1 Introduction

In the last decades a plethora of rigorous analytical studies concerning water flows exhibiting vorticity have appeared. The vorticity is a fundamental property of a fluid flow, measuring the local spin or rotation of a fluid element (thereby producing a rotational or swirling motion), see [8]. (Constant) vorticity models wave-current interactions. Spectacular and impressive for observer examples of wave-current interactions are those at the eastern coast of South Africa (where 6 m high sea waves from the south-west meeting the Agulhas current cause many oil tankers wreckages) and those at the Columbia River entrance where the wave height doubles in just a few hours, see [27]. The vertical structure of the current profile whose description is realized by the vorticity determines the strength of the interaction, see [27, 42, 45]. Zero vorticity models irrotational flows as well as currents which are uniform with depth. The simplest rotational setting being that of linearly sheared currents of constant non-zero vorticity [31]. Real flows are very rarely irrotational anywhere, but for many flows the vorticity is very small almost everywhere, and these may therefore be modelled by assuming irrotationality. For the situation of irrotational three-dimensional gravity water flows, Iooss and Plotnikov [25, 26] have showed the existence of double periodic waves using Nash-Moser theory.

We will focus our attention to rotational water waves with constant non-zero vorticity. Scientific research in the rotational water waves direction began in the 19th century. In 1809 Gerstner [19] constructed a solution in Lagrangian coordinates describing an explicit family of periodic two-dimensional travelling rotational gravity water waves with non-zero vorticity. We refer the reader to the studies of Constantin [6] and Constantin, Strauss [12] in which it is presented a rigorous analysis of Gerstner’s results and to the studies of Constantin [6, 8] and Henry [23] for modern presentation of Gerstner’s wave solution. In the work of Constantin and Strauss [12] the existence of large-amplitude periodic water waves over two-dimensional flows with arbitrary (continuous) vorticity was proved. In [13] extended the result to water flows with discontinuous vorticity. This type of vorticity is of practical relevance and can be observed in regions where there is a rapid change of the current strength, cf. [27]. For thorough studies of water flows with a discontinuous vorticity distribution we refer the reader to the papers [10, 11, 13, 24, 33, 41].

It is known that (constant) vorticity models wave-current interactions, [8, 46] being also a prerequisite for the emergence of critical layers, see [14, 15, 17, 46]. In [9] Constantin proved that the mere existence of a surface gravity wave train in a flow of constant non-zero vorticity with a bounded velocity field implies the two-dimensionality of the flow beneath: the velocity field, the pressure, and the free surface present no variation in the direction orthogonal to the wave propagation direction. Moreover, the vorticity points in the direction orthogonal to the direction of wave propagation. Wahlén [47] obtained a similar rigidity-type result under the assumption that the free surface has a travelling character in both horizontal directions and that the flow beneath is time-independent. In [35] Martin proved that the two-dimensionality of the flow holds true if one assumes only a constant (non-vanishing) vorticity vector while allowing a general time dependence of the surface wave and of the flow beneath. The case of solitary water waves was considered in the works of Craig [16] and Stuhlmeier [44].

Liouville-type results concerning the steady Euler equations with or without geophysical effects and the steady Navier–Stokes equations in two or three dimensions were obtained in [2,3,4, 20,21,22, 29, 36, 43]. In [39] have been investigated the time-dependent three-dimensional Euler and Navier–Stokes equations with a free surface assuming the presence of an interface (playing the role of an internal wave) that arises as the result of discontinuous density stratification. Liouville-type results for time-dependent three-dimensional stratified geophysical water flows over variable bottom in the \(\beta \)-plane approximation were obtained in [40]. In this paper we consider non-geophysical water flows. The presence of an interface is a distinctive feature, compared with [9, 35, 38, 44, 47]. Some of the results obtained at [39] are as follows: assuming that the vorticity vectors in the two layers are constant, non-vanishing and parallel to each other, it was proven that bounded solutions to the three-dimensional equations are essentially two-dimensional. More precisely, the free surface, the interface, the pressure and the velocity field present no variations in the direction orthogonal to the direction of motion. In addition, one of the horizontal components of the velocity field is constant throughout the flow and the vorticity also points in the direction orthogonal to the wave propagation direction. A natural question arises: will these results remain true in the case of surface tension influence? This work is devoted to this question.

For the effects of the surface tension on wave trains at the surface of water in a flow with constant non-zero vorticity we refer the reader to [7, 34]. In this paper considering the water flows with effects of surface tension, under the assumption that the vorticity vectors in the two layers are constant, non-vanishing and parallel to each other we prove that the solutions to the three-dimensional equations of stratified steady inviscid water waves problem are essentially two-dimensional. Also we prove that the free surface, the interface, the pressure and the velocity field present no variations in the direction orthogonal to the direction of motion.

