2}\int _{-h_-}^0 y^-(x_2)^2\textrm{d}x_2>0, \end{aligned}$$
(3.15a)
$$\begin{aligned} \partial _K Y^+(c, k)=- \frac{1}{y^+(0)^2}\int _0^{h_+} y^+(x_2)^2\textrm{d}x_2<0. \end{aligned}$$
(3.15b)

Similarly, we integrate (3.14a) on \([-h_-, x_2]\) and (3.14b) on \([x_2, h_+]\), and then divide it by \(y^\pm (x_2)\) respectively. We obtain that

$$\begin{aligned} & \frac{d}{dx_2}\Big (\frac{\partial _K y^-(x_2)}{y^-(x_2)}\Big )=\frac{1}{y^-(x_2)^2}\int _{-h_-}^{x_2}y^-(x_2')^2\textrm{d}x_2' >0, \qquad x_2\in [-h_-, 0].\qquad \qquad \end{aligned}$$
(3.16a)
$$\begin{aligned} & \frac{d}{dx_2}\Big (\frac{\partial _K y^+(x_2)}{y^+(x_2)}\Big )=-\frac{1}{y^+(x_2)^2}\int _{x_2}^{h_+}y^+(x_2')^2\textrm{d}x_2' <0, \qquad x_2\in [0, h_+].\qquad \qquad \end{aligned}$$
(3.16b)

These imply that

$$\begin{aligned} & \partial _{KK}Y^-(c, k)=-2\int _{-h_-}^0 \frac{y^-(x_2)^2}{y^-(0)^2}\Big (\frac{\partial _Ky^-(0)}{y^-(0)}-\frac{\partial _Ky^-(x_2)}{y^-(x_2)} \Big )\textrm{d}x_2<0.\qquad \quad \end{aligned}$$
(3.17a)
$$\begin{aligned} & \partial _{KK}Y^+(c, k)=2\int _0^{h_+} \frac{y^+(x_2)^2}{y^+(0)^2}\Big (\frac{\partial _Ky^+(0)}{y^+(0)}-\frac{\partial _Ky^+(x_2)}{y^+(x_2)} \Big )\textrm{d}x_2>0. \end{aligned}$$
(3.17b)

Therefore, for \(c\in \mathbb {R}{\setminus } \Big ( U^-\big ((-h_-, 0])\cup U^+([0, h_+)\big )\Big )\),

$$\begin{aligned} \partial _{KK}F(c, k, \epsilon )=-\epsilon \partial _{KK} Y^+(c, k)\big (U^+(0)-c\big )^2+\partial _{KK} Y^-(c, k)\big (U^-(0)-c\big )^2<0. \end{aligned}$$

Suppose that \(c\in \mathbb {R}{\setminus } \Big ( U^-\big ((-h_-, 0])\cup U^+([0, h_+)\big )\Big )\). We differentiate (3.7a) in K to obtain that

$$\begin{aligned} \partial _{Kc} Y^+(c, k)=\frac{2}{y^+(0)^2}\int _0^{h_+}\frac{(U^+)''}{(U^+-c)^2}(y^+)^2\big (\frac{\partial _K y^+(0)}{y^+(0)}-\frac{\partial _K y^+(x_2)}{y^+(x_2}\big )\textrm{d}x_2. \end{aligned}$$

Using (3.16b), we conclude that

$$\begin{aligned} {{\,\textrm{sgn}\,}}(\partial _{Kc} Y^+)={{\,\textrm{sgn}\,}}\big ((U^+)''\big ). \end{aligned}$$
(3.18a)

A similar computation leads to

$$\begin{aligned} {{\,\textrm{sgn}\,}}(\partial _{Kc} Y^-)=-{{\,\textrm{sgn}\,}}\big ((U^-)''\big ). \end{aligned}$$
(3.18b)

We consider the function

$$\begin{aligned} \partial _{Kc} F= & \partial _{Kc}Y^-\big (U^-(0)-c\big )^2-2\big (U^-(0)-c\big )\partial _KY^-\nonumber \\ & +\epsilon \Big (-\partial _{Kc}Y^+\big (U^+(0)-c \big )^2+2\big (U^+(0)-c \big )\partial _KY^+ \Big ). \end{aligned}$$
(3.19)

Suppose that \(c\le a\), \((U^+)''>0\), and \((U^-)''>0\). Then, based on (3.19), we use (3.15) and (3.18) to obtain that \(\partial _{Kc}F(c, k, \epsilon )<0\). Lemma 2.5, Lemma 2.6, and Lemma 3.1 imply that \(\partial _{Kc}F\) is well defined at \(c=a\) and then \(\partial _{Kc}F(a, k, \epsilon )<0\). A similar proof of statement (3) for the case of \(c\ge b\) and both \((U^+)''<0\) and \((U^-)''<0\) is similar. \(\square \)

As |k| getting small from infinity, there might exist singular modes. Lemma 3.3 implies that we can use the monotonicity of \(\partial _K F(c, k, \epsilon )\) and \(\partial _c F(c, k, \epsilon )\) in K and reduce some computations to the case of \(k=0\). Hence, it is worth paying closer attention to the special case of \(k=0\). When \(k=0\), \(y^\pm (x_2)\) which are solutions to (2.2a) and (2.12) have explicit representations. A direction computation shows that, if \(c=U^+(h_+)\),

$$\begin{aligned} y^+(x_2)=\frac{U^+(x_2)-U^+(h_+)}{(U^+)'(h_+)}. \end{aligned}$$
(3.20)

If \(c\in \mathbb {R}\setminus U^+\big ([0, h_+]\big )\),

$$\begin{aligned} y^+(x_2)=\big (U^+(h_+)-c\big )\big (U^+(x_2)-c\big )\int _{x_2}^{h_+}\frac{-1}{\big (U^+(x_2')-c \big )^2}\textrm{d}x_2'. \end{aligned}$$
(3.21)

If \(c=U^-(-h_-)\),

$$\begin{aligned} y^-(x_2)=\frac{U^-(x_2)-U^-(-h_-)}{(U^-)'(-h_-)}. \end{aligned}$$
(3.22)

If \(c\in \mathbb {R}\setminus U^-([-h_-, 0])\),

$$\begin{aligned} y^-(x_2)=\big (U^-(x_2)-c \big )\big (U^-(-h_-)-c \big )\int _{-h_-}^{x_2}\frac{1}{(U^--c)^2}\textrm{d}x_2. \end{aligned}$$
(3.23)

In the next lemma, we address the possible locations of neutral limiting modes for given \(\epsilon \in (0, 1)\) and small \(\epsilon \).

Lemma 3.4

Assume that \(U^\pm \) satisfy (1.3), and \((U^\pm )''\ne 0\). Let

$$\begin{aligned} & \mathcal {E}:=\{U^+(h_+), U^+(0), U^-(-h_-), U^-(0)\}, \quad \nonumber \\ & \mathcal {I}:=U^+\big ((0, h_+)\big )\cap U^-\big ((-h_-, 0)\big ), \end{aligned}$$
(3.24)

Then the following hold:

  1. (1)

    Suppose that \(\epsilon \in (0, 1)\) and there is a sequence of unstable modes \((c_n, k_n)\) converges to \((c_\infty , k_\infty )\in \mathbb {R}^2\) as \(n\rightarrow \infty \), and

    $$\begin{aligned} g(1-\epsilon )+\frac{\sigma }{\rho ^-}k_\infty ^2>0. \end{aligned}$$
    (3.25)

    Then the following hold.

    1. (a)

      If \(\mathcal {I}=\emptyset \), then \(c_\infty \in [a, b]{\setminus } \Big (U^-\big ((-h_-, 0)\big )\cup U^+\big ((0, h_+) \big )\Big )\). Under additional assumption that \(U^+(0)=U^-(0)\), \(c_\infty \in \mathcal {E}{\setminus } \{U^\pm (0)\}\).

    2. (b)

      Assume \(\mathcal {I}\ne \emptyset \). Then \(c_\infty \in \mathcal {E}\cup \mathcal {I}\) if \(U^+(0)\ne U^-(0)\). And if \(U^+(0)= U^-(0)\), \(c_\infty \in \mathcal {E}\cup \mathcal {I}{\setminus } \{U^\pm (0)\}\). In addition, if \((U^+)'' (U^-)''>0\), then \(c_\infty \in \{a, b\}\), where a, b are defined in (1.7).

  2. (2)

    Suppose that \(\mathcal {I}= \emptyset \). There exists \(\epsilon _0>0\) such that for any \(\epsilon \in (0, \epsilon _0)\), if an unstable sequence \((c_n, k_n)\rightarrow (c_\infty , k_\infty )\in \mathbb {R}^2\) as \(n\rightarrow \infty \), (3.25) holds, and we further assume one of the following hold,

    1. (a)

      \((U^-)', (U^-)''>0\), and \(\max _{[0, h_+]} U^+(x_2)<U^-(-h_-)\); or

    2. (b)

      \((U^-)', (U^-)''>0\), and \(\min _{[0, h_+]}U^+(x_2)>U^-(0)\),

    then \(c_\infty \in \mathcal {E}\).

Proof

Suppose that \(\epsilon \in (0, 1)\) is given. Lemma 3.1 implies that

$$\begin{aligned} F(c_\infty , k_\infty , \epsilon )=\lim _{n\rightarrow \infty }F(c_n, k_n, \epsilon )=0. \end{aligned}$$

To address the possible locations of the neutral limiting mode for each \(\epsilon >0\), we need to consider the imaginary part of \(F(c_\infty , k_\infty , \epsilon )\).

$$\begin{aligned} F_I(c_\infty , k_\infty , \epsilon )= & -\epsilon Y^+_I(c_\infty , k_\infty )\big (U^+(0)-c_\infty \big )^2\nonumber \\ & +Y^-_I(c_\infty , k_\infty )\big (U^-(0)-c_\infty \big )^2. \end{aligned}$$
(3.26)

Since (3.25) holds, Theorem 1.1 implies that \(c_\infty \in [a, b]\), where a and b are defined in (1.7). We shall discuss the cases as following.

* Case (1a). \(\mathcal {I}=\emptyset \). Since \((U^\pm )''\ne 0\), according to Lemma 2.5 and Lemma 2.6, \(Y^-_I\ne 0\) if \(c\in U^-\big ((-h_-, 0] \big )\). And \(Y_I^-\equiv 0\) if \(c\in \mathbb {R}{\setminus } U^-\big ((-h_-, 0]\big )\), \(Y^+_I\ne 0\) if \(c\in U^+\big ([0, h_+) \big )\), and \(Y^+_I\equiv 0\) if \(c\in \mathbb {R}{\setminus } U^+\big ([0, h_+)\big )\). Hence, \(F_I(c_\infty , k_\infty , \epsilon )\ne 0\) for \(c_\infty \in U^-\big ((-h_-, 0)\big )\cup U^+\big ((0, h_+)\big )\). Particularly, if we further assume that \(U^+(0)=U^-(0)\), then (2.23) and (2.27) imply that \(F(c_\infty , k, \epsilon )=-g(1-\epsilon )-\frac{\sigma }{\rho ^-}k^2< 0\) for all \(k\in \mathbb {R}\) and \(\epsilon \in (0, 1)\). Hence, in this case, \(c\notin U^\pm (0)\).

* Case (1b). \(\mathcal {I}\ne \emptyset \). It is clear that \(c_\infty \in \mathcal {E}\cup \mathcal {I}\) by considering \(F_I(c_\infty , k_\infty , \epsilon )=0\). We know that \(c_\infty \ne U^\pm (0)\) if \(U^+(0)=U^-(0)\). Let \(x_{c_\infty }^-:=(U^-)^{-1}(c_\infty )\) and \(x_{c_\infty }^+:=(U^+)^{-1}(c_\infty )\). Based on (2.24) and (2.28), if \(c_\infty \in \mathcal {I}\), (3.26) can be rewritten as

$$\begin{aligned} F_I(c_\infty , k_\infty , \epsilon )&=\epsilon \frac{\pi (U^+)''(x_{c_\infty }^+)y^+(x_{c_\infty })^2}{|(U^+)'(x_{c_\infty }^+)||y^+(0)|^2}\big (U^+(0)-c_\infty \big )^2\\&\quad +\frac{\pi (U^-)''(x_{c_\infty }^-)y^-(x_{c_\infty })^2}{|(U^-)'(x_{c_\infty }^-)||y^-(0)|^2}\big (U^-(0)-c_\infty \big )^2. \end{aligned}$$

Hence, if \({{\,\textrm{sgn}\,}}\big ((U^-)''\big )={{\,\textrm{sgn}\,}}\big ((U^+)''\big )\), \(F_I(c_\infty , k_\infty , \epsilon )\ne 0\). Statement (1) is proved.

2. Now we consider \(\mathcal {I}=\emptyset \) and will prove \(c_\infty \in \mathcal {E}\) under the assumption of (2a) or (2b). To prove (2), we will show that in the following cases, there exists \(\epsilon _0>0\) such that for any \(\epsilon \in (0, \epsilon _0)\), if \(F(c, k, \epsilon )=0\) then \(\partial _c F(c, k, \epsilon )\ne 0\) under some conditions. Let \(K:=k^2\) and \(K_\infty =k_\infty ^2\).