2 Statement of the problem

In this work we consider stratified inviscid water wave problem with two layers. The fluid domain is bounded below by a bottom boundary \(z=-d\), above by a free surface, denoted \(z=\tilde{\eta }\) and is split by an interface \(z=\eta \) in two regions of different densities, \(\tilde{\rho }\) and \(\rho \) respectively. In the first part of the work we will study steadily propagating waves in the direction of the x axis. A layer adjacent to the bottom and a layer adjacent to the free surface, respectively are written as

$$\begin{aligned}{} & {} D_\eta :=\left\{ (x,y,z):-d\le z\le \eta (x-ct)\right\} , \\{} & {} D_{\eta ,\tilde{\eta }}:=\left\{ (x,y,z):\eta (x-ct)\le z\le \tilde{\eta }(x-\tilde{c}t)\right\} , \end{aligned}$$

where \(\eta , \tilde{\eta }\in C^1(\mathbb {R})\), c and \(\tilde{c}\) are positive constants.

We denote with \({\textbf {u}}(x,y,z,t)=\left( u(x,y,z,t),v(x,y,z,t),w(x,y,z,t)\right) \) (respectively \(\tilde{{\textbf {u}}}(x,y,z,t)=\left( \tilde{u}(x,y,z,t),\tilde{v}(x,y,z,t),\tilde{w}(x,y,z,t)\right) \)) the velocity field in \(D_\eta \) (respectively in \(D_{\eta ,\tilde{\eta }}\)), P(x, y, z, t) (respectively \(\tilde{P}(x,y,z,t)\)) the pressure in \(D_\eta \) (respectively in \(D_{\eta ,\tilde{\eta }}\)) and with g the gravitational acceleration.

The motion of an incompressible and inviscid water flow is governed in \(D_\eta \) by the Euler equations

$$\begin{aligned} \begin{array}{lll} u_t+uu_x+vu_y+wu_z&{}=-\frac{P_x}{\rho }\\ \\ v_t+uv_x+vv_y+wv_z&{}=-\frac{P_y}{\rho }\\ \\ w_t+uw_x+vw_y+ww_z&{}=-\frac{P_z}{\rho }-g \end{array} \end{aligned}$$
(1)

and the equation of mass conservation

$$\begin{aligned} u_x+v_y+w_z=0. \end{aligned}$$
(2)

Likewise, in \(D_{\eta ,\tilde{\eta }}\) the flow motion obeys the equations

$$\begin{aligned} \begin{array}{lll} \tilde{u}_t+\tilde{u}\tilde{u}_x+\tilde{v}\tilde{u}_y+\tilde{w}\tilde{u}_z&{}=-\frac{\tilde{P}_x}{\tilde{\rho }}\\ \\ \tilde{v}_t+\tilde{u}\tilde{v}_x+\tilde{v}\tilde{v}_y+\tilde{w}\tilde{v}_z&{}=-\frac{\tilde{P}_y}{\tilde{\rho }}\\ \\ \tilde{w}_t+\tilde{u}\tilde{w}_x+\tilde{v}\tilde{w}_y+\tilde{w}\tilde{w}_z&{}=-\frac{\tilde{P}_z}{\tilde{\rho }}-g \end{array} \end{aligned}$$
(3)

and

$$\begin{aligned} \tilde{u}_x+\tilde{v}_y+\tilde{w}_z=0. \end{aligned}$$
(4)

The water wave problem is completely specified by the kinematic boundary conditions: on the free surface \(z=\tilde{\eta }(x-\tilde{c}t)\)

$$\begin{aligned} \tilde{w}=(\tilde{u}-\tilde{c})\tilde{\eta }_x, \end{aligned}$$
(5)

on the interface \(z=\eta (x-ct)\)

$$\begin{aligned} \begin{array}{ll} \tilde{w}=(\tilde{u}-c)\eta _x,\\ \\ w=(u-c)\eta _x \end{array} \end{aligned}$$
(6)

and on the bed \(z=-d\)

$$\begin{aligned} w=0. \end{aligned}$$
(7)

The balance of forces at the interface is encoded in the continuity of the pressure across \(z=\eta (x-ct)\), that is

$$\begin{aligned} P(x,y,\eta (x-ct),t)=\tilde{P}(x,y,\eta (x-ct),t) \ \ \text {for all} \ \ x,y,t. \end{aligned}$$
(8)

Lastly, the dynamic boundary condition, states the continuity of the pressure across the free surface \(z=\tilde{\eta }(x-\tilde{c}t)\), that is

$$\begin{aligned} \tilde{P}=P_{atm}-\sigma \cdot \frac{\tilde{\eta }_{xx}}{(1+\tilde{\eta }_x^2)^{3/2}}, \end{aligned}$$
(9)

where \(P_{atm}\) denotes the constant atmospheric pressure and \(\sigma >0\) is the coefficient of surface tension. Here we assume that \(\tilde{\eta }\in C^2(\mathbb {R})\).

Liouville-type results for non-stratified capillary-gravity water wave problem was obtained by Martin in [34], where was proved that wave trains can propagate steadily in the direction of the x axis only if the flow is two-dimensional. The two-layer gravity water flows were studied in [39], where a general time-dependence was taken into account.