Case (a). \((U^-)'>0, (U^-)''>0\), and \(\max _{[0, h_+]} U^+(x_2)<U^-(-h_-)\). Assume that

$$\begin{aligned} c\in \big (\max _{[0, h_+]} U^+(x_2), U^-(-h_-) \big ). \end{aligned}$$
(3.27)

Since \((U^-)''>0\), (3.18) implies that \(\partial _{Kc} Y^-<0\). By (3.15a), \(\partial _K Y^->0\). Hence,

$$\begin{aligned} \partial _{Kc} F(c, k, 0)=\big ((U^-(0)-c)^2\big )\partial _{kc} Y^-(c, k)-2\big (U^-(0)-c\big )\partial _K Y^-(c, k)<0. \end{aligned}$$

This implies that \(\partial _c F(c, k, 0)<\partial _c F(c, 0, 0)\) for all \(k>0\). If \(c=U^-(-h_-)\), \(y^-(x_2)\) is in the form of (3.22). Using (3.7b), we compute that

$$\begin{aligned} & \big (U^-(0)-U^-(-h_-)\big )^2\partial _c Y^-\big (U^-(-h_-), 0\big )\nonumber \\ & \quad =\int _{-h_-}^0 (U^-)'' \textrm{d}x_2=(U^-)'(0)-(U^-)'(-h_-), \end{aligned}$$
(3.28)
$$\begin{aligned} & Y^-\big (U^-(-h_-), 0\big )=\frac{(U^-)'(0)}{U^-(0)-U^-(-h_-)}. \end{aligned}$$
(3.29)

Hence, we obtain that

$$\begin{aligned} \partial _c F\big (U^-(-h_-), 0, 0 \big )=-(U^-)'(-h_-)<0. \end{aligned}$$
(3.30)

Similarly, if \(c\in \mathbb {R}\setminus U^-([-h_-, 0])\), \(y^-(x_2)\) is in the form of (3.23). Then we compute

$$\begin{aligned} & \big (U^-(0)-c\big )^2 \partial _c Y^-(c, 0)=\frac{1}{\big ( \int _{-h_-}^0\frac{1}{(U^--c)^2}dx_2\big )^2}\int _{-h_-}^0 (U^-)''\nonumber \\ & \qquad \bigg (\int _{-h_-}^{x_2}\frac{1}{(U^--c)^2}\textrm{d}x_2'\bigg )^2\textrm{d}x_2\nonumber \\ & \quad \le \int _{-h_-}^0 (U^-)''dx_2 =(U^-)'(0)-(U^-)'(-h_-), \end{aligned}$$
(3.31)
$$\begin{aligned} & Y^-(c, 0)= \frac{(U^-)'(0)}{U^-(0)-c}+\frac{1}{\big (U^-(0)-c\big )^2\int _{-h_-}^0 \frac{1}{(U^--c)^2}dx_2}. \end{aligned}$$
(3.32)

Then, we obtain that

$$\begin{aligned} \partial _c F(c, 0, 0)\le -(U^-)'(-h_-)-\frac{2}{\big (U^-(0)-c\big )\int _{-h_-}^0 \frac{1}{(U^--c)^2}\textrm{d}x_2}<0. \end{aligned}$$
(3.33)

Hence, \(\partial _c F(c, 0, 0)<0\) if \(c\le U^-(h_-)\). By (3.12), there exists \(\epsilon _0>0\) such that \(\partial _c F(c, k, \epsilon )=\partial _c F(c, k, 0)+O(\epsilon )< \partial _c F(c, 0, 0)+O(\epsilon )<0\) for any \(\epsilon \in (0, \epsilon _0)\) and c satisfies (3.27). In fact, this result does not require that \(F(c, k, \epsilon )=0\).

Case (b). \((U^-)'>0\), \((U^-)''>0\), and \(U^-(0)<\min _{[0, h_+]}U^+(x_2)\). Assume that

$$\begin{aligned} c\in \big (U^-(0), \min _{[0, h_+]}U^+(x_2)\big ). \end{aligned}$$
(3.34)

Since \(F(c, k, \epsilon )=0\), by the smoothness of F in \(\epsilon \), we obtain that

$$\begin{aligned} F(c, k, \epsilon )=F(c, k, 0)+O(\epsilon ). \end{aligned}$$

Hence, a direction computation leads to

$$\begin{aligned} Y^-(c, k)=\frac{g+\frac{\sigma }{\rho ^-}k^2+O(\epsilon )}{\big (U^-(0)-c\big )^2}+\frac{(U^-)'(0)}{U^-(0)-c}. \end{aligned}$$

Then we compute that

$$\begin{aligned} \partial _c F(c, k, \epsilon )=\big (U^-(0)-c\big )^2\partial _c Y^-(c, k)+(U^-)'(0)-2\frac{g+\frac{\sigma }{\rho ^-}k^2+O(\epsilon )}{U^-(0)-c}+O(\epsilon ). \end{aligned}$$

Using the assumption that \((U^-)'>0\), the fact that \(\partial _c Y^->0\) if \((U^-)''>0\) (see (3.7b)), and the smallness of \(\epsilon \), we have \(\partial _c F(c, k, \epsilon )>0\) in this case.

We let

$$\begin{aligned} \mathcal {N}:=[a, b]\setminus \Big (U^-\big ([-h_-, 0] \big )\cup U^+\big ([0, h_+] \big )\Big ). \end{aligned}$$

By the proof in (1), to prove (2), it remains to consider \(c_\infty \in \mathcal {N}\). We suppose that for some \(\epsilon \in (0, \epsilon _0)\), there is a unstable sequence \((c_n, k_n)\rightarrow (c_\infty , k_\infty )\in \mathbb {R}^2\). \(\epsilon _0\) is chosen by the previous discussion. Lemma 3.2 implies that \(k_\infty \) can not be sufficiently large. Hence, \(\epsilon _0\) can be chosen independent of \(k_\infty \) by the smoothness of F in \((c, k, \epsilon )\in \mathcal {N}\times \mathbb {R}\times (0, 1)\). Since \(F(c, k, \epsilon )\) is analytic near \(c_\infty \in \mathcal {N}\), we use the Cauchy-Riemann equation to compute the \(2\times 2\) Jacobian matrix of \(D_c F\).

$$\begin{aligned} D_c F\big (c_\infty , k_\infty , \epsilon \big )=\begin{pmatrix} \partial _{c_R} F_R & \partial _{c_I} F_R\\ \partial _{c_R} F_I & \partial _{c_I} F_I \end{pmatrix}\Big | _{(c_\infty , k_\infty , \epsilon )}= \partial _c F(c_\infty , k_\infty , \epsilon ) I_{2\times 2}. \end{aligned}$$

For both case (a) and (b), \(\partial _c F(c_\infty , k_\infty , \epsilon )\ne 0\). Hence, by the Implicit Function Theorem, there exists a smooth complex-valued function \(\mathcal {C}(k)\) such that all the roots of \(F(\cdot , \cdot , \epsilon )\) near \((c_\infty , k_\infty )\) are in the form of \(\big (\mathcal {C}(k), k\big )\) and \(\mathcal {C}(k_\infty )=c_\infty \). We will show that \(\mathcal {C}(k)\in \mathbb {R}\) to complete the proof. Since \(F_R\) is smooth for \(c\in \mathcal {N}\) and \(\partial _{c_R} F_R(c_\infty , k_\infty , \epsilon )=\partial _c F(c_\infty , k_\infty , \epsilon )\ne 0\), we apply the Implicit Function Theorem on \(F_R\). Then there exists a smooth real-valued function \(\mathcal {C}_1(k)\) for k near \(k_\infty \) such that \(\mathcal {C}_1(k_\infty )=c_\infty \) and \(F_R\big (\mathcal {C}_1(k), k, \epsilon \big )=0\). Since \(c_\infty \in \mathcal {N}\), \(F_I(c, k, \epsilon )=0\) near \(c_\infty \). Hence, by the uniqueness of solutions obtained by the Implicit Function Theorem, \(\mathcal {C}(k)=\mathcal {C}_1(k)\in \mathbb {R}\) near \(c_\infty \). Hence, there is no unstable mode near \((c_\infty , k_\infty )\). The lemma is proved. \(\square \)

According to Lemma 3.4, for each \(\epsilon \in (0, 1)\), neutral limiting modes might happen at the endpoints of the range of \(U^\pm \) or the intersection of the range of \(U^\pm \). In either case, (c, k) is a singular mode. By Remark 3.1, we notice that the number of eigenvalues \(\lambda =-ikc\) may be changed if \(c^\pm (k)\) obtained in Lemma 3.2 touches the bounds of \(U^\pm \) as |k| decreases. On the other hand, if we can rule out the existence of any singular modes, then \(c^\pm (k)\) can be extended to all \(k\in \mathbb {R}\). We would like to study how the eigenvalue distributes when |k| gets smaller. The rest of this section includes two main directions. One is to prove the stability of shear flows by ruling out all singular modes under some certain conditions. The other one is to seek instability occurring near singular modes for general \(\epsilon \ge 0\). In the following lemma, we provide some sufficient conditions for the non-existence of singular modes.

Lemma 3.5

Assume that \(U^\pm \) satisfies (1.3), \((U^\pm )''\ne 0\), and \(\epsilon \in (0, 1)\). Let a, b be defined in (1.7), \(\mathcal {E}, \mathcal {I}\) be defined in (3.24), and

$$\begin{aligned} m(c):= \epsilon \int _0^{h_+}\big (U^+(x_2)-c\big )^2\textrm{d}x_2+\int _{-h_-}^0\big (U^-(x_2)-c\big )^2\textrm{d}x_2. \end{aligned}$$
(3.35)

If

$$\begin{aligned} & \min _{c\in \{U^+(0), U^+(h_+)\}} \int _{-h_-}^0 \frac{1}{(U^--c)^2}\textrm{d}x_2>\frac{1}{g}, \quad \nonumber \\ & \quad \min _{c\in \{U^-(0), U^-(h_-)\}}\int _0^{h_+}\frac{1}{(U^+-c)^2}\textrm{d}x_2>\frac{\epsilon }{g}, \end{aligned}$$
(3.36)

and one of the following holds:

  1. (a)

    \(\mathcal {I}=\emptyset \) and \(\max _{c\in \mathcal {E}} m(c)<\frac{\sigma }{\rho ^-}\);

  2. (b)

    \(\mathcal {I}\ne \emptyset \), \(\max _{c\in \{a, b\}}m(c)<\frac{\sigma }{\rho ^-}\), and \((U^+)''(U^-)''>0\).

Then for each \(k>0\), \(c^\pm (k)\) obtained in Lemma 3.2 can be extended to be even and analytic functions for all \(k\in \mathbb {R}\) and the following hold:

$$\begin{aligned} F(c^\pm (k), k, \epsilon )=0, \quad c^+(k)>b, \quad c^-(k)<a, \quad k\partial _c F(k, c^\pm (k))>0. \end{aligned}$$
(3.37)

Proof

According to (2.23–2.25), \(\big (U^-(0)-c\big )^2\partial _k^j Y^-(c, k), \;j\in \mathbb {N}\) is well defined and \(C^1\) near \(c=U^-(0)\), \(U^-(-h_-)\). Similarly, (2.27–2.29) imply that \(\big (U^+(0)-c\big )^2\partial _k^j Y^+(c, k), \; j\in \mathbb {N}\) is well defined and \(C^1\) near \(c=U^+(0)\) and \(U^+(h_+)\). In what follows, we prove the non-existence of singular neutral mode under the assumption in the statement.

* Case 1. \(\mathcal {I}= \emptyset \) and \(\max _{c\in \mathcal {E}}m(c)<\frac{\sigma }{\rho ^-}\). We will show that there is no singular mode at \(c\in \mathcal {N}:=[a, b]{\setminus } \Big (U^-\big ((-h_-, 0)\big )\cup U^+\big ((0, h_+)\big ) \big )\).

Step 1. We will show that \(\partial _K F(c, 0, \epsilon )<0\) for \(c\in \mathcal {E}\) if \(\max _{c\in \mathcal {E}} m(c)<\frac{\sigma }{\rho ^-}\). A direct computation leads to

$$\begin{aligned} \partial _K F(c, k, \epsilon )= \big (U^-(0)-c\big )^2\partial _K Y^-(c, k)-\epsilon \big (U^+(0)-c\big )^2\partial _K Y^+(c, k)-\frac{\sigma }{\rho ^-}.\nonumber \\ \end{aligned}$$
(3.38)

Suppose that \(K=0\). If \(c=U^+(h_+)\), \(y^+(x_2)\) is in the form of (3.20). We use (3.15b) to compute

$$\begin{aligned} -\epsilon \partial _K Y^+\big (U^+(h_+), 0\big ) \big (U^+(0)-c\big )^2= \epsilon \int _0^{h_+}\big (U^+(x_2)-U^+(h_+)\big )^2\textrm{d}x_2.\qquad \quad \end{aligned}$$
(3.39)

For \(c\in \mathbb {R}{\setminus } U^+\big ([0, h_+]\big )\), \(y^+(x_2)\) has the representation of (3.21). We obtain that

$$\begin{aligned} -\epsilon \partial _K Y^+(c, 0) \big (U^+(0)-c\big )^2= & \epsilon \frac{1}{\big (\int _0^{h_+} \frac{-1}{(U^+-c)^2}dx_2\big )^2}\int _0^{h_+}\big (U^+(x_2)-c\big )^2\nonumber \\ & \Big (\int _{x_2}^{h_+}\frac{-1}{(U^+-c)^2}dx_2'\Big )^2\textrm{d}x_2,\nonumber \\\le & \epsilon \int _0^{h_+}\big (U^+(x_2)-c\big )^2\textrm{d}x_2. \end{aligned}$$
(3.40)

Similarly, if \(c=U^-(-h_-)\), \(y^-(x_2)\) is in the form of (3.22). Using (3.15a), we compute

$$\begin{aligned} \partial _K Y^-\big (U^-(-h_-), 0\big )\big (U^-(0)-U^-(-h_-)\big )^2=\int _{-h_-}^0 \big (U^-(x_2)-U^-(-h_-)\big )^2\textrm{d}x_2.\nonumber \\ \end{aligned}$$
(3.41)

For any \(c\in \mathbb {R}\setminus U^-\big ([-h_-, 0]\big )\), using (3.23), we obtain that

$$\begin{aligned} \partial _K Y^-(c, 0)\big (U^-(0)-c\big )^2= & \frac{1}{\big (\int _{-h_-}^0 \frac{1}{(U^--c)^2}\textrm{d}x_2\big )^2}\int _{-h_-}^0\big (U^-(x_2)-c\big )^2\nonumber \\ & \big (\int _{-h_-}^{x_2}\frac{1}{(U^--c)^2}\textrm{d}x_2'\big )^2dx_2 \nonumber \\\le & \int _{-h_-}^0 \big (U^-(x_2)-c\big )^2\textrm{d}x_2. \end{aligned}$$
(3.42)

Since \(Y^-(c, k)\) has a logarithmic singularity near \(c=U^-(0)\), we have

$$\begin{aligned} \lim _{c\rightarrow U^-(0)}\partial _K Y^-(c, 0)\big (U^-(0)-c\big )^2=0. \end{aligned}$$
(3.43)

Similarly, if c is near \(U^+(0)\), we obtain that

$$\begin{aligned} \lim _{c\rightarrow U^+(0)}-\epsilon \partial _K Y^+(c, 0)\big (U^+(0)-c\big )^2=0. \end{aligned}$$
(3.44)

Now, we compute \(\partial _K F(c, 0, \epsilon )\) for each \(c\in \mathcal {E}\). If \(c=U^-(-h_-)\notin U^+([0, h_+])\), we use the formula (3.38) and estimates (3.41) and (3.40) to compute

$$\begin{aligned} \partial _K F(c, 0, \epsilon )\le&\int _{-h_-}^0\big ( U^-(x_2)-U^-(-h_-)\big )^2\textrm{d}x_2\\&+\epsilon \int _0^{h_+}\big (U^+(x_2)-U^-(-h_-)\big )^2\textrm{d}x_2-\frac{\sigma }{\rho ^-},\\ =&m\big (U^-(-h_-)\big )-\frac{\sigma }{\rho ^-}<0. \end{aligned}$$

According to Lemma 3.3, we have \(\partial _K F(c, k, \epsilon )<\partial _K F(c, 0, \epsilon )\) for \(c\in \mathbb {R}{\setminus } \big (U^-((-h_-, 0))\cup U^+((0, h_+))\big )\). Hence, \(\partial _K F\big (U^-(-h_-), k, \epsilon \big )<\partial _K F\big (U^-(-h_-), 0, \epsilon \big )<0\).

If \(c=U^-(0)\notin U^+([0, h_+])\), we use (3.17b), (3.43), and (3.40) to compute that

$$\begin{aligned} \partial _K F\big (U^-(0), k, \epsilon \big )=&-\epsilon \partial _K Y^+\big (U^-(0), k\big )\big (U^+(0)-c\big )^2\\&\le -\epsilon \partial _K Y^+\big (U^-(0), 0\big )\big (U^+(0)-c\big )^2\\ \le&\epsilon \int _0^{h_+}\big (U^+(x_2)-U^-(0)\big )^2\textrm{d}x_2<m\big (U^-(0)\big )<0. \end{aligned}$$

Similarly, using the above estimates and logarithmic singularity of \(\partial _K Y^\pm \) near \(U^\pm (0)\) respectively, one may check that for all \(c\in \mathcal {E}\),

$$\begin{aligned} \partial _K F(c, k, \epsilon )\le m(c)<0. \end{aligned}$$

Step 2. We will show that \(F(c, 0, \epsilon )<0\) if \(c\in \mathcal {E}\). If \(c\notin U^+([0, h_+])\), using (3.21), we compute

$$\begin{aligned} Y^+(c, 0)=\frac{(U^+)'(0)}{U^+(0)-c}-\frac{1}{\big (U^+(0)-c\big )^2\int _0^{h_+}\frac{1}{(U^+-c)^2}\textrm{d}x_2}. \end{aligned}$$

If \(c=U^-(-h_-)\notin U^+([0, h_+])\), we use (3.22), (3.32), and (3.36) to obtain that

$$\begin{aligned} F\big (U^-(-h_-), 0, \epsilon \big )=-g+\epsilon \frac{1}{\int _0^{h_+}\frac{1}{(U^+-c)^2}\textrm{d}x_2}<0. \end{aligned}$$
(3.45)

If \(c=U^-(-h_-)=U^+(0)\) or \(c=U^-(-h_-)=U^+(h_+)\), then \(F(c, 0, \epsilon )=-g<0\). Following a similar argument for \(c=U^\pm (0)\) and \(U^+(h_+)\), we have \(F(c, 0, \epsilon )<0\) for \(c\in \mathcal {E}\) and \(\mathcal {I}=\emptyset \).

Combining the results from step 1 and step 2, we obtain that \(F(c, k, \epsilon )<0\) for \(c\in \mathcal {E}\) and all \(k\in \mathbb {R}\) if \(\mathcal {I}=\emptyset \) by Lemma 3.3 and the logarithmic singularity of \(Y^+(c, k)\) near \(c=U^+(0)\)(resp. \(Y^-(c, k)\) near \(c=U^-(0)\)). In other words, in this case, there is no singular mode at \(c\in \mathcal {E}\).

Now, we first consider the case of \(\max _{[-h_-, 0]} U^-<\min _{[0, h_+]} U^+\). Let \(k_0\) be defined in Lemma 3.2. There are exactly two wave speed \(c^\pm (k_0)\in \mathbb {C}\) such that \(F\big (c^\pm (k_0), k_0, \epsilon \big )=0\). According to Lemmas 2.5 and 2.6, \(F_I(c, k, \epsilon )\ne 0\) if \(c\in U^-\big ((-h_-, 0)\big )\cup U^+\big ((0, h_+)\big )\). Because of the non-existence of singular modes at \(c\in \mathcal {E}\), \(F(c, k, \epsilon )\ne 0\) for all \(c\in U^-\big ([-h_-, 0]\big )\cup U^+\big ([0, h_+]\big )\). By the compactness and continuity of F, there exist open sets \(B_{1, 2}\subset {\mathbb {C}}\) satisfying \(U^-([-h_-, 0])\subset B_1\) and \(U^+([0, h_+])\subset B_2\) such that \(F(c, k, \epsilon )\ne 0\) whenever \(c\in B_1\cup B_2\) and \(k\in \mathbb {R}\). By (2.24), (2.25), and (2.16), we notice that if c is real and |c| is large, the leading term of \(\big (U^-(0)-c \big )^2 Y^-(c, k)\) is \(\big (U^-(0)-c\big )^2k\coth (kh_-)\). Similarly, we use (2.28), (2.29), and (2.19) to obtain the leading term of \(\big (U^+(0)-c\big )^2Y^+(c, k)\) is \(-\big (U^+(0)-c\big )^2k\coth (kh_+)\). Hence, for \(|k|\in [0, k_0]\), \(F(c, k, \epsilon )\) behaves like a quadratic function in c with a positive uniformly bounded leading coefficient if c is real and |c| is large. Therefore, there exists \(B_c>0\), such that if \(F(c, k, \epsilon )=0\) and \(|k|\in [0, k_0]\), then \(|c|<B_c\). We choose \(B_3\subset \mathbb {C}\) be a disk centered at the origin with radius larger than \(B_c\) such that \(c^\pm (k_0)\in B_3\) and \(B_{1, 2}\subset B_3\). Then \(F(c, k, \epsilon )\ne 0\) for all \(c\in \partial B_3\) and \(|k|\in [0, k_0]\). Let \(\Omega \subset B_3\setminus (B_1\cup B_2)\) containing \((\mathbb {R}\cap B_3)\setminus (B_1\cup B_2)\) such that \(\partial \Omega \) is sufficiently close to \(\partial B_3\), \(\partial B_2\), and \(\partial B_1\). Then \(\Omega \) can be chosen such that \(F(c, k, \epsilon )\ne 0\) for all \(c\in \partial \Omega \) and \(|k|\in [0, k_0]\). We consider

$$\begin{aligned} n(k):=\int _{\partial \Omega } \frac{\partial _c F}{F}(c, k, \epsilon )\textrm{d}c. \end{aligned}$$
(3.46)

Since \(c^\pm (k_0)\) are the unique roots of \(F(\cdot , k_0, \epsilon )=0\), \(n(k_0)=2\). Since n(k) is continuous in k, \(n(k)=2\) for all \(|k|\in [0, k_0]\). Hence, for all \(k\in \mathbb {R}\), \(n(k)\equiv 2\). Moreover, by Remark 3.1, we obtain that \(c^\pm (k)\) obtained in Lemma 3.2 can be extended to be even and analytic functions for all \(k\in \mathbb {R}\) and they are the only roots of \(F(\cdot , k, \epsilon )=0\). Since \(F(c, k, \epsilon )\in \mathbb {R}\) for all \(c\in \mathbb {R}{\setminus } \big (U^-([-h_-, 0])\cup U^+([0, h_+])\big )\), \(\partial _c F(c^\pm (k), k, \epsilon )\) does not change sign and theirs signs are the same as the ones given in Lemma 3.2.

Let us consider the remaining case of \(\max _{[-h_-, 0]} U^-(x_2)=\min _{[0, h_+]} U^+(x_2)\). Since \(F(c, k, \epsilon )\ne 0\) for all \(k\in \mathbb {R}\) and \(c\in [\min U^-, \max U^+]\), there exists \(B_4\subset \mathbb {C}\) containing \([\min U^-, \max U^+]\) such that \(F(c, k, \epsilon )\ne 0\) for all \(k\in \mathbb {R}\) and \(c\in B_4\). Then, there exists a big enough \(\Omega _+\subset \{c\ge \max U^+\}{\setminus } B_4\) containing \(c^+(k_0)\) which is a bounded region such that \(F(c, k, \epsilon )\ne 0\) for all \(c\in \partial \Omega _+\) and \(|k|\in [0, k_0]\). We can apply (3.46) and a similar argument as above on such \(\Omega _+\) to obtain that \(c^+(k)\) which is obtained in Lemma 3.2 can be extended to be an even and analytic function for all \(k\in \mathbb {R}\). Similarly, we can choose \(\Omega _-\) big enough containing \(c^-(k_0)\) such that \(F(c, k, \epsilon )\ne 0\) on \(\partial \Omega _-\) and obtain that \(c^-(k)\) can be extended to all \(k\in \mathbb {R}\). Moreover, let \(\tilde{\Omega }\) be a large bounded region containing \(c^\pm (k_0)\) and apply (3.46) on \((\tilde{\Omega }{\setminus } B_4)\cap \{c_I>-\alpha \}\) where \(0<\alpha \ll 1\) and \((\tilde{\Omega }{\setminus } B_4)\cap \{c_I<\alpha \}\) respectively. We can obtain that \(c^\pm (k)\) are the only roots of \(F(\cdot , k, \epsilon )=0\) for all \(k\in \mathbb {R}\).

* Case 2. \(\mathcal {I}\ne \emptyset \), \(\max _{c\in \{a, b\}}m(c)<\frac{\sigma }{\rho ^-}\), and \((U^+)''(U^-)''>0\). Following the same argument of Case 1, it remains to consider \(c\in \mathcal {I}\). We notice that \(F_I(c, k, \epsilon )\ne 0\) for any \(c\in \mathcal {I}\) if \({{\,\textrm{sgn}\,}}(Y^-_I)=-{{\,\textrm{sgn}\,}}(Y_I^+)\). Lemma 2.5 and Lemma 2.6 imply that if \((U^-)''(U^+)''>0\), then \(F(c, k, \epsilon )\ne 0\) for \(c\in \mathcal {I}\). The rest of the proof can be completed by using the same continuation argument of n(k) as the one in Case 1.

\(\square \)

We notice that \(c=a\) is a point on the boundary of the domain of analyticity of \(F(\cdot , k, \epsilon )\). Even more, it is possible that it corresponds to a neutral limiting mode. If it is, there might be bifurcation occuring near this point. Hence, it is worth discussing the properties of \(F(c, k, \epsilon )\) at this special point. In the next lemma, we shall study the behavior of \(\partial _c F\) near \(c=a\) for any \(\epsilon \ge 0\).

Lemma 3.6

Assume that \(U^\pm \in C^6\), \((U^\pm )'\), and \((U^\pm )''>0\). For each \(\epsilon \ge 0\), if it that either holds,

  1. (1)

    \(a=U^+(0)\le U^-(-h_-)\), or or

  2. (2)

    \(a=U^-(-h_-)<U^+(0)\) and

    $$\begin{aligned} & -(U^-)'(-h_-)+\epsilon (U^+)'(h_+)<2\epsilon \Big (\big (U^+(0)-U^-(-h_-)\big )\nonumber \\ & \quad \int _0^{h_+}\frac{1}{(U^+-c)^2}\textrm{d}x_2 \Big )^{-1}, \end{aligned}$$
    (3.47)

then \(\partial _c F(a, k, \epsilon )<0\) for all \(k\in \mathbb {R}\).

Proof

Let \(K:=k^2\). Lemma 3.3 implies that \(\partial _c F(a, K, \epsilon )<\partial _c F(a, 0, \epsilon )\) for all \(K\ge 0\). We shall compute \(\partial _c F(a, 0, \epsilon )\) in the following cases:

* Case 1. \(a=U^-(-h_-)\). We apply (3.28) and (3.29) to compute \(\partial _c F(a, 0, \epsilon )\). If \(U^-(-h_-)=U^+(0)\), Lemma 2.6 implies that both \(\big (U^+(0)-c\big )Y^+\) and \(\partial _c Y^+\big (U^+(0)-c\big )^2\) vanish at \(c=U^+(0)\). Since \((U^-)'>0\), we compute (3.8) to obtain that

$$\begin{aligned} \partial _c F(a, 0, \epsilon )= -(U^-)'(-h_-)-\epsilon (U^+)'(0)<0, \quad \forall \epsilon \ge 0. \end{aligned}$$

If \(U^-(-h_-)\notin U^+([0, h_+])\), we use (3.21) and (3.7a) to compute that

$$\begin{aligned} -\big (U^+(0)-a\big )^2\partial _c Y^+(a, 0)=&\frac{1}{\big (\int _0^{h_+} \frac{1}{(U^+-a)^2}\textrm{d}x_2 \big )^2}\int _0^{h_+} (U^+)'' \\&\bigg (\int _{x_2}^{h_+} \frac{1}{(U^+-a)^2}\textrm{d}x_2' \bigg )^2 \textrm{d}x_2\\ \le&\int _0^{h_+} (U^+)''(x_2) dx_2 =(U^+)'(h_+)-(U^+)'(0),\\ Y^+(a, 0)=&\frac{(U^+)'(0)}{U^+(0)-a}-\frac{1}{\big (U^+(0)-a\big )^2\int _0^{h_+}\frac{1}{(U^+-a)^2}\textrm{d}x_2}. \end{aligned}$$

Using (3.47), this implies that

$$\begin{aligned}&\partial _c F(a, 0, \epsilon )\le -(U^-)'(-h_-)\\&\quad +\epsilon \Big ((U^+)'(h_+)-\frac{2}{\big (U^+(0)-U^-(-h_-) \big )\int _0^{h_+}\frac{1}{(U^+-U^-(-h_-))^2}\textrm{d}x_2} \Big )<0. \end{aligned}$$

* Case 2. \(a=U^+(0)\). It remains to consider \(U^+(0)\notin U^-([-h_-, 0])\). Using (3.31) and (3.32), we compute

$$\begin{aligned}&\partial _c F(a, 0, \epsilon )\le -(U^-)'(-h_-) -\frac{2}{\big (U^-(0)-U^+(0)\big )\int _{-h_-}^0\frac{1}{\big (U^--U^+(0) \big )^2}\textrm{d}x_2}\\&\quad -\epsilon (U^+)'(0)<0. \end{aligned}$$

\(\square \)

Remark 3.2

When \((U^\pm )'>0\), \((U^\pm )''\ne 0\), and \(a=U^-(-h_-)<U^+(0)\), there exists \(\epsilon _0>0\) such that (3.47) holds for all \(\epsilon \in [0, \epsilon _0)\). Particularly, when \(\epsilon =0\), the result in Lemma 3.6 coincides the result for the capillary gravity water wave linearized at monotone convex shear flows.