In the second part of this work we will study travelling water waves that steadily propagate in the xy-plane. A layer adjacent to the bottom and a layer adjacent to the free surface, respectively are written as

$$\begin{aligned}{} & {} D_\eta :=\left\{ (x,y,z):-d\le z\le \eta (x,y)\right\} , \\{} & {} D_{\eta ,\tilde{\eta }}:=\left\{ (x,y,z):\eta (x,y)\le z\le \tilde{\eta }(x,y)\right\} , \end{aligned}$$

where \(\eta , \tilde{\eta }\in C^1(\mathbb {R}^2)\). We denote with \({\textbf {u}}(x,y,z)=\left( u(x,y,z),v(x,y,z),w(x,y,z)\right) \) (respectively \(\tilde{{\textbf {u}}}(x,y,z)=\left( \tilde{u}(x,y,z),\tilde{v}(x,y,z),\tilde{w}(x,y,z)\right) \)) the velocity field in \(D_\eta \) (respectively in \(D_{\eta ,\tilde{\eta }}\)) and P(x, y, z) (respectively \(\tilde{P}(x,y,z)\)) the pressure in \(D_\eta \) (respectively in \(D_{\eta ,\tilde{\eta }}\)). The Euler equations in \(D_\eta \) are the form

$$\begin{aligned} \begin{array}{lll} uu_x+vu_y+wu_z&{}=-\frac{P_x}{\rho }\\ \\ uv_x+vv_y+wv_z&{}=-\frac{P_y}{\rho }\\ \\ uw_x+vw_y+ww_z&{}=-\frac{P_z}{\rho }-g \end{array} \end{aligned}$$
(10)

and in \(D_{\eta ,\tilde{\eta }}\)

$$\begin{aligned} \begin{array}{lll} \tilde{u}\tilde{u}_x+\tilde{v}\tilde{u}_y+\tilde{w}\tilde{u}_z&{}=-\frac{\tilde{P}_x}{\tilde{\rho }}\\ \\ \tilde{u}\tilde{v}_x+\tilde{v}\tilde{v}_y+\tilde{w}\tilde{v}_z&{}=-\frac{\tilde{P}_y}{\tilde{\rho }}\\ \\ \tilde{u}\tilde{w}_x+\tilde{v}\tilde{w}_y+\tilde{w}\tilde{w}_z&{}=-\frac{\tilde{P}_z}{\tilde{\rho }}-g, \end{array} \end{aligned}$$
(11)

where g is the gravitational acceleration as in the problem formulated above. The equations of mass conservations in \(D_{\eta }\) and \(D_{\eta ,\tilde{\eta }}\), respectively, are specified by the formulas (2) and (4). The kinematic boundary conditions, stating the impermeability of the boundaries: on the free surface \(z=\tilde{\eta }(x,y)\) we require

$$\begin{aligned} \tilde{w}=\tilde{u}\tilde{\eta }_x+\tilde{v}\tilde{\eta }_y, \end{aligned}$$
(12)

on the interface \(z=\eta (x,y)\)

$$\begin{aligned} \begin{array}{ll} \tilde{w}=\tilde{u}\eta _x+\tilde{v}\eta _y,\\ \\ w=u\eta _x+v\eta _y \end{array} \end{aligned}$$
(13)

hold, and on the bed \(z=-d\) it holds that

$$\begin{aligned} w=0. \end{aligned}$$
(14)

The balance of forces at the interface is encoded in the continuity of the pressure across \(z=\eta (x,y)\), that is

$$\begin{aligned} P(x,y,\eta (x,y))=\tilde{P}(x,y,\eta (x,y)) \ \ \text {for all} \ \ x,y. \end{aligned}$$
(15)

The dynamic boundary condition across the free surface \(z=\tilde{\eta }(x,y)\) has the form

$$\begin{aligned} \tilde{P}=P_{atm}-\sigma \frac{(1+\tilde{\eta }^2_y)\tilde{\eta }_{xx}-2\tilde{\eta }_x\tilde{\eta }_y\tilde{\eta }_{xy}+(1+\tilde{\eta }^2_x)\tilde{\eta }_{yy}}{(1+\tilde{\eta }_x^2+\tilde{\eta }_y^2)^{3/2}}, \end{aligned}$$
(16)

where \(P_{atm}\) denotes the constant atmospheric pressure and \(\sigma >0\) is the coefficient of surface tension. We then assume that \(\tilde{\eta }\in C^2(\mathbb {R}^2)\). The explanations regarding (16) and derivation of this formula can be read in [28].

Two-dimensionality of non-stratified gravity and capillary-gravity water flows was studied by Wahlén in the work [47], where was proved non-existence of three-dimensional travelling water waves steadily propagating in the xy-plane. Rigidity-type results for stratified water wave problem with flat bottom and flat surface were obtained by Chen et al. in [5].

Throughout the paper for both formulated above problems we denote by \(\Omega \) and \(\tilde{\Omega }\) the vorticity vectors in \(D_{\eta }\) and \(D_{\eta ,\tilde{\eta }}\) respectively, defined as the curl of the velocity field, that is,

$$\begin{aligned} \begin{array}{ll} \Omega =\left( w_y-v_z, u_z-w_x,v_x-u_y\right) =:\left( \Omega _1,\Omega _2,\Omega _3\right) \ \ \text {in} \ \ D_\eta ,\\ \\ \tilde{\Omega }=\left( \tilde{w}_y-\tilde{v}_z, \tilde{u}_z-\tilde{w}_x,\tilde{v}_x-\tilde{u}_y\right) =:\left( \tilde{\Omega }_1,\tilde{\Omega }_2,\tilde{\Omega }_3\right) \ \ \text {in} \ \ D_{\eta ,\tilde{\eta }}. \end{array} \end{aligned}$$
(17)