According to Lemma 3.6, we can apply bifurcation analysis near \(c=a\) and seek instability if \(F(a, k, \epsilon )=0\). We introduce a gravitational strength \(g_*\) which tells us when a bifurcation may happen at \(c=a\). In the following lemma, we first study the roots of \(F(a, \cdot , \epsilon )=0\) based on the value of g relative to \(g_*\). Recall that a is defined in (1.7)

Lemma 3.7

Assume that \(U^\pm \) satisfies (1.3) and \((U^\pm )''\ne 0\). Let

$$\begin{aligned} g_*=\max \{F(a, k, \epsilon )+g\; |\; k\in \mathbb {R}\}, \end{aligned}$$
(3.48)

then the following hold:

  1. (1)

    \(g_*> 0\) for any \(\epsilon >0\).

  2. (2)

    If \(g>g_*\), then \(F(a, k, \epsilon )<0\) for all \(k\in \mathbb {R}\) and \(\epsilon \ge 0\).

  3. (3)

    If one of the following cases holds,

    1. (a)

      \(a=U^+(0)\notin U^-([-h_-, 0])\) and

      $$\begin{aligned} 0<\frac{1}{g(1-\epsilon )}<\int _{-h_-}^0 \frac{1}{\big ( U^-(x_2)-U^+(0)\big )^2}\textrm{d}x_2, \end{aligned}$$
      (3.49)
    2. (b)

      \(a=U^+(h_+)\notin U^-([-h_-, 0])\) and

      $$\begin{aligned} 0<\frac{1}{g(1-\epsilon )}<\int _{-h_-}^0 \frac{1}{\big ( U^-(x_2)-U^+(h_+)\big )^2}\textrm{d}x_2 \end{aligned}$$
      (3.50)
    3. (c)

      \(a=U^+(0)\in U^-([-h_-, 0])\) or \(U^+(h_+)\in U^-([-h_-, 0])\),

    4. (d)

      \(a=U^-(-h_-)\) or \(U^-(0)\),

    then the following hold.

    1. (i)

      If \(0<g=g_*\), then there exists a unique \(k_*>0\) such that \(F(a, \pm k_*, \epsilon )=0, \partial _k F(a, \pm k_*, \epsilon )=0\), and \(F(a, k, \epsilon )<0\) for all \(|k|\ne k_*\).

    2. (ii)

      If \(g\in (0, g_*)\), then there exist \(k_*^+>k_*^->0\) such that

      $$\begin{aligned} F(a, k, \epsilon )<0, \; |k|\notin (k_*^-, k_*^+);&\quad F(a, k, \epsilon )>0,\; |k|\in (k_*^-, k_*^+);\\ F(a, \pm k_*^\pm , \epsilon )=0,&\quad {\mp } \partial _k F(a, k_*^\pm , \epsilon )>0. \end{aligned}$$

Proof

Since \(Y^\pm (c, k)\) are comparable to \({\mp } k\) as |k| becomes large, \(F(a, k, \epsilon )\) behaves like a quadratic function of k with a negative leading coefficient. Hence, \(g_*\) exists. If \(a=U^-(-h_-)\), \(Y^-(a, 0)\) is in the form of (3.32). If \(a=U^-(0)\), (2.22) implies that \(\big (U^-(0)-a\big )^2Y^-(a, 0)=0\). If \(a\notin U^-([-h_-, 0])\), then one can use (3.32) to compute \(F(a, 0, \epsilon )\). If \(a=U^+(0)\), (2.26) implies that \(\big (U^+(0)-a\big )^2Y^+(a, 0)=0\). If \(a=U^+(h_+)\), we apply (3.20) to compute

$$\begin{aligned} Y^+(a, 0)=\frac{(U^+)'(0)}{U^+(0)-a}. \end{aligned}$$
(3.51)

If \(a\notin U^+([0, h_+])\), we apply (3.21) to obtain that

$$\begin{aligned} Y^+(a, 0)=\frac{(U^+)'(0)}{U^+(0)-a}+\frac{1}{\big (U^+(0)-a\big )^2\int _0^{h_+}\frac{1}{(U^+-a)^2}\textrm{d}x_2}. \end{aligned}$$
(3.52)

Using these formulas, one can obtain that

$$\begin{aligned} g_*\ge F(a, 0, \epsilon )+g\ge \epsilon g>0. \end{aligned}$$

If \(a=U^+(0)\notin U^-([-h_-, 0])\),

$$\begin{aligned} F(a, 0, \epsilon )=g(\epsilon -1)+\frac{1}{\int _{-h_-}^0 \frac{1}{(U^--a)^2}\textrm{d}x_2}. \end{aligned}$$
(3.53)

Hence, \(F(a, 0, \epsilon )<0\) if (3.49) holds. If \(a=U^+(h_+)\notin U^-([-h_-, 0])\), \(F(a, 0, \epsilon )\) is in the form of (3.53) and \(F(a, 0, \epsilon )<0\) if (3.50) holds. If \(a=U^-(-h_-)\notin U^+([0, h_+])\),

$$\begin{aligned} F(a, 0, \epsilon )=g(\epsilon -1)-\epsilon \Big (\int _0^{h_+}\frac{1}{(U^+-a)^2}\textrm{d}x_2\Big )^{-1}<0. \end{aligned}$$
(3.54)

If \(a=U^-(-h_-)=U^+(0)\) or \(U^-(-h_-)=U^+(h_+)\), then \(F(a, 0, \epsilon )=g(\epsilon -1)<0\). If \(a=U^-(0)\notin U^+([0, h_+])\), then \(F(a, 0, \epsilon )\) is in the form of (3.54) and \(F(a, 0, \epsilon )<0\). If \(a=U^-(0)=U^+(0)\) or \(U^-(0)=U^+(h_+)\), then \(F(a, 0, \epsilon )=g(\epsilon -1)<0\). The rest of the proof for (3) follows directly from the concavity of \(F(a, k, \epsilon )\) in \(K=k^2\) by (3.12) and the definition of \(g_*\). \(\square \)

In the next lemma, we analyze the roots of \(F(\cdot , k, \epsilon )=0\) near \(c=a\) if \(F(a, k_*, \epsilon )=0\) for some \(k_*>0\) based on the different properties of F at \((a, k_*)\). This is a local result near \((a, k_*)\) for all \(\epsilon \ge 0\).

Lemma 3.8

Assume that \(U^\pm \) satisfies (1.3) and \((U^\pm )''>0\), and \(F(a, k_*, \epsilon )=0\) for some \(k_*>0, \epsilon \ge 0\). If \(\partial _c F(a, k_*, \epsilon )<0\), then there exist \(\delta >0\) and a unique \(\mathcal {C}^-\in C^{1, \alpha }\big ([k_*-\delta , k_*+\delta ], \mathbb {C} \big )\) for any \(\alpha \in [0, 1)\) such that

$$\begin{aligned} \mathcal {C}^-(k_*)=a, \quad F(\mathcal {C}^-(k), k, \epsilon )=0 \quad \forall |k|\in [k_*-\delta , k_*+\delta ]. \end{aligned}$$

And \(\mathcal {C}^-(k)\) satisfies the following:

  1. (1)

    If \(\partial _k F(a, k_*, \epsilon )=0\), then \((\mathcal {C}^-)'(k_*)=0\), \(\mathcal {C}^-_I\equiv 0\) and \(\mathcal {C}^-(k)<a\) for all \(0<|k-k_*|<\delta \). Moreover, \(\pm k_*\) are the only roots of \(F(a, \cdot , \epsilon )=0\).

  2. (2)

    If \(\pm \partial _k F(a, k_*, \epsilon )>0\), then \(\pm (\mathcal {C}^-_R)'(k_*)>0\). Moreover,

    $$\begin{aligned} \mathcal {C}_R^-(k)<a,\; \mathcal {C}_I^-(k)=0, \; \forall 0<\pm (k_*-k)\le \delta ,\\ \mathcal {C}_R^-(k)>a, \; \mathcal {C}_I^-(k)>0, \; \forall 0<\pm (k-k_*)\le \delta . \end{aligned}$$

Proof

Recall that \(\mathcal {I}\) is defined in (3.24). According to Lemmas 2.5 and 2.6, \(F_I\) is not continuous at \(c\in U^-\big ( (-h_-, 0]\big )\cup U^+\big ([0, h_+) \big )\subset \mathbb {C}\) in general. Hence, we extend \(F(c, k, \epsilon )\) by letting

$$\begin{aligned} \tilde{F}(c, k, \epsilon )=\tilde{F}_R(c, k, \epsilon )+i\tilde{F}_I(c, k, \epsilon )\in \mathbb {C}, \quad \tilde{F}=F \; \text {for}\; c_I\ge 0, \end{aligned}$$
(3.55)

be a \(C^{1, \alpha }\) extension of F into a neighborhood of \((a, k)\in \mathbb {C}\times \mathbb {R}\) for any \(\alpha \in [0, 1)\) and each \(\epsilon \ge 0\). By Lemma 3.1, since Cauchy-Riemann equations hold and \(F(c, k, \epsilon )\in \mathbb {R}\) for all \(c<a\), the \(2\times 2\) Jacobian matrix of \(D_c \tilde{F}\) satisfies

$$\begin{aligned} D_c \tilde{F}(a, k_*, \epsilon )=\begin{pmatrix} \partial _{c_R} \tilde{F}_R & \partial _{c_I}\tilde{F}_R \\ \partial _{c_R} \tilde{F}_I & \partial _{c_I} \tilde{F}_I \end{pmatrix}\Big |_{\big (a, k_*, \epsilon \big )}=\partial _c F(a, k_*, \epsilon )I_{2\times 2}. \end{aligned}$$

Since \(\partial _c F(a, k_*, \epsilon )<0\), we apply the Implicit Function Theorem and obtain a \(C^{1, \alpha }\) complex-valued function \(\mathcal {C}^-(k)\) and \(\delta >0\) for any \(\alpha \in [0, 1)\) such that \(\mathcal {C}^-(k_*)=a\) and \(\tilde{F}(\mathcal {C}^-(k), k, \epsilon )=0\) on \([k_*-\delta , k_*+\delta ]\). We know that \(F_R\in C^1\) and \(\partial _{c_R} F_R(a, k_*, \epsilon )=\partial _c F(a, k_*, \epsilon )<0\). According to the Implicit Function Theorem, there exist \(\tilde{\delta }>0\) and a \(C^{1, \alpha }\) real-valued function \(\tilde{\mathcal {C}}(k)\) for any \(\alpha \in [0, 1)\) on \([k_*-\tilde{\delta }, k_*+\tilde{\delta }]\) such that \(\tilde{\mathcal {C}}(k_*)=a\) and \(F_R(\tilde{\mathcal {C}}(k), k, \epsilon )=0\). Since F is real-valued if \(c\le a\), the uniqueness of solutions by the Implicit Function Theorem ensures that \(\mathcal {C}^-(k)=\tilde{\mathcal {C}}(k)\in \mathbb {R}\) if \(\tilde{\mathcal {C}}(k)\le a\). Moreover, \(F\big (\mathcal {C}^-(k), k, \epsilon \big )=\tilde{F}\big (\mathcal {C}^-(k), k, \epsilon \big )=0\).

We first consider the case of \(\partial _k F(a, k_*, \epsilon )=0\) with \(k_*>0\). The uniqueness of \(k_*\) is ensured by (3.12). Since \(\partial _{c_R} F_R(a, k_*, \epsilon )=\partial _c F(a, k_*, \epsilon )<0\) and \(F(a, k, \epsilon )\) is concave in \(K=k^2\), we obtain that

$$\begin{aligned} F_R(c, k_*, \epsilon )>0, \; \forall 0<a-c\ll 1, \quad F_R(a, k, \epsilon )<0, \; \forall k\in \mathbb {R}^+\setminus \{k_*\}. \end{aligned}$$

Then there exist k near \(k_*\) and c near a with \(c<a\) such that \(F_R(c, k, \epsilon )=0\). Moreover, the uniqueness of solutions by the Implicit Function Theorem ensures that \(c=\tilde{\mathcal {C}}(k)\). Therefore, \(\mathcal {C}^-(k)=\tilde{\mathcal {C}}(k)<a\) for \(k\ne k_*\) close to \(k_*\).

Now, we consider the case of \(\partial _k F(a, k_*, \epsilon )>0\). The proof for the case of \(\partial _k F(a, k_*, \epsilon )<0\) is similar. Since \(\partial _c F(a, k_*, \epsilon )<0\),

$$\begin{aligned} (\mathcal {C}^-)'(k_*)=-\frac{\partial _k F(a, k_*, \epsilon )}{\partial _c F(a, k_*, \epsilon )}>0. \end{aligned}$$

Hence, \(\mathcal {C}_R^-(k)>a\) for \(k\in (k_*, k_*+\delta ]\). By the Mean Value Theorem, there exists \(\gamma \) between 0 and \(\mathcal {C}_I^-(k)\) such that

$$\begin{aligned} 0=F_I(\mathcal {C}^-(k), k, \epsilon )=F_I(\mathcal {C}^-_R(k), k, \epsilon )+\mathcal {C}^-_I(k)\partial _{\mathcal {C}_I}F_I\big (\mathcal {C}_R^-(k)+i\gamma , k, \epsilon \big ). \end{aligned}$$

By the \(C^{1, \alpha }\) regularity of F and \(\mathcal {C}^-(k)\), we obtain that

$$\begin{aligned} \mathcal {C}^-_I(k)&=-\frac{F_I\big (\mathcal {C}_R^-(k), k, \epsilon \big )}{\partial _{\mathcal {C}_I} F_I\big (\mathcal {C}_R^-(k)+i\gamma , k, \epsilon \big )}\\&=-\frac{Y^-_I\big (\mathcal {C}_R^-(k), k, \big )\big (U^-(0)-\mathcal {C}_R^-(k) \big )^2-\epsilon Y_I^+\big (\mathcal {C}_R^-(k), k, \big )\big (U^+(0)-\mathcal {C}_R^-(k)\big )^2}{\partial _{\mathcal {C}_I}F_I\big (\mathcal {C}_R^-(k), k, \epsilon \big )+O(|\mathcal {C}_I^-(k)|^\alpha )}\\&=-\frac{Y^-_I\big (\mathcal {C}_R^-(k), k, \big )\big (U^-(0)-a+O(|k-k_*|)\big )^2-\epsilon Y_I^+\big (\mathcal {C}_R^-(k), k, \big )\big (U^+(0)-a+O|k-k_*|\big )^2}{\partial _cF(a, k_*, \epsilon )+O(|k-k_*|^\alpha )}. \end{aligned}$$

Since \((U^\pm )''>0\), using (2.24) and (2.28), we have that the numerator is always positive for small \(\delta \). This implies that \(\mathcal {C}_I^-(k)>0\) when \(0<k-k_*\ll 1\). \(\square \)

Remark 3.3

  1. 1).

    If \(\partial _c F(a, k_*, \epsilon )>0\), to obtain the same results above, the assumption in Lemma 3.8(2) should be changed to \(\pm \partial _k F(a, k_*, \epsilon )<0\). 2). If \((U^\pm )''<0\), the signs in the result of Lemma 3.8(2) should be reversed.

In what follows, we complete the picture of the branch \(c^-(k)\) by combining the Lemmas 3.5–3.8 and finish the proof of Theorems 1.3 and 1.4.

Proof of Theorem 1.3 - 1.4

Theorem 1.3 follows directly from Lemma 3.7 (2) and Remark 3.1 (2). To prove Theorem 1.4(1), one notices that by Lemma 3.7 (3)(i) and Lemma 3.8, there exist \(\delta >0\) and a unique \(\mathcal {C}^-(k)\) on \([k_*-\delta , k_*+\delta ]\) such that \(F(\mathcal {C}^-(k), k, \epsilon )=0\) and \(\mathcal {C}^-(k_*)=a\). Let \(k_0\) be the one in Theorem 1.2(1). Based on the argument above (3.46) in the proof of Lemma 3.5, there exists \(R>0\) such that \(F(k, c, \epsilon )\ne 0\) for all \(k\in [-k_0, k_0]\) and \(|c|>R\). Suppose \(\Omega \) be a disk on the complex plane centered at \(\frac{-R-1+\mathcal {C}^-(k_*+\frac{\delta }{4})}{2}\) with radius \(\frac{|-R-1-\mathcal {C}^-(k_*+\frac{\delta }{4})|}{2}\). According to Theorem 1.1, \(F(c, k, \epsilon )\ne 0\) for all \(k\in \mathbb {R}\) on \(\partial \Omega {\setminus }\{-R-1, \mathcal {C}^-(k_*+\frac{\delta }{4})\}\). Now, we consider n(k) in (3.46) with \(\Omega \) defined as above. Since n(k) is continuous and \(n(k_0)=1\), \(n(k)\equiv 1\) on \(|k|\in [k_*+\frac{\delta }{2}, k_0]\). Hence, \(c^-(k)\) can be extended to an analytic function for all \(|k|\in (k_*, \infty )\). A similar argument can be applied on \(|k|\in (0, k_*)\).

It remains to prove Theorem 1.4(2)(ii) under the assumption (b). Suppose that \(a=U^-(-h_-)\) and \(U^-(-h_-)\notin U^+([0, h_+])\). Then (3.45) shows that \(g_*\ge F(U^-(-h_-), 0, \epsilon )+g>0\) if \(\epsilon >0\). Similarly, one may check that \(F(U^+(0), 0, \epsilon )+g>0\) by using (3.23) if \(a=U^+(0)\notin U^-([-h_-, 0])\) and \(\epsilon >0\). In the case of \(a=U^-(-h_-)=U^+(0)\), since \(F(a, 0, \epsilon )+g=0\), \(g_*=0\) if and only if \(\partial _K F(a, 0, \epsilon )\le 0\), where \(K:=k^2\). The proof of the theorem is completed. \(\square \)

4 Ocean-Air Model

In this section, we study the spectral distribution for the case of \(0\le \epsilon \ll 1\). We shall treat the interface problem as a perturbation to the capillary-gravity water wave problem. According to Theorem 1.1, all the unstable modes must have wave speed lie in a big upper semicircle (1.9) if (1.8) is satisfied. We first locate all the possible locations of neutral limiting modes inside [a, b] where a, b are defined in (1.7) as \(\epsilon \) tends to zero, and then seek unstable modes nearby when \(\epsilon \) is perturbed. We give a complete picture of the eigenvalue distribution for a special case of \((U^\pm )'>0, (U^\pm )''\ne 0\), and \(U^-(0)<U^+(0)\). In the next lemma, we first address the possible locations of neutral limiting mode as \(\epsilon \) tends to 0. We recall \(\mathcal {I}\) defined in (3.24).

Lemma 4.1

Assume that \(U^\pm \) satisfy (1.3) and \(\mathcal {I}=\emptyset \). If a sequence of unstable modes \((c_n, k_n, \epsilon _n)\) with \(\epsilon _n>0\) converges to \((c_*, k_*, 0)\) as \(n \rightarrow \infty \), then \(c_*=\lim _{n\rightarrow \infty }c_n \in \{U^-(-h_-)\}\cup U^+([0, h_+]){\setminus } U^-(0)\) or

$$\begin{aligned}&\Omega :=\left\{ c_R+ic_I, c_R\in \mathbb {R}, c_I\in \mathbb {R}^+| \left( c_R-\frac{U^-(-h_-)+U^-(0)}{2}\right) ^2\right. \\&\quad \left. +c_I^2<\left( \frac{U^-(-h_-)-U^-(0)}{2}\right) ^2\right\} . \end{aligned}$$

Proof

Recall \(F(c, k, \epsilon )\) defined in (3.1). Lemma 3.1 implies that \(\lim _{n\rightarrow \infty }F(c_n, k_n, \epsilon _n)=F(c_*, k_*, 0)=0\). According to Theorem 1.1 in [11], \(c_*\in (\mathbb {R}\setminus U^-((-h_-, 0]))\cup \Omega \). Since \(F(c_n, k_n, \epsilon _n)=0\), Theorem 1.1 implies that \(c_n\) stays in the upper semicircle (1.9). By the definition of \((c_n, k_n, \epsilon _n)\), for any \(0<\alpha \ll 1\), there exists \(N_0>0\) such that whenever \(n>N_0\), \(\epsilon _n, |c_n-c_*|, |k_n-k_*|<\alpha \). Hence, \(c_*\in \Big ([a, b]{\setminus } U^-\big ((-h_-, 0]\big ) \Big )\cup \Omega \), where a and b are defined in (1.7). We shall discuss the following possible cases to shrink this region.

Case 1. \(U^-(0)<U^+(0)\). Suppose that \(c_*\in \big (U^-(0), U^+(0)\big )\). By Remark 3.1, when \(\epsilon =0\), \(c^+(k)\) obtained in Lemma 3.2 can be extended to all \(k\in \mathbb {R}\). This shows that \(c_*\) is a simple root of \(F(c, k_*, 0)\) and according to (3.10), \(\partial _c F(c_*, k_*, 0)>0\). Since \(F(c, k, \epsilon )\) is analytic near \(c=c*\), there exist \(\delta , \epsilon _0>0\) and a smooth function \(\mathcal {C}(k, \epsilon ): [k_*-\delta , k_*+\delta ]\times [-\epsilon _0, \epsilon _0]\rightarrow \mathbb {C}\) such that

$$\begin{aligned} F\big (\mathcal {C}(k, \epsilon ), k, \epsilon \big )=0, \quad \mathcal {C}(k_*, 0)=c_*. \end{aligned}$$

By the uniqueness of solutions ensured by the Implicit Function Theorem and the definition of the sequence \(\{(c_n, k_n, \epsilon _n)\}\), there exists \(N>0\) such that for all \(n\ge N\), \(\mathcal {C}(k_n, \epsilon _n)=c_n\). This implies that \(\mathcal {C}_I(k_n, \epsilon _n)>0\).

Since \(c_*\notin U^-([-h_-, 0])\cup U^+([0, h_+])\), \(F_R\in C^\infty \) and \(\partial _{c_R}F_R(c_*, k_*, 0)=\partial _c F(c_*, k_*, 0)>0\). Then the Implicit Function Theorem implies that there exists a smooth real-valued function \(\tilde{C}(k, \epsilon )\) for k near \(k_*\) and \(\epsilon \) near 0 such that

$$\begin{aligned} \tilde{C}(k_*, 0)=c_*, \quad F_R\big (\tilde{C}(k, \epsilon ), k, \epsilon \big )=0. \end{aligned}$$

Since \(F_I(c, k, \epsilon )=0\) if \(c\in \big (U^-(0), U^+(0) \big )\), the uniqueness of solutions by the Implicit Function Theorem implies that \(\mathcal {C}(k, \epsilon )=\tilde{C}(k, \epsilon )\in \mathbb {R}\) if \(\tilde{C}\in \big (U^-(0), U^+(0)\big )\). Hence, \(\mathcal {C}(k_n, \epsilon _n)\in \mathbb {R}\) for all \(n>N\), which contradicts to the previous argument that \(\mathcal {C}_I(k_n, \epsilon _n)>0\). Therefore, \(c_*\notin \big (U^-(0), U^+(0)\big )\).

Case 2. \(U^+(h_+)\le U^-(-h_-)\). It suffices to consider \(U^+(h_+)<U^-(-h_-)\). According to Remark 3.1, \(\partial _c F(c_*, k_*, 0)<0\). By a similar argument as above, the lemma is proved. \(\square \)

This lemma implies that instability might occur near \(U^-(-h_-)\) and \(U^+([0, h_+])\) for \(0<\epsilon \ll 1\). We shall analyze the bifurcation near \(U^+(h_+)\) and \(U^+(0)\) and give a complete picture of the eigenvalue distribution when \(0<\epsilon \ll 1\). The strategy can be adapted to analyze some other cases including all the cases of \((U^\pm )', (U^\pm )''\ne 0\) and \(\mathcal {I}\ne \emptyset \).

Lemma 4.2

Suppose that \(U^\pm \) satisfy (1.3), \((U^\pm )'> 0, U^-(0)<U^+(0)\), \((U^\pm )''\ne 0\). Assume that \(F\big (U^+(h_+), k_0, 0\big )=0\) for some \(k_0> 0\). Then the following hold:

  1. (1)

    Suppose that \(\partial _k F\big (U^+(h_+), k_0, 0 \big )\ne 0\). Then there exist \(\epsilon _0>0\) such that for each \(\epsilon \in (0, \epsilon _0)\) there exist \(\delta _0>0\), \(k(\epsilon )\in C^{1, \beta }\big ([0, \epsilon _0], \mathbb {R}^+\big )\), and \(\mathcal {C}^+(k)\in C^{1, \alpha }\big ([k(\epsilon )-\delta _0, k(\epsilon )+\delta _0] \big )\) for any \(\alpha , \beta \in [0, 1)\) such that

    $$\begin{aligned}&F\big (U^+(h_+),\pm k(\epsilon ), \epsilon \big )=0,&k(0)=k_0, \\&F\big (\mathcal {C}^+(k), \pm k, \epsilon \big )=0, \; \\&\qquad \forall |k|\in [k(\epsilon )-\delta _0,\;&k(\epsilon )+\delta _0], \quad \mathcal {C}^+\big (\pm k(\epsilon )\big )=U^+(h_+). \end{aligned}$$

    And for \(c_I\in [0, U^+(h_+)-U^+(0)]\) and \(|c_R-U^+(h_+)|\le U^+(h_+)-U^+(0)\), \(\mathcal {C}^+(k)\) is the unique root of \(F(\cdot , k, \epsilon )=0\) with \(|k|\in [k(\epsilon )-\delta _0, k(\epsilon )+\delta _0]\). Moreover, the following hold.