We will assume that \(\Omega \) and \(\tilde{\Omega }\) are constant and parallel to each other. The evolution of the vorticity is governed by the equations

$$\begin{aligned} \Omega _t+\left( {\textbf {u}}\cdot \nabla \right) \Omega =\left( \Omega \cdot \nabla \right) {\textbf {u}} \ \ \text {and} \ \ \tilde{\Omega }_t+\left( \tilde{{\textbf {u}}}\cdot \nabla \right) \tilde{\Omega }=\left( \tilde{\Omega }\cdot \nabla \right) \tilde{{\textbf {u}}}, \end{aligned}$$

(see [8, 32]), which reduced to

$$\begin{aligned} \left( \Omega \cdot \nabla \right) {\textbf {u}}=0 \ \ \text {and} \ \ \left( \tilde{\Omega }\cdot \nabla \right) \tilde{{\textbf {u}}}=0, \end{aligned}$$
(18)

given that \(\Omega \) and \(\tilde{\Omega }\) are constant.

3 Main results

A key observation on this Section is that each component of the velocity \(\textbf{u}\) and of the velocity \(\tilde{\textbf{u}}\) is harmonic:

$$\begin{aligned} \begin{array}{ll} \Delta u=\Delta v=\Delta w=0 \ \ \text {in} \ \ D_{\eta }, \\ \ \\ \Delta \tilde{u}=\Delta \tilde{ v}=\Delta \tilde{ w}=0 \ \ \text {in} \ \ D_{\eta ,\tilde{\eta }}. \end{array} \end{aligned}$$

This follows by taking the curl of (17) and using the equations of mass conservation (2) and (4). For details about properties of harmonic functions we refer to read [1].

3.1 Liouville-type results for the water wave problem (1–9)

Theorem 1

Assuming that the velocity field (u, v, w) is bounded throughout the flow and that the vorticity vector \(\Omega \) is constant and non-vanishing it follows that its vertical component \(\Omega _3\), vanishes.

Proof

We perform a proof by contradiction and so assume that \(\Omega _3\ne 0\). Then from (18) we have that

$$\begin{aligned} \Omega _1w_x+\Omega _2w_y+\Omega _3w_z=0. \end{aligned}$$

Since \(\Omega _3\ne 0\) we conclude that w is constant in a non-horizontal direction. The boundary condition (7) entails now that w vanishes in a rectangular box completely contained in the fluid domain. Arguing as in [9] we make use of the analiticity of the harmonic function w and conclude that \(w=0\) throughout the entire fluid domain. Another direct consequence of the finding \(w\equiv 0\) (when considering the definition of the vorticity vector) is that

$$\begin{aligned} u_z=\Omega _2 \ \ \text { and} \ \ v_z=-\Omega _1 \end{aligned}$$
(19)

which imply the existence of two functions \(u_1=u_1(x,y,t)\), \(v_1=v_1(x,y,t)\) with

$$\begin{aligned} \begin{array}{ll} u(x,y,z,t)=u_1(x,y,t)+\Omega _2z,\\ \\ v(x,y,z,t)=v_1(x,y,t)-\Omega _1z. \end{array} \end{aligned}$$
(20)

From \(w = 0\) we also obtain (via the equation of mass conservation (2)) the existence of a stream function \(\psi =\psi (x,y,t)\) such that

$$\begin{aligned} u_1=\psi _y, \ \ v_1=-\psi _x. \end{aligned}$$
(21)

Note that the first two components of the vorticity equation can be written as

$$\begin{aligned} \begin{array}{ll} \Omega _1u_x+\Omega _2u_y+\Omega _3u_z=0,\\ \\ \Omega _1v_x+\Omega _2v_y+\Omega _3v_z=0. \end{array} \end{aligned}$$
(22)

Making now use of the relation \(\Omega _3=v_x-u_y\), of the equations (22) and of the property (21) of the stream function \(\psi \), we obtain the system

$$\begin{aligned} \begin{array}{ll} \Omega _1\psi _{xy}+\Omega _2\psi _{yy}+\Omega _2\Omega _3=0,\\ \\ -\Omega _1\psi _{xx}-\Omega _2\psi _{xy}-\Omega _1\Omega _3=0,\\ \\ \psi _{xx}+\psi _{yy}=-\Omega _3. \end{array} \end{aligned}$$
(23)

Working under the hypothesis that \(\Omega _1^2+\Omega _2^2>0\) we can conclude from (23) that

$$\begin{aligned} \psi _{xx}=-\frac{\Omega _1^2\Omega _3}{\Omega _1^2+\Omega _2^2}, \ \ \psi _{xy}=-\frac{\Omega _1\Omega _2\Omega _3}{\Omega _1^2+\Omega _2^2}, \ \ \psi _{yy}=-\frac{\Omega _2^2\Omega _3}{\Omega _1^2+\Omega _2^2}. \end{aligned}$$