    1. (a)

      If \(\partial _k F\big ( U^+(h_+), k_0, 0\big )<0\), then

      1. (i)

        \(k\) \('(0)>0\) and \(k(\epsilon )>k_0\);

      2. (ii)

        \(\mathcal {C}^+(k)>U^+(h_+)\) for any \(|k|\in \big (k(\epsilon ), k(\epsilon )+\delta _0]\) and

        $$\begin{aligned} (U^+)''\mathcal {C}^+_I(k)<0,\quad \mathcal {C}_R^+(k)<U^+(h_+),\; \forall |k|\in [k(\epsilon )-\delta _0, k(\epsilon )\big ). \end{aligned}$$
    2. (b)

      If \(\partial _k F\big (U^+(h_+), k_0, 0\big )>0\), then

      1. (i)

        \(k'(0)<0\) and \(k(\epsilon )<k_0\);

      2. (ii)

        \(\mathcal {C}^+(k)>U^+(h_+)\) for any \( |k|\in [k(\epsilon )-\delta _0, k(\epsilon )\big )\) and

        $$\begin{aligned} (U^+)''\mathcal {C}^+_I(k)<0, \quad \mathcal {C}_R^+(k)<U^+(h_+),\; |k|\in \big ( k(\epsilon ), k(\epsilon )+\delta _0]. \end{aligned}$$
  2. (2)

    Suppose that \(\partial _k F\big (U^+(h_+), k_0, 0\big )=0\). Then there exist \(\epsilon _1>0\) such that for each \(\epsilon \in (0, \epsilon _1]\) there exist \(k_2(\epsilon )>k_0>k_1(\epsilon )\ge 0\), \(\delta >0\), and \(\mathcal {C}_{1, 2}^+\in C^{1, \alpha }\big ([k_{1, 2}(\epsilon )-\delta , k_{1, 2}(\epsilon )+\delta ], \mathbb {C}\big )\) for any \(\alpha \in [0, 1)\) such that

    $$\begin{aligned}&F\big (U^+(h_+), \pm k_{1, 2}(\epsilon ), \epsilon \big )=0,\\&F\big (\mathcal {C}_{1, 2}^+(k), \pm k, \epsilon \big )=0, \; \forall |k|\in [k_{1, 2}(\epsilon )-\delta , k_{1, 2}(\epsilon )+\delta ], \quad \\&\quad \mathcal {C}^+_{1, 2}\big (\pm k_{1, 2}(\epsilon )\big )=U^+(h_+). \end{aligned}$$

    Moreover, \(\mathcal {C}_{1, 2}^+(k)\) satisfy that

    $$\begin{aligned}&\mathcal {C}_1^+(k)>U^+(h_+), \; \forall |k|\in [k_1(\epsilon )-\delta , k_1(\epsilon )\big ), \quad \mathcal {C}_2^+(k)>U^+(h_+), \;\\&\quad \forall |k|\in \big (k_2(\epsilon ), k_2(\epsilon )+\delta ], \\&\mathcal {C}_{1 _R}^+(k)<U^+(h_+), \quad (U^+)''\mathcal {C}_{1 _I}^+(k)<0, \; \forall |k|\in \big (k_1(\epsilon ), k_1(\epsilon )+\delta ],\\&\mathcal {C}_{2 _R}^+(k)<U^+(h_+), \quad (U^+)''\mathcal {C}_{2 _I}^+(k)<0, \; \forall |k|\in [k_2(\epsilon )-\delta , k_2(\epsilon )\big ). \end{aligned}$$

    In addition, for \(|c_R-U^+(h_+)|<U^+(h_+)-U^+(0)\) and \(c_I\in [0, U^+(h_+)-U^+(0)]\), \(\mathcal {C}^+_1\) is the unique root of \(F(\cdot , k, \epsilon )=0\) with \(|k|\in [k_1(\epsilon )-\delta , k_1(\epsilon )+\delta ]\), \(\mathcal {C}^+_2\) is the unique root of \(F(\cdot , k, \epsilon )=0\) with \(|k|\in [k_2(\epsilon )-\delta , k_2(\epsilon )+\delta ]\).

Proof

Let \(K:=k^2\). We shall prove the case of \(\partial _K F\big (U^+(h_+), k_0, 0\big )<0\). The proof for the case of \(\partial _K F\big (U^+(h_+), k_0, 0\big )>0\) is similar. Since \(\partial _k F\big (U^+(h_+), k_0, 0\big )<0\) and \(k_0>0\), \(\partial _K F\big (U^+(h_+), k_0, 0\big )<0\). And there exist \(0<\epsilon _0\ll 1\) and a real-valued smooth function \(K(\epsilon )\) such that \(F\big (U^+(h_+), K(\epsilon ), \epsilon \big )=0\) for all \(\epsilon \in [-\epsilon _0, \epsilon _0]\) and \(K(0)=k_0^2\). We compute that

$$\begin{aligned} \partial _\epsilon F(c, k, \epsilon )=-\big (U^+(0)-c\big )^2Y^+(c, k)+(U^+)'(0)\big (U^+(0)-c\big )+g. \end{aligned}$$
(4.1)

We apply (3.51) to obtain that

$$\begin{aligned} \partial _\epsilon F\big (U^+(h_+), 0, 0\big )=g>0 \end{aligned}$$
(4.2)

According to (3.15b),

$$\begin{aligned} \partial _{\epsilon K}F\big (U^+(h_+), k, \epsilon \big )=-\big (U^+(0)-U^+(h_+) \big )^2\partial _K Y^+\big ( U^+(h_+), k\big )>0, \end{aligned}$$

for all \(k\in \mathbb {R}\) and \(\epsilon \ge 0\). Hence, \(\partial _\epsilon F\big ( U^+(h_+), k_0, 0\big )>\partial _\epsilon F\big (U^+(h_+), 0, 0\big )>0\). Using the assumption that \(\partial _K F\big (U^+(h_+), k_0, 0\big )<0\), we obtain that

$$\begin{aligned} K'(0)=-\frac{\partial _\epsilon F}{\partial _K F}\big (U^+(h_+), k_0, 0 \big )>0. \end{aligned}$$

Then \(\epsilon _0\) can be chosen small enough such that \(K'(\epsilon )>0\) for all \(\epsilon \in [0, \epsilon _0]\).

For each \(\epsilon \in (0, \epsilon _0]\), we consider \(F\big (U^+(h_+), k(\epsilon ), \epsilon \big )\), where \(k(\epsilon )=\sqrt{K(\epsilon )}\). We extend \(F(c, k, \epsilon )\) by letting \(\tilde{F}\) be defined in (3.55). Then \(\tilde{F}\) is a \(C^{1, \alpha }\) extension of F into a neighborhood of \(\big (U^+(h_+), k(\epsilon )\big )\in \mathbb {C}\times \mathbb {R}\) for any \(\alpha \in [0, 1)\). Since \(F(c, k, \epsilon )\in \mathbb {R}\) for all \(c\ge U^+(h_+)\), we use Cauchy-Riemann equations to compute

$$\begin{aligned} & D_c \tilde{F}\big (U^+(h_+), k(\epsilon ), \epsilon )=\begin{pmatrix} \partial _{c_R} \tilde{F}_R & \partial _{c_I}\tilde{F}_R \\ \partial _{c_R} \tilde{F}_I & \partial _{c_I} \tilde{F}_I \end{pmatrix}\Big |_{\big (U^+(h_+), k(\epsilon ), \epsilon \big )}\nonumber \\ & \quad =\partial _c F\big (U^+(h_+), k(\epsilon ), \epsilon \big )I_{2\times 2}. \nonumber \\ \end{aligned}$$
(4.3)

We apply the Mean Value Theorem and the smoothness of \(K(\epsilon )\) to obtain that

$$\begin{aligned} \partial _cF\big (U^+(h_+), k(\epsilon ), \epsilon \big )=&\partial _c F\big (U^+(h_+), k(\epsilon ), 0\big )+\partial _{c\epsilon } F\big (U^+(h_+), k(\epsilon ), \theta _1\big )\epsilon \\ =&\partial _c F\big (U^+(h_+), k_0, 0\big ) \\&+\partial _{Kc} F\big (U^+(h_+), k_0+\theta _2, 0\big )\big (K(\epsilon )-k_0^2\big )+O(\epsilon )\\ =&\partial _c F\big (U^+(h_+), k_0, 0\big ) +O(\epsilon ), \end{aligned}$$

where \(\theta _1\in (0, \epsilon )\) and \(\theta _2\in \big (0, k(\epsilon )-k_0\big )\). By (3.11), \(\partial _c F\big (U^+(h_+), k_0, 0\big )>0\). Since \(\epsilon _0 \ll 1\),

$$\begin{aligned} \partial _c F\big (U^+(h_+), k(\epsilon ), \epsilon \big )>0, \quad \forall \epsilon \in [0, \epsilon _0]. \end{aligned}$$
(4.4)

Therefore, there exist \(0<\beta \ll 1\) and a \(C^{1, \alpha }\) function \(\mathcal {C}^+(k):[k(\epsilon )-\beta , k(\epsilon )+\beta ]\rightarrow \mathbb {C}\) for any \(\alpha \in [0, 1)\) such that

$$\begin{aligned}&\tilde{F}\big (\mathcal {C}^+(k), k, \epsilon \big )=0, \quad \mathcal {C}^+\big (k(\epsilon )\big )=U^+(h_+), \nonumber \\&\quad \mathcal {C}_R^+(k)> U^-(0). \end{aligned}$$
(4.5)

Since \(F_R\in C^1\) and \(\partial _{c_R}F_R\big (U^+(h_+), k(\epsilon ), \epsilon \big )=\partial _c F\big (U^+(h_+), k(\epsilon ), \epsilon \big )>0\), there is a \(C^1\) real-valued function \(\tilde{C}(k)\) for k near \(k(\epsilon )\) such that

$$\begin{aligned} \tilde{C}\big (k(\epsilon )\big )=U^+(h_+), \quad F_R\big (\tilde{C}(k), k, \epsilon \big )=0. \end{aligned}$$

Since \(F_I(c, k, \epsilon )=0\) for all \(c\ge U^+(h_+)\), the uniqueness of solutions by the Implicit Function Theorem ensure that \(\tilde{C}(k)=\mathcal {C}^+(k)\in \mathbb {R}\) if \(\tilde{C}\ge U^+(h_+)\).

By the Mean Value Theorem and smoothness of \(K(\epsilon )\), we use the smallness of \(\epsilon _0\) to obtain that

$$\begin{aligned} \partial _K F\big ( U^+(h_+), k(\epsilon ), \epsilon \big ) =\partial _K F\big (U^+(h_+), k_0, 0\big )+O(\epsilon )<0. \end{aligned}$$
(4.6)

Hence, we compute

$$\begin{aligned} \tilde{C}'\big (k(\epsilon )\big )=(\mathcal {C}^+)'\big (k(\epsilon ) \big )=-\frac{2k(\epsilon )\partial _K F}{\partial _c F}\big (U^+(h_+), k(\epsilon ), \epsilon \big )>0. \end{aligned}$$

Hence, \(\tilde{C}(k)=\mathcal {C}^+(k)>U^+(h_+)\) for k slightly larger than \(k(\epsilon )\). When \(k\in [k(\epsilon )-\beta , k(\epsilon )\big )\), \(\mathcal {C}^+_R(k)<U^+(h_+)\). In this case, we consider

$$\begin{aligned} 0=\tilde{F}_I\big (\mathcal {C}^+(k), k, \epsilon \big )=F_I\big (\mathcal {C}_R^+(k), k, \epsilon \big )+\mathcal {C}_I^+(k)\partial _{c_I}\tilde{F}_I\big (\mathcal {C}_R^+(k)+i\theta , k, \epsilon \big ), \end{aligned}$$

where \(|\theta |\in \big (0, \mathcal {C}_I^+(k)\big )\). then

$$\begin{aligned} \mathcal {C}_I^+(k)=&-\frac{F_I\big (\mathcal {C}_R^+(k), k, \epsilon \big )}{\partial _{c_I}\tilde{F}_I\big (\mathcal {C}_R^+(k)+i\theta , k, \epsilon \big )}\\ =&- \frac{\big (U^-(0)-\mathcal {C}_R^+(k)\big )^2Y_I^-\big (C_R^+(k), k \big )-\epsilon \big (U^+(0)-\mathcal {C}_R^+(k)\big )^2Y_I^+\big (\mathcal {C}_R^+(k), k\big )}{\partial _{c_I}\tilde{F}_I\big (\mathcal {C}_R^+(k), k, \epsilon \big )+O\big (|\mathcal {C}_I^+(k)|^\alpha \big )+O(\epsilon )}. \end{aligned}$$

By (2.24), (4.5), and (2.28) \(Y_I^-\big (C_R^+(k), k \big )=0\). Hence,

$$\begin{aligned} \mathcal {C}_I^+(k)=&\frac{\epsilon \big (U^+(0)-\mathcal {C}_R^+(k)\big )^2Y_I^+\big (\mathcal {C}_R^+(k), k\big )}{\partial _{c_I}F_I\big (U^+(h_+), k(\epsilon ), \epsilon \big )+O\big ( |k-k(\epsilon )|^\alpha \big )}\\ =&\frac{\epsilon \big (U^+(0)-U^+(h_+)+O(|k-k(\epsilon )|)\big )^2Y_I^+\big (\mathcal {C}_R^+(k), k\big )}{\partial _c F\big (U^+(h_+), k(\epsilon ), \epsilon \big )+O\big ( |k-k(\epsilon )|^\alpha \big )}. \end{aligned}$$

By (2.28), the statement (1) is proved.

Now we consider the case of \(\partial _k F\big (U^+(h_+), k_0, \epsilon \big )=0\). Since \(\partial _\epsilon F\big (U^+(h_+), k_0, \epsilon \big )>0\), there exist \(0<\delta \ll 1\) and \(\Gamma (k)\in C^\infty ([k_0-\delta , k_0+\delta ], \mathbb {R})\) such that

$$\begin{aligned} F\big (U^+(h_+), k, \Gamma (k)\big )=0, \; \Gamma (k_0)=0. \end{aligned}$$

Since \(\partial _K F\big (U^+(h_+), k_0, \epsilon \big )=0\), we use (3.12) to obtain that

$$\begin{aligned} \partial _{KK}\Gamma (k_0)=-\frac{\partial _{KK}F}{\partial _\epsilon F}\big (U^+(h_+), k_0, \epsilon \big )>0. \end{aligned}$$

Hence, \(\Gamma (k)\ge 0\) on \([k_0-\delta , k_0+\delta ]\). Let \(\epsilon _1:=\sup _{|k|\in [k_0-\delta , k_0+\delta ]}\Gamma (k)\). Then for each \(\epsilon \in (0, \epsilon _1)\), there exist \(k_2(\epsilon )>k_0>k_1(\epsilon )> 0\) such that

$$\begin{aligned} F\big (U^+(h_+), k_2(\epsilon ), \epsilon \big )=F\big (U^+(h_+), k_1(\epsilon ), \epsilon \big )=0. \end{aligned}$$

Thanks to (3.12), we have

$$\begin{aligned} \partial _K F\big (U^+(h_+), k_2(\epsilon ), \epsilon \big )<0, \quad \partial _K F\big (U^+(h_+), k_1(\epsilon ), \epsilon \big )>0. \end{aligned}$$

For such \(\epsilon \), we apply the bifurcation analysis near \((U^+(h_+), k_1(\epsilon ), \epsilon \big )\) and \((U^+(h_+), k_2(\epsilon ), \epsilon \big )\) respectively, which is similar as the proof for the case of \(\partial _K F\big (U^+(h_+), k_0, \epsilon \big )<0\). The proof is completed. \(\square \)

In the following, we study the case where \(U^+(h_+)\notin U^-([[-h_-, 0])\) is the maximum speed of both fluids (in particular, if \(\big (U^+(h_+), 0\big )\) is a neutral mode in the capillary gravity water wave problem linearized at monotone shear flow \(\big (U^-(x_2), 0 \big )\), we discuss the possible bifurcation near \(c=U^+(h_+)\)):

Lemma 4.3

Suppose that \(U^\pm \) satisfy (1.3), \(U^-(0)<U^+(0)\), \((U^\pm )'> 0\), \((U^+)''\ne 0\). Assume that \(F\big (U^+(h_+), 0, 0 \big )=0\). Then the following hold:

  1. (1)

    There exist \(\epsilon _0>0\) and \(\bar{C}(\epsilon )\in C^{1, \alpha }([0, \epsilon _0], \mathbb {C})\) for any \(\alpha \in [0, 1)\) such that

    $$\begin{aligned} F\big (\bar{C}(\epsilon ), 0, \epsilon \big ) =0, \; \forall \epsilon \in [0, \epsilon _0], \quad \bar{C}(0)=U^+(h_+). \end{aligned}$$

    Moreover, \(\bar{C}\) satisfies that

    $$\begin{aligned} \bar{C}_R(\epsilon )<U^+(h_+), \quad (U^+)''\bar{C}_I(\epsilon )<0,\; \forall \epsilon \in (0, \epsilon _0]. \end{aligned}$$
  2. (2)

    Let \(K:=k^2\). If \(\partial _K F\big (U^+(h_+), 0, 0\big )\le 0\), then there exist \(\epsilon _1>0\) and \(k(\epsilon )\in C^\infty ([0, \epsilon _1], \mathbb {R}^+\cup \{0\})\) such that for each \(\epsilon \in (0, \epsilon _1]\),

    $$\begin{aligned} F\big (U^+(h_+), \pm k(\epsilon ), \epsilon \big )=0, \quad k(\epsilon )>0,\; \text {and}\; k(0)=0. \end{aligned}$$

    Moreover, there exist \(\delta _0>0\) and \(\mathcal {C}^+\in C^{1, \beta }([k(\epsilon )-\delta _0, k(\epsilon )+\delta _0], \mathbb {C})\) for any \(\beta \in [0, 1)\) such that

    $$\begin{aligned}&F\big (\mathcal {C}^+(k), \pm k, \epsilon \big )=0, \quad \mathcal {C}^+\big ( k(\epsilon )\big )=U^+(h_+),\quad \mathcal {C}^+(k)>U^+(h_+), \;\\&\quad \forall |k|\in \big (k(\epsilon ), k(\epsilon )+\delta _0], \\&\quad (U^+)''\mathcal {C}_I^+(k)<0, \quad \mathcal {C}_R^+(k)<U^+(h_+), \; \forall |k|\in [k(\epsilon )-\delta _0, k(\epsilon )\big ). \end{aligned}$$

    In addition, for \(c_I\in [0, U^+(h_+)-U^+(0)]\), \(|c_R-U^+(h_+)|<U^+(h_+)-U^+(0)\), \(\mathcal {C}^+(k)\) is the only root of \(F(\cdot , k, \epsilon )=0\) with \(|k|\in [k(\epsilon )-\delta _0, k(\epsilon )+\delta _0]\).

Proof

We extend \(F(c, k, \epsilon )\) into a neighborhood of \(\big (U^+(h_+), 0\big )\in \mathbb {C}\times \mathbb {R}\) by letting \(\tilde{F}\) be defined in (3.55). Using Cauchy-Riemann equations, \(D_c\tilde{F}\big (U^+(h_+), 0, 0\big )\) is in the form of (4.3). By (3.11), \(\partial _c F\big (U^+(h_+), 0, 0\big )>0\). Hence, there exist \(0<\epsilon _0\ll 1\) and a \(C^{1, \alpha }\) function \(\bar{C}(\epsilon ):[-\epsilon _0, \epsilon _0]\rightarrow \mathbb {C}\) for any \(\alpha \in [0, 1)\) such that

$$\begin{aligned} \tilde{F}\big (\bar{C}(\epsilon ), 0, \epsilon \big )=0, \quad \bar{C}(0)=U^+(h_+). \end{aligned}$$

By a similar argument as the proof in Lemma 4.2, we obtain that, \(F\big ( \bar{C}(\epsilon ), 0,\epsilon \big )=\tilde{F}\big (\bar{C}(\epsilon ), 0, \epsilon \big )\). Using (4.2), we obtain that \((\bar{C})'(0)=-\frac{\partial _\epsilon F}{\partial _c F}\big (U^+(h_+), 0, 0\big )<0\). Then, for each \(\epsilon \in (0, \epsilon _0)\), \(\bar{C}_R(\epsilon )<U^+(h_+)\). Moreover, from the Mean Value Theorem, for such \(\epsilon \), there exists \(|\theta |\in (0, \epsilon )\) such that

$$\begin{aligned} 0=\tilde{F}_I\big (\bar{C}(\epsilon ), 0, \epsilon \big )=F_I\big (\bar{C}_R(\epsilon ), 0, \epsilon \big )+\bar{C}_I(\epsilon )\partial _{c_I}\tilde{F}_I\big (\bar{C}_R(\epsilon )+i\theta , 0, \epsilon \big ). \end{aligned}$$

Similar as the argument in the proof of Lemma 4.2, using the \(C^{1, \alpha }\) regularity of F and \(\bar{C}(\epsilon )\), we obtain that

$$\begin{aligned} \bar{C}_I(\epsilon )=&-\frac{F_I\big (\bar{C}_R(\epsilon ), 0, \epsilon \big )}{\partial _{c_I}\tilde{F}_I\big (\bar{C}_R(\epsilon )+i\theta , 0, \epsilon \big )} \\ =&\frac{\epsilon Y_I^+\big (\bar{C}_R(\epsilon ), 0\big )\big (U^+(0)-U^+(h_+)+O(\epsilon )\big )^2}{\partial _c F\big (U^+(h_+), 0, 0\big )+O(\epsilon )}. \end{aligned}$$

The proof of statement (1) is completed.

Let \(K:=k^2\). To prove the statement (2), we notice that the proof for the case of \(\partial _K F\big (U^+(h_+), 0, 0\big )<0\) is similar as the one in Lemma 4.2. It remains to consider the case of \(\partial _K F\big (U^+(h_+), 0, 0\big )=0\). By (4.2), there exist \(0<K_0\ll 1\) and \(\Gamma \in C^\infty ([-K_0, K_0], \mathbb {R})\) such that

$$\begin{aligned} F\big (U^+(0), k, \Gamma (K)\big )=0, \quad \Gamma (0)=0. \end{aligned}$$

Moreover, by a direct computation, \(\partial _{KK}\Gamma (0)=-\frac{\partial _{KK}F}{\partial _\epsilon F}\big (U^+(h_+), 0, 0 \big )>0\). Let \(k_0:=\sqrt{K_0}\). For any \(k\in (0, k_0)\), we have \(\Gamma (K)>0\) and \(\Gamma '(K)>0\). Let \(\epsilon _0:=\sup _{(0, k_0)}\Gamma (k)\), \(K(\epsilon ):=\Gamma ^{-1}(\epsilon )>0\), and \(k(\epsilon ):=\sqrt{K(\epsilon )}\). Then for each \(\epsilon \in (0, \epsilon _0)\),

$$\begin{aligned}&F\big (U^+(h_+), k(\epsilon ), \epsilon \big )=0,\\ \partial _c F\big (U^+(h_+), k(\epsilon ),&\epsilon \big )=\partial _c F\big (U^+(h_+), 0, 0\big )+O(\epsilon )>0. \end{aligned}$$

Moreover, we use (3.15b) to compute that

$$\begin{aligned} \partial _K F\big (U^+(h_+), k(\epsilon ), \epsilon \big )=&\partial _K F\big (U^+(h_+), k(\epsilon ), 0\big )-\epsilon \partial _K Y^+\big (U^+(0)-U^+(h_+)\big )^2\\<&\partial _K F\big (U^+(h_+), 0, 0\big )+O(\epsilon )<0. \end{aligned}$$

The rest of the proof is similar as the proof for Lemma 4.2(1a). \(\square \)

Using a similar strategy, we can obtain results for small \(\epsilon \) if the one fluid free boundary problem has a neutral mode at \(\big (U^+(0), k_0\big )\) with \(k_0>0\) or \(\big (U^+(0), 0\big )\).

Lemma 4.4

Suppose that \(U^\pm \) satisfy (1.3), \((U^\pm )'>0\), \(U^-(0)<U^+(0)\), \((U^\pm )''\ne 0\). Assume that there exists some \(k_0>0\) such that \(F\big (U^+(0), k_0, 0\big )=0\). Then the following hold:

  1. (1)

    Suppose that \(\partial _k F\big (U^+(0), k_0, 0\big )\ne 0\). Then there exist \(\epsilon _0>0\) such that for each \(\epsilon \in (0, \epsilon _0)\), there exist \(\delta _0>0\), \(k(\epsilon )\in C^{1, \alpha }([0, \epsilon _0], \mathbb {R}^+)\), and \(\mathcal {C}^+(k)\in C^{1, \alpha }\big ([k(\epsilon )-\delta _0, k(\epsilon )+\delta _0]\big )\) for any \(\alpha \in [0, 1)\) such that

    $$\begin{aligned} F\big (U^+(0), \pm k(\epsilon ), \epsilon \big )=0,&\quad k(0)=k_0, \\ F\big (\mathcal {C}^+(k), \pm k, \epsilon \big )=0, \; \forall |k|\in [k(\epsilon )-\delta _0,\;&k(\epsilon )+\delta _0], \text {and} \; \mathcal {C}^+\big (k(\epsilon )\big )=U^+(0). \end{aligned}$$

    Let \(\rho :=\min \{U^+(h_+)-U^+(0), U^+(0)-U^-(0)\}\). For \(c_I\in [0, \rho ]\) and \(|c_R-U^+(0)|\le \rho \), \(\mathcal {C}^+(k)\) is the only root of \(F(\cdot , k, \epsilon )=0\) with \(|k|\in [k(\epsilon )-\delta _0, k(\epsilon )+\delta _0]\). Moreover, the following hold.

    1. (a)

      If \(\partial _k F\big (U^+(0), k_0, 0\big )<0\), then

      1. (i)

        \(k'(\epsilon )>0\) and \(k(\epsilon )>k_0\);

      2. (ii)

        \(\mathcal {C}^+(k)<U^+(0)\) for any \(|k|\in [k(\epsilon )-\delta _0, k(\epsilon )\big )\) and

        $$\begin{aligned} (U^+)''\mathcal {C}_I^+(k)<0,\quad \mathcal {C}_R^+(k)>U^+(0), \; \forall |k|\in \big (k(\epsilon ), k(\epsilon )+\delta _0]. \end{aligned}$$
    2. (b)

      \(If \partial _k F\big (U^+(0), k_0, 0\big )>0\), then

      1. (i)

        \(k'(\epsilon )<0\) and \(k(\epsilon )<k_0\);

      2. (ii)

        \(\mathcal {C}^+(k)<U^+(0)\) for any \(|k|\in \big (k(\epsilon ), k(\epsilon )+\delta _0]\) and

        $$\begin{aligned} (U^+)''\mathcal {C}_I^+(k)<0 , \quad \mathcal {C}_R^+(k)>U^+(0), \; \forall |k|\in [k(\epsilon )-\delta _0, k(\epsilon )\big ). \end{aligned}$$
  2. (2)

    Suppose that \(\partial _k F\big (U^+(0), k_0, 0\big )=0\). Then there exist \(\epsilon _1>0\) such that for each \(\epsilon \in (0, \epsilon _1]\), there exist \(k_2(\epsilon )>k_0>k_1(\epsilon )\ge 0\), \(\delta >0\), and \(C^{1, \alpha }\) functions \(\mathcal {C}_{1, 2}:[k_{1, 2}(\epsilon )-\delta , k_{1, 2}(\epsilon )+\delta ]\rightarrow \mathbb {C}\) for any \(\alpha \in [0, 1)\) such that

    $$\begin{aligned} F\big (U^+(0), \pm k_{1, 2}(\epsilon ), \epsilon \big )=0, \quad \mathcal {C}_{1, 2}^+\big (\pm k_{1, 2}(\epsilon )\big )=U^+(0), \\ F\big (\mathcal {C}_{1, 2}^+(k), \pm k, \epsilon \big )=0, \quad \forall |k|\in [k_{1, 2}(\epsilon )-\delta , k_{1, 2}(\epsilon )+\delta ]. \end{aligned}$$

    In addition, \(\mathcal {C}_{1, 2}^+(k)\) satisfy that

    $$\begin{aligned}&\mathcal {C}^+_1(k)<U^+(0),\; \forall |k|\in \big (k_1(\epsilon ), k_1(\epsilon )+\delta ], \quad \\&\quad \mathcal {C}^+_2(k)<U^+(0), \;\forall |k|\in [k_2(\epsilon )-\delta , k_2(\epsilon )\big ), \\&\quad \mathcal {C}^+_{1_ R}(k)>U^+(0), \quad (U^+)''\mathcal {C}^+_{1_I}(k)<0, \; \forall |k|\in [k_1(\epsilon )-\delta , k_1(\epsilon )\big ),\\&\quad \mathcal {C}^+_{2_ R}(k)>U^+(0), \quad (U^+)''\mathcal {C}^+_{2_I}(k)<0, \; \forall |k|\in \big (k_2(\epsilon ), k_2(\epsilon )+\delta ]. \end{aligned}$$

    Let \(\rho :=\min \{U^+(h_+)-U^+(0), U^+(0)-U^-(0)\}\). Then for \(|c_R-U^+(0)|<\rho \) and \(c_I\in [0, \rho ]\), \(\mathcal {C}_1^+(k)\) is the unique root of \(F(\cdot , k, \epsilon )=0\) with \(|k|\in [k_1(\epsilon )-\delta , k_1(\epsilon )+\delta ]\) and \(\mathcal {C}_2^+(k)\) is the unique root of \(F(\cdot , k, \epsilon )=0\) with \(|k|\in [k_2(\epsilon )-\delta , k_2(\epsilon )+\delta ]\).