Therefore, the stream function \(\psi \) is written as

$$\begin{aligned} \psi (x,y,t)=Ax^2+Bxy+Cy^2+a(t)x+b(t)y+c(t), \end{aligned}$$

for some functions a, b, c and with

$$\begin{aligned} A=-\frac{\Omega _1^2\Omega _3}{2(\Omega _1^2+\Omega _2^2)}, \ \ B= -\frac{\Omega _1\Omega _2\Omega _3}{\Omega _1^2+\Omega _2^2}, \ \ C=-\frac{\Omega _2^2\Omega _3}{\Omega _1^2+\Omega _2^2}. \end{aligned}$$

Since the boundedness of u and v entails the boundedness of \(u_1\) and \(v_1\), we have that the functions

$$\begin{aligned} \begin{array}{ll} (x,y,t)\rightarrow \psi _x(x,y,t) = 2Ax+By + a(t), \\ \ \\ (x,y,t)\rightarrow \psi _y(x,y,t) = Bx+2Cy+b(t) \end{array} \end{aligned}$$

are bounded as functions of (x, y). Consequently, \(A=B=C=0\) and, hence, the stream function simplifies to

$$\begin{aligned} \psi (x,y,t) = a(t)x +b(t)y +c(t). \end{aligned}$$

Applying now the decomposition (20) of u and v we have that for all (x, y, z, t) it holds that

$$\begin{aligned} u(x,y,z,t) = b(t)+\Omega _2z \ \ \text{ and } \ \ \ v(x,y,z,t)=-a(t)-\Omega _1z, \end{aligned}$$

which yields that \(\Omega _3=v_x-u_y=0\). But, the latter is a contradiction with the assumption \(\Omega _3\ne 0\). Hence, \(\Omega _3=0\).

Since the previous contradiction arose from the assumption that \(\Omega ^2_1+\Omega _2^2>0\) it remains to consider the case \(\Omega _1=\Omega _2=0\). In this case, from (19) it follows that

$$\begin{aligned} u_z=v_z=0. \end{aligned}$$
(24)

We assume for the sake of contradiction that u is not a constant function. Then, the maximum of u is achieved on the boundary of the fluid domain. Due to the Hopf boundary point lemma (see [18]), the maximum of u cannot be achieved on the bed \(z=-d\). Therefore the maximum of u is assumed on the interface \(z=\eta (x-ct)\). Let \((x_0,y_0,\eta (x_0-ct_0))\) be a point on the interface where u achieves its maximum at some time \(t_0\). But then, owing to (24), we obtain that

$$\begin{aligned} u(x_0,y_0,\eta (x_0-ct_0),t_0)=u(x_0,y_0,-d,t_0), \end{aligned}$$

which shows that u also takes on its maximum on the bed \(z=-d\). The previous conclusion is a contradiction. Hence, u is a constant function throughout the fluid domain. Analogously, we show that v is also a constant within the water flow. Thus, as before, \(\Omega _3 = v_x -u_y = 0\) which is again a contradiction with the assumption \(\Omega _3\ne 0\). Hence, \(\Omega _3=0\). \(\square \)

Remark 1

Due to the invariance of the water wave problem under rotations around the z-axis, we can assume without loss of generality that one of the horizontal components of the vorticity vector vanishes. We consider the case \(\Omega _1=0\), \(\Omega _2\ne 0\). Then, since \(\Omega \) and \(\tilde{\Omega }\) are parallel, it follows that \(\tilde{\Omega }_1=0\) and \(\tilde{\Omega }_2\ne 0\). Moreover, since \(\Omega _3=0\), we also have that \(\tilde{\Omega }_3=0\).

Theorem 2

Assume that the tuples \((\eta , u, v, w, P)\) in \(D_{\eta }\) and \((\tilde{\eta },\tilde{u},\tilde{v},\tilde{w}, \tilde{P})\) in \(D_{\eta , \tilde{\eta }}\), respectively, represent a bounded solutions of the water wave problem (1) – (9) with constant non-vanishing vorticity vector \(\Omega \) in \(D_{\eta }\) and \(\tilde{\Omega }\) in \(D_{\eta ,\tilde{\eta }}\), respectively. Then v,  \(\tilde{v}\) are constants, u, \(\tilde{u}\), w, \(\tilde{w}\), P and \(\tilde{P}\) present no variations in the y-direction.

Proof

Remark 1 and the vorticity Eq. (18) yield that

$$\begin{aligned} \begin{array}{ll} u_y=v_y=w_y=0, \\ \\ \tilde{u}_y=\tilde{v}_y=\tilde{w}_y=0. \end{array} \end{aligned}$$
(25)

Moreover, \(\Omega _1=w_y-v_z=0\), \(\Omega _3=v_x-u_y=0\) and (25) demand that \(v_x\equiv 0\) and \(v_z\equiv 0\). The latter inferences show that the horizontal component of the velocity, v, is only a function of the time t. From the second of the Euler’s equations (1), we conclude that throughout the domain \(D_{\eta }\) it holds that \(P_y=-\rho v'(t)\), which yields the existence of a function \((x, z) \longrightarrow f (x, z)\) such that

$$\begin{aligned} P(x,y,z,t)=-\rho v'(t)y+f(x,z) \ \ \text {for all} \ \ x,y,z,t, \end{aligned}$$

with the property that (x, y, z) lies in the fluid domain at time t. Fixing x, z and t we infer from the boundedness of the function \(y \longrightarrow P(x, y, z,t)\) that \(v'(t)=0\) for all t. Thus, v is constant and \(P_y\) vanishes within \(D_{\eta }\). Analogously, we can show that \(\tilde{v}\) is constant and \(\tilde{P}_y=0\) in \(D_{\eta , \tilde{\eta }}\). \(\square \)

3.2 Liouville-type results for the water wave problem (2), (4), (10–16)

Theorem 3

Assuming that the velocity field (u, v, w) is bounded throughout the flow and that the vorticity vector \(\Omega \) is constant and non-vanishing it follows that its vertical component \(\Omega _3\), vanishes.