Lemma 4.5

Suppose that \(U^\pm \) satisfy (1.3), \(U^-(0)<U^+(0)\), \((U^\pm )'>0\), \((U^+)''<0\). Assume that \(F\big (U^+(0), 0, 0\big )=0\). Then the following hold:

  1. (1)

    There exist \(\epsilon _0>0\) and \(C^{1, \alpha }\) function \(\bar{C}(\epsilon ):[0, \epsilon _0]\rightarrow \mathbb {C}\) for any \(\alpha \in [0, 1)\) such that

    $$\begin{aligned}&F\big (\bar{C}(\epsilon ), 0, \epsilon \big )=0,\quad \bar{C}(0)=U^+(0),\\&\bar{C}(\epsilon )\in \big (U^-(0), U^+(0)\big ), \; \forall \epsilon \in (0, \epsilon _0]. \end{aligned}$$
  2. (2)

    Let \(K:=k^2\). If \(\partial _K F\big (U^+(0), 0, 0\big )\le 0\), then there exist \(\epsilon _1>0\) and smooth function \(k(\epsilon ):[0, \epsilon _1]\rightarrow \mathbb {R}\) such that for each \(\epsilon \in (0, \epsilon _1]\),

    $$\begin{aligned} F\big (U^+(0), \pm k(\epsilon ), \epsilon \big )=0,\quad k(\epsilon )>0,\; \text {and}\; k(0)=0. \end{aligned}$$

    In addition, there exist \(\delta _0>0\) and \(C^{1, \beta }\) function \(\mathcal {C}^+(k):[k(\epsilon )-\delta _0, k(\epsilon )+\delta _0]\rightarrow \mathbb {C}\) for any \(\beta \in [0, 1)\) such that

    $$\begin{aligned}&F\big (\mathcal {C}^+(k), \pm k, \epsilon \big )=0,\quad \mathcal {C}^+\big (k(\epsilon )\big )=U^+(0),\quad \mathcal {C}^+(k)<U^+(0), \;\\&\quad \forall |k|\in [k(\epsilon )-\delta _0, k(\epsilon )\big ), \\&(U^+)''\mathcal {C}_I^+(k)<0, \quad \mathcal {C}_R^+(k)>U^+(0), \; \forall |k|\in \big (k(\epsilon ), k(\epsilon )+\delta _0]. \end{aligned}$$

    Let \(\rho :=\min \{U^+(0)-U^-(0), U^+(h_+)-U^+(0)\}\). For \(c_I\in [0, \rho )\), \(|c_R-U^+(0)|<\rho \), \(\mathcal {C}^+(k)\) is the unique root of \(F(\cdot , k, \epsilon )=0\) with \(|k|\in [k(\epsilon )-\delta _0, k(\epsilon )+\delta _0]\).

Now, we are in the position of proving Theorem 1.5. The proof is based on a continuation argument of a holomorphic function over an appropriately chosen region in the complex plane.

Proof of Theorem 1.5

When \(k=0, \epsilon =0\) and \(c>U^-(0)\), \(y^-\) is in the form of (3.32). Then \(F(c, 0, 0)=0\) corresponds to

$$\begin{aligned} \left( \int _{-h_-}^0 \frac{1}{(U^--c)^2}dx_2\right) ^{-1}=g. \end{aligned}$$

Hence, \(c_0\) is the only root of \(F(\cdot , 0, 0)=0\) in \(\{c_R>U^-(0)\}\).

We first consider the case of \(c_0>U^+(h_+)\). By (3.11),

$$\begin{aligned} F(c, 0, 0)<F\big (U^+(h_+), 0, 0\big )<F(c_0, 0, 0)=0, \; \forall c\in \big (U^-(0), U^+(h_+)\big ). \nonumber \\ \end{aligned}$$
(4.7)

There exists \(0<\epsilon _0\ll 1\) such that for each \(\epsilon \in (0, \epsilon _0)\), \(|F(c, 0, \epsilon )|>0\) where \(c\in \{U^+(0), U^+(h_+) \}\). Let m(c) be defined in (3.35). Then (3.36) holds for small \(\epsilon \). Let \(c^+(k)\) be the one obtained in Lemma 3.2. Then, Lemma 3.5 implies that \(c^+(k)\) can be extended for all \(k\in \mathbb {R}\) and it is the only singular and non-singular mode in \(\{c_R>U^-(0)\}\). Statement (1) is proved.

Next, we shall prove statement (2). Let \(K:=k^2\). Suppose that \(c_0\in \big (U^+(0), U^+(h_+)]\). Let \(c^+_0(k)\) be the \(c^+(k)\) obtained in Lemma 3.2 when \(\epsilon =0\). By Remark 3.1, \(c^+_0(k)\) can be extended for all \(k\in \mathbb {R}\). Then there exists \(k_0>0\) such that \(c^+_0(k_0)=U^+(h_+)\). We use (3.12) and compute that

$$\begin{aligned}&\partial _K F\big (U^+(h_+), k_0, 0\big )<\partial _K F\big (U^+(h_+), 0, 0\big )\\&\quad =\big (U^-(0)-U^+(h_+) \big )\partial _K Y^-\big (U^+(h_+), 0\big )-\frac{\sigma }{\rho ^-}. \end{aligned}$$

Using (3.42) and the assumption (1.14), we obtain that

$$\begin{aligned} \partial _K F\big (U^+(h_+), 0, 0\big )\le \int _{-h_-}^0 \big (U^-(x_2)-U^+(h_+)\big )^2\textrm{d}x_2-\frac{\sigma }{\rho ^-}<0. \end{aligned}$$

This implies that

$$\begin{aligned} \partial _K F\big (U^+(h_+), k_0, 0\big )<0. \end{aligned}$$
(4.8)

For \(\epsilon =0\), we use (3.11) to obtain that

$$\begin{aligned} (c^+_0)'(k_0)=-\frac{2k \partial _K F}{\partial _c F}\big ( U^+(h_+), k_0, 0\big )>0. \end{aligned}$$

Since \(c^+_0(k)\) is analytic in \(k\in \mathbb {R}\), this implies that \(k_0\) is the unique root of \(F\big (U^+(h_+), \cdot , 0\big )=0\). By Lemma 4.2, there exist \(\delta _0>0\), \(k(\epsilon )\in C^{1, \beta }\big ([0, \epsilon _1], \mathbb {R}^+\big )\), and \(\mathcal {C}^+(k)\in C^{1, \alpha }\big ([k(\epsilon )-\delta _0, k(\epsilon )+\delta _0], \mathbb {C} \big )\) for any \(\alpha , \beta \in [0, 1)\) such that \(k(\epsilon )>k_0\), \(F\big (U^+(h_+), k(\epsilon ), \epsilon \big )=0\), \(F\big (\mathcal {C}^+(k), k, \epsilon \big )=0\), and \(\mathcal {C}^+(k(\epsilon ))=U^+(h_+)\). Since \((U^+)''<0\), \(\mathcal {C}_I^+(k)>0\), \(\mathcal {C}_R^+(k)<U^+(h_+)\) for any \(k\in [k(\epsilon )-\delta _0, k(\epsilon )\big )\) and \(\mathcal {C}^+(k)>U^+(h_+)\) for any \(k\in \big (k(\epsilon ), k(\epsilon )+\delta _0]\). By Remark 3.1, \(c^+(k)\) obtained in Lemma 3.2 can be extended to be an analytic function for all \(|k|>k(\epsilon )\) and \(C^{1, \alpha }\) for any \(\alpha \in [0, 1)\) in \(k\ge k(\epsilon )\) such that \(F\big (c^+(k), k, \epsilon \big )=0\). The uniqueness of solutions by the Implicit Function Theorem ensures that \(\mathcal {C}^+(k)=c^+(k)\) for any \(k\in \big (k(\epsilon ), k(\epsilon )+\delta _0]\). Moreover, there exists an open \(B_0\) satisfying \(\big (U^+(h_+), k(\epsilon )\big )\in B_0\subset \mathbb {C}\) such that \(F(c, k, \epsilon )\ne 0\) for any \(c\in \overline{B_0}\) and \(|k|\in [k(\epsilon )-\delta _0, k(\epsilon )-\frac{\delta _0}{4}]\). By the smallness of \(\epsilon _1\), (4.6) holds. Then (3.12) implies that \(k(\epsilon )\) is the unique root of \(F\big (U^+(h_+), \cdot , \epsilon \big )=0\) for all \(k\in \mathbb {R}\). We use (3.42), the assumption (1.14), \((U^-)'>0\), and \(U^-(0)<U^+(0)\) to compute that

$$\begin{aligned} \partial _K F(c, 0, 0)=&\big (U^-(0)-c\big )^2\partial _K Y^-(c, 0)-\frac{\sigma }{\rho ^-}\\ \le&\int _{-h_-}^0 \big (U^-(x_2)-c\big )^2\textrm{d}x_2-\frac{\sigma }{\rho ^-}\\ \le&\int _{-h_-}^0 \big (U^-(x_2)-U^+(h_+)\big )^2\textrm{d}x_2\\&-\frac{\sigma }{\rho ^-}<0, \; \forall c\in \big (U^-(0), U^+(h_+)\big ). \end{aligned}$$

Hence, by (3.12), for any \(k\in \mathbb {R}\) and \(c\in \big (U^-(0), U^+(h_+)\big )\), \(\partial _K F(c, k, 0)<\partial _K F(c, 0, 0)<0\). Since \(0<\epsilon _1\ll 1\), for any \(\epsilon \in [0, \epsilon _1)\), \(\partial _K F(c, k, \epsilon )<0\) for all \(k\in \mathbb {R}\) and \(c\in \big (U^-(0), U^+(h_+)\big )\). Since \(F(c, k, \epsilon )=F(c, k, 0)+\epsilon g\) and \(c_0>U^+(0)\), using the smallness of \(\epsilon _1\) and (4.7), we obtain that

$$\begin{aligned} F(c, k, \epsilon )<0, \; \forall \epsilon \in [0, \epsilon _1), \; \forall k\in \mathbb {R}, \; \forall c\in \big (U^-(0), U^+(0)]. \end{aligned}$$
(4.9)

Since \(F_I(c, k, \epsilon )\ne 0\) for any \(c\in U^+\big ((0, h_+)\big )\), \(F(c, k, \epsilon )\ne 0\) for any \(c\in U^+\big ([0, h_+]\big )\), \(|k|\le k(\epsilon )-\frac{\delta _0}{4}\). Hence, there exists an open set \(B_1\) satisfying that \(U^+([0,h_+])\subset B_1\subset \mathbb {C}\) and \(B_1\cap B_0\ne \emptyset \) such that \(F(c, k, \epsilon )\ne 0\) for any \(c\in B_1\) and \(|k|\le k(\epsilon )-\frac{\delta _0}{4}\). Because of (4.9) and Lemma 4.1, there exists \(\gamma >0\) such that \(|F(c, k, 0)|> 0\) for any \(c\in B_2:=\{U^+(0)-\gamma \le c_R\le U^+(0)+\gamma \}\) and \(k\in \mathbb {R}\). The smallness of \(\epsilon _1\) implies that \(|F(c, k, \epsilon )|> 0\) for \(c\in B_2\) and \(k\in \mathbb {R}\). Let \(B_3:=[U^+(0), U^+(h_+)]\times [0, U^+(h_+)-U^-(0)]\subset \mathbb {C}\). Consider \(\Omega :=B_3{\setminus } (B_0\cup B_1\cup B_2)\). Thanks to Theorem 1.1, we obtain that \(F(c, k, \epsilon )\ne 0\) for any \(c\in \partial \Omega \) and \(|k|\le k(\epsilon )-\frac{\delta _0}{2}\).

Now, we consider n(k) which is defined in (3.46). Lemma 4.2 implies that \(n\big (k(\epsilon )\big )=1\). Since n(k) is continuous for \(|k|\le k(\epsilon )-\frac{\delta _0}{2}\), \(n(k)\equiv 1\) for \(|k|\le k(\epsilon )-\frac{\delta _0}{2}\). Lemma 4.2 shows that \(\mathcal {C}^+(k)\) is the unique zero of \(F(\cdot , k, \epsilon )\) for \(|k|\in [k(\epsilon )-\delta _0, k(\epsilon )+\delta _0]\). Hence, \(n(k)\equiv 1\) for all \(|k|\le k(\epsilon )\). This implies that \(\mathcal {C}^+(k)\) can be extended to be a \(C^{1, \alpha }\) function (for any \(\alpha \in [0, 1)\)) for all \(|k|\le k(\epsilon )\) and analytic except at \(k=k(\epsilon )\). Let \(c^+(k)=\mathcal {C}^+(k)\) for \(|k|<k(\epsilon )\). Statement (2) is proved. The proof for statement (3) is similar. \(\square \)

The proof of Theorem 1.6 is very similar as above. One can consider

$$\begin{aligned} g_\#:=\max \{F(c, k, 0)+g\}. \end{aligned}$$

Then, 1.6 (1) directly follows from Theorem 1.1 in [11] and a similar proof as the one for Theorem 1.5 (1). The proof for Theorem 1.6 (2) is also followed by Implicit function Theorem and the standard degree theory. It is similar as the one for Theorem 1.5 (2) and we skip the details.