Proof

We perform a proof by contradiction and so assume that \(\Omega _3\ne 0\). Arguing as in the proof of Theorem 1 we obtain the existence of two functions \(u_1=u_1(x,y)\), \(v_1=v_1(x,y)\) with

$$\begin{aligned} \begin{array}{ll} u(x,y,z)=u_1(x,y)+\Omega _2z,\\ \\ v(x,y,z)=v_1(x,y)-\Omega _1z. \end{array} \end{aligned}$$
(26)

Since

$$\begin{aligned} \Delta _{(x,y,z)} u=\Delta _{(x,y)} u_1, \ \ \ \ \Delta _{(x,y,z)} v=\Delta _{(x,y)} v_1, \end{aligned}$$

and

$$\begin{aligned} \Delta _{(x,y,z)} u=\Delta _{(x,y,z)} v=0, \end{aligned}$$

we conclude that

$$\begin{aligned} \Delta _{(x,y)} u_1=\Delta _{(x,y)} v_1=0, \end{aligned}$$

i.e. \(u_1\) and \(v_1\) are harmonic functions. The boundedness of u and v entails the boundedness of \(u_1\) and \(v_1\). So, \(u_1\) and \(v_1\) are bounded harmonic functions in \(\mathbb {R}^2\), which allows one to appeal to the Liouville theorem for harmonic functions (see [1]) to conclude that \(u_1\) and \(v_1\) are constants. Applying now the decomposition (26) of u and v we have that for all (x, y, z) it holds that

$$\begin{aligned} u(x,y,z) = \Omega _2z +u_1\ \ \text{ and } \ \ \ v(x,y,z)=-\Omega _1z+v_1, \end{aligned}$$
(27)

where \(u_1\) and \(v_1\) are constants. Using (27) we obtain that \(\Omega _3=v_x-u_y=0\). But, the latter is a contradiction with the assumption \(\Omega _3\ne 0\). Hence, \(\Omega _3=0\). \(\square \)

Remark 2

Due to the invariance of the water wave problem under rotations around the z-axis, we can assume without loss of generality that one of the horizontal components of the vorticity vector vanishes. We consider the case \(\Omega _1=0\), \(\Omega _2\ne 0\). In this case, since \(\Omega \) and \(\tilde{\Omega }\) are parallel, it follows that \(\tilde{\Omega }_1=0\) and \(\tilde{\Omega }_2\ne 0\). Moreover, since \(\Omega _3=0\), we also have that \(\tilde{\Omega }_3=0\). So, \(\Omega =(0,\Omega _2,0)\), \(\tilde{\Omega }=(0,\tilde{\Omega }_2,0)\) and the vorticity equations delivers

$$\begin{aligned} \begin{array}{ll} u_y=v_y=w_y=0,\\ \\ \tilde{u}_y=\tilde{v}_y=\tilde{w}_y=0. \end{array} \end{aligned}$$
(28)

Remark 3

While the continuity of the pressure across the interface \(z=\eta (x, y)\) is equivalent to the balance of forces at the location, it is not to be expected that the pressure is also continuously differentiable on \(z=\eta (x, y)\). Throughout the paper, we will make the assumption that the pressure is nowhere continuously differentiable on \(z=\eta (x, y)\).

To establish the next result of this work we shall need a Liouville-type theorem for elliptic equations of the form

$$\begin{aligned} a_{ij}\partial _i\partial _jf+b_i\partial _if+cf=0, \end{aligned}$$
(29)

where we have used Einstein’s summation convention. The following lemma is sufficient for our needs.

Lemma 1

(see Krylov [30], Corollary 2.9.3) Let \(a_{ij}\), \(b_i\) \((i,j=1,...,n)\) and c be continuous functions on \(\mathbb {R}^n\). Assume that the functions \(a_{ij}\) and \(b_i\) are bounded, that the matrix \((a_{ij}({\textbf {x}}))\) is symmetric and non-negative for any \({\textbf {x}}\in \mathbb {R}^n\) and that

$$\begin{aligned} \sup \limits _{{\textbf {x}}\in \mathbb {R}^n}{c({\textbf {x}})}<0. \end{aligned}$$

Let \(f\in C^2(\mathbb {R}^n)\) be a bounded solution of the elliptic Eq. (29). Then \(f\equiv 0\).

Theorem 4

Assume that \(\tilde{\eta }\in C^3(\mathbb {R}^2)\), first and second order partial derivatives of the function \(\tilde{\eta }\) are bounded and the tuples \((\eta , u, v, w, P)\) in \(D_{\eta }\), \((\tilde{\eta },\tilde{u},\tilde{v},\tilde{w},\tilde{P})\) in \(D_{\eta , \tilde{\eta }}\), respectively, represent a bounded solution of the water wave problem (2), (4), (10–16) with constant non-vanishing vorticity vector \(\Omega \) in \(D_{\eta }\), \(\tilde{\Omega }\) in \(D_{\eta ,\tilde{\eta }}\), respectively. Assume also the lack of differentiability of the pressure on the interface, and that the condition on the surface

$$\begin{aligned} \sup \limits _{(x,y)\in \mathbb {R}^2}\tilde{P}_z(x,y,\tilde{\eta }(x,y))<0 \end{aligned}$$
(30)

holds. Then v,  \(\tilde{v}\) are constant, u, \(\tilde{u}\), w, \(\tilde{w}\), P, \(\tilde{P}\), the free surface \(z=\tilde{\eta }(x,y)\) and the interface \(z=\eta (x, y)\) present no variations in the y-direction.

Proof

From (28), since \(\Omega _3=0\) we obtain that \(v_x\equiv 0,\) while from \(\Omega _1=0\) we infer that \(v_z\equiv 0\). We obtained that all spatial derivatives of the function v vanish, which allowed us to infer that v is constant. Then from the second of the Euler’s equations (10), we conclude that throughout the domain \(D_{\eta }\) it holds that

$$\begin{aligned} -\frac{P_y}{\rho }=0. \end{aligned}$$

Thus, \(P_y\) vanishes within \(D_{\eta }\). Analogously, we can show that \(\tilde{v}\) is constant and \(\tilde{P}_y=0\) in \(D_{\eta ,\tilde{\eta }}\).

We claim now that

$$\begin{aligned} P_z(x,y,\eta (x,y))\ne \tilde{P}_z(x,y,\eta (x,y)) \ \ \text {for all} \ \ x,y,t. \end{aligned}$$
(31)

To prove the claim, we assume that there are \((x_0, y_0)\) such that

$$\begin{aligned} P_z(x_0,y_0,\eta (x_0,y_0))=\tilde{P}_z(x_0,y_0,\eta (x_0,y_0)). \end{aligned}$$
(32)

Differentiating with respect to x in the relation (15) and taking into account (32) we obtain that

$$\begin{aligned} P_x(x_0,y_0,\eta (x_0,y_0))=\tilde{P}_x(x_0,y_0,\eta (x_0,y_0)). \end{aligned}$$
(33)

Recalling that \(P_y=\tilde{P}_y=0\) we have from (32) and (33) that the pressure is continuously differentiable at the point \((x_0,y_0,\eta (x_0,y_0))\), which is a contradiction with the assumption made in Remark 3 on the lack of continuous differentiability of the pressure across the interface. Hence, the claim made in (31) is proved.

We differentiate now with respect to y in the relation (15) and obtain

$$\begin{aligned} \begin{array}{ll} P_y\mid _{z=\eta (x,y)}+P_z\mid _{z=\eta (x,y)}\eta _y(x,y)=\tilde{P}_y\mid _{z=\eta (x,y)}+\tilde{P}_z\mid _{z=\eta (x,y)}\eta _y(x,y). \end{array} \end{aligned}$$

Making use of the claim (31) and recalling that \(P_y=\tilde{P}_y=0\) we get from the latter relation that

$$\begin{aligned} \eta _y(x,y)=0 \ \ \text { for all } \ \ x, y. \end{aligned}$$

It remains to prove that

$$\begin{aligned} \tilde{\eta }_y(x,y)=0 \ \ \text { for all } \ \ x, y. \end{aligned}$$
(34)

To establish (34) we will use the idea proposed by E. Wahlén in [47].

Differentiating the dynamic boundary condition (16) with respect to y and taking into account the vanishing of \(\tilde{P}_y\) we can write an equation of the form (29) for \(f=\tilde{\eta }_y\), where \(\partial _1=\partial _x\), \(\partial _2=\partial _y\) and

$$\begin{aligned} (a_{ij})=\dfrac{\sigma }{(1+\tilde{\eta }^2_x+\tilde{\eta }^2_y)^{3/2}}\left( \begin{array}{ll} 1+\tilde{\eta }^2_y &{} \ -\tilde{\eta }_x\tilde{\eta }_y \\ \\ -\tilde{\eta }_x\tilde{\eta }_y &{} \ 1+\tilde{\eta }^2_x \end{array} \right) . \end{aligned}$$

This matrix is non-negative. The coefficient c is given by \(c(x,y)=\tilde{P}_z(x,\tilde{\eta }(x,y))\). An application of Lemma 1 therefore shows that \(\tilde{\eta }_y\) vanishes identically. \(\square \)

3.3 Further remarks

In this subsection we provide some comments on the results obtained in the work and illustrate the main results with an example.

Remark 4

For the tuples \((\eta , u, v, w, P)\) in \(D_{\eta }\), \((\tilde{\eta },\tilde{u},\tilde{v},\tilde{w},\tilde{P})\) in \(D_{\eta , \tilde{\eta }}\), respectively, to represent a bounded solution of the water wave problem (2), (4), (10–16) with constant non-vanishing vorticity vector \((0,\Omega _2,0)\) in \(D_{\eta }\), \((0,\tilde{\Omega }_2,0)\) in \(D_{\eta ,\tilde{\eta }}\), respectively, it is necessary that the conditions

$$\begin{aligned} u_z\ne 0, \ \ \ \tilde{u}_z\ne 0 \end{aligned}$$

are satisfied.

Indeed, if we assume that \(u_z=0\), then \(\Omega _2=-w_x\), which taking into account that \(w_y=0\) implies that there exists a function \(z\longrightarrow \zeta (z)\) such that

$$\begin{aligned} w(x,y,z)=-\Omega _2 x+\zeta (z) \ \text {for all} \ \ x, y,z. \end{aligned}$$

Fixing z we infer from the boundedness of the function \(x\rightarrow w(x,y,z)\) that \(\Omega _2=0\). But, the latter is a contradiction with the assumption \(\Omega _2\ne 0\). Hence, \(u_z\ne 0\). Analogously, we can show that \(\tilde{u}_z\ne 0\).

Example 1

Now we consider an example of a flow with stratification and constant vorticity satisfying the three-dimensional water wave problem. Let the velocity field in \(D_{\eta }\) by defined by

$$\begin{aligned} u(x,y,z)=z, \ \ \ \ v(x,y,z)\equiv 0, \ \ \ \ w(x,y,z) \equiv 0, \end{aligned}$$
(35)

and the velocity field in \(D_{\eta ,\tilde{\eta }}\) by defined by

$$\begin{aligned} \tilde{u}(x,y,z)=z, \ \ \ \ \tilde{v}(x,y,z)\equiv 0, \ \ \ \ \tilde{w}(x,y,z)\equiv 0. \end{aligned}$$
(36)

We note that \(\Omega =(0,1,0)\) and \(\tilde{\Omega }=(0,1,0)\). Also we note that the equations of mass conservation (2) and (4) are satisfied in \(D_{\eta }\) and \(D_{\eta ,\tilde{\eta }}\) respectively. The Euler equations in \(D_{\eta }\) are equivalent with

$$\begin{aligned} P_x=0, \ \ P_y=0, \ \ P_z=-\rho g, \end{aligned}$$

which yields the formula for the pressure in \(D_{\eta }\):

$$\begin{aligned} P(x,y,z)=-\rho g z+P_0 \end{aligned}$$
(37)

for some constant \(P_0\), and analogously, the pressure in \(D_{\eta ,\tilde{\eta }}\) is given as

$$\begin{aligned} \tilde{P}(x,y,z)=-\tilde{\rho } gz+\tilde{P}_0 \end{aligned}$$
(38)

for some constant \(\tilde{P}_0\). Utilizing now the balance of forces at the interface, we determine the shape of the interface \(\eta \), namely

$$\begin{aligned} \eta (x,y)=\frac{\tilde{P}_0-P_0}{g(\tilde{\rho }-\rho )}. \end{aligned}$$
(39)

Hence,

$$\begin{aligned} u{\eta }_x+v\eta _y=\tilde{u}\eta _x+\tilde{v}\eta _y=0. \end{aligned}$$
(40)

From the kinematic boundary condition (12) the function \(\tilde{\eta }\) should satisfy the condition

$$\begin{aligned} \tilde{u}{\tilde{\eta }}_x=0. \end{aligned}$$
(41)

Replacing \(\tilde{u}\) from (36) in (41) we obtain

$$\begin{aligned} \tilde{\eta }{\tilde{\eta }}_x=0. \end{aligned}$$
(42)

Integrating (42) with respect to x, we get that there exists a function \(y\longrightarrow \xi (y)\) such that

$$\begin{aligned} \frac{\tilde{\eta }^2(x,y)}{2}=\xi (y), \end{aligned}$$

where we obtain that

$$\begin{aligned} \tilde{\eta }(x,y)=\pm (2\xi (y))^{1/2} \end{aligned}$$

and can conclude that the function \(\tilde{\eta }\) does not depend of the variable x and hence \(\tilde{\eta }_x=0\). Then the dynamic boundary condition at the free surface yields

$$\begin{aligned} \tilde{\sigma }\dfrac{\tilde{\eta }''(y)}{\sqrt{1+\tilde{\eta }'^2(y)}}-\tilde{\rho }g\tilde{\eta }(y)=P_{atm}-\tilde{P}_0. \end{aligned}$$
(43)

The function

$$\begin{aligned} \tilde{\eta }(y)=\dfrac{\tilde{P}_0-P_{atm}}{\tilde{\rho }g} \end{aligned}$$
(44)

is the solution of the differential Eq. (43) that satisfies the condition \(\tilde{\eta }_y=0\) (this condition has been established in the Theorem 4).

We infer from (35), (40) and (41) that the kinematic boundary conditions at the bottom, at the interface and at the free surface are satisfied. Thus, the velocity field (35–36), the pressure (37–38), the interface (39) and the free surface (44) constitute a solution to the water wave problem (2), (4), (10–16).

For more details about the explicit solutions to the full nonlinear water wave problem we refer to read the work [37] in which are presented the examples of solutions for explicitly given free surface.