# Large-amplitude steady downstream water waves

Adrian Constantin, Walter Strauss and Eugen Vărvărucă Faculty of Mathematics, University of Vienna, Oskar-Morgenstern-Platz 1, 1090 Vienna, Austria Brown University, Department of Mathematics and Lefschetz Center for Dynamical Systems, Box 1917, Providence, RI 02912, USA Faculty of Mathematics, “Al. I. Cuza” University, Bdul. Carol I, nr. 11, 700506 Iaşi, Romania
###### Abstract.

We study wave-current interactions in two-dimensional water flows of constant vorticity over a flat bed. For large-amplitude periodic traveling waves that propagate at the water surface in the same direction as the underlying current (downstream waves), we prove explicit uniform bounds for their amplitude. In particular, our estimates show that the maximum amplitude of the waves becomes vanishingly small as the vorticity increases without limit. We also prove that the downstream waves on a global bifurcating branch are never overhanging, and that their mass flux and Bernoulli constant are uniformly bounded.

Keywords: water waves, vorticity, Dirichlet-Neumann operator, Hilbert transform.

AMS Subject Classifications (2010): 76B15, 42B37, 35B50.

## 1. Introduction

Wave-current interactions are ubiquitous since typically a non-trivial mean flow, a current, underlies surface water waves. Sheared underlying currents are indicated by the presence of non-trivial vorticity. The primary sources of currents are winds of long duration [22]. In particular, in shallow regions with nearly flat beds, such as on the continental shelves, systematic studies of the velocity profiles of wind-generated currents have shown that they are accurately described as flows with constant vorticity [20].

In this paper we consider two-dimensional inviscid steady waves with constant vorticity. Inviscid theory is the usual framework for studying water waves that are not close to breaking because the most significant effects of viscosity in the open sea produce wave-amplitude reduction, as well as diffusion of the deeper motions, over time scales and length scales (wave periods and wavelengths) that are far larger than those of the dynamical processes at the surface [18]. The choice of constant vorticity is not merely a mathematical simplification. Indeed, when waves propagate at the surface of water over a nearly flat bed, for waves that are long compared to the mean water depth, it is the existence of a non-zero mean vorticity that is more important than its specific distribution (see the discussion in [18]). Moreover, in contrast to the substantial research literature on steady three-dimensional irrotational waves, it turns out that flows of constant vorticity are inherently two-dimensional (see the discussion in [6, 49]), with the vorticity correlated with the direction of wave-propagation.

The presence of a non-uniform underlying current is experimentally known to drastically alter the behavior of surface waves, when compared with irrotational waves which travel from their region of generation through water that is either quiescent or in uniform flow. The nature of a two-dimensional wave-current interaction notably depends on the directionality of the vertical shear of the current profile in relation to the direction of wave propagation. Here we distinguish between favorable currents, which are sheared in the same direction as that of the wave propagation (downstream waves), and adverse currents, which are sheared in the direction opposite to that of wave propagation (upstream waves). Field data, laboratory experiments and numerical simulations (see [11, 12, 23, 24, 27, 36, 37, 38, 40]) lead to the conjecture that an adverse current will shorten the wavelength, increasing the wave height and the wave steepness, to the extent that bulbous waves appear with overhanging wave profiles (see [18, 34]). On the other hand, for waves propagating downstream, the favorable current appears to lengthen the wavelength and flatten the wave out, so that its slopes are less steep.

At the present time the state-of-the-art to derive a priori bounds on surface water waves of large amplitude lags behind the experimental and numerical developments, and is to a large extent confined to irrotational deep-water flows. In the latter setting [4, 39] it was proven that an increase in wave height results in wave profiles that are not symmetric about their mean level, unlike the small-amplitude sinusoidal waves familiar from linear theory. Instead, the crests become higher and the troughs flatter to the extent that, for a given wavelength, there exists a limiting wave, the so-called ‘wave of greatest height’ or ’wave of extreme form’, which is on the verge of breaking. This extreme wave is distinguished by the fact that in the reference frame moving with the wave the water comes to rest at its peaked crest with included angle , as conjectured by Stokes in 1880 and finally proved about a century later in [3] (see also [46]). In general, at the wave crest of a traveling surface wave in irrotational flow with no underlying current, the fluid particles are moving forward at a speed less than the wave speed (see [5, 15]). As the wave profile approaches the wave of greatest height the horizontal particle velocity at the crest approaches the wave speed until these two velocities become equal in the limiting wave. A further increase in wave height will cause the fluid particles to overtake the wave itself, and breaking will ensue (see [29]).

On the other hand, while it has long been known (see [25]) that formal expansions indicate that a uniform vorticity distribution may accommodate such limiting wave forms and does not alter considerably the shape of their crest (namely, a symmetric peak with an included angle of , as in the case without vorticity), progress towards rigorous results has been much more difficult (see [45, 47]). Apart from their interest in their own right, a priori bounds on smooth waves of large amplitude are a necessary prerequisite for the existence of rotational waves of extreme form.

Let us briefly discuss the present state of rigorous mathematical investigations of rotational waves of large amplitude that are not perturbative, taking advantage instead of structural properties of the equations. For flows with general vorticity but without stagnation points the existence of surface waves of large amplitude, with profiles that can be represented as graphs of smooth functions, was established in [13]. For flows of constant vorticity an alternative geometric approach [16] accommodates stagnation points as well as the possibility of overhanging profiles. A number of qualitative studies of rotational waves of large amplitude are also available. They include symmetry results for wave profiles that are monotone between successive crests and troughs [8, 9], results about the location of the point of maximal horizontal fluid velocity [16, 41], and bounds on the maximum slope of the waves [35]. Although such non-perturbative results are not restricted to waves of small amplitude, their validity has typically required some extraneous information, such as the absence of stagnation points, the assumption that the wave profile is a graph or the assumption that some specific bounds (on the velocity or on the amplitude) hold. Thus the lack of a priori bounds for flows with vorticity has impeded substantial progress towards a comprehensive theory for waves of large amplitude.

The main aim of this paper is to derive a uniform bound of the amplitude of downstream waves, the existence of which has recently been proved in [16]. The approach in [16] relies on a formulation, first introduced there, of the governing equations for steady water waves with constant vorticity as a one-dimensional nonlinear pseudo-differential equation (2.3a) with a scalar constraint (2.3b). This formulation permits the presence of stagnation points in the flow as well as overhanging wave profiles. While from a mathematical point of view it appears at first sight to be dismayingly complicated, it has a variational structure that warrants a profound analysis, enabling us to establish, by means of the analytic theory of global bifurcation in a suitable function space, the existence of two solution curves that contain waves of large amplitude [16], one of this curves consisting of upstream waves, and the other of downstream waves. While the results in [16] left open several possibilities concerning the properties of the waves corresponding to points on this global curve as the parameter along the curve tends to , such as, for example, those that waves could become overhanging, or that either the parameters in the problem or suitable norms of the solution could increase without bound, these issues are settled in the present paper in the case of downstream waves. A fine analysis of the system (2.3) that uncovers some unexpected structures leads to the following main result, entirely consistent with the numerical computations in [23, 19].

###### Theorem 1.

Consider the waves with constant favorable vorticity lying on the bifurcation curve which is parametrized by . Then

(i) The waves are not overhanging.

(ii) The wave amplitude (elevation difference between crest and trough) is uniformly bounded along the bifurcation curve, with an explicit bound depending only on the vorticity and the wave period.

(iii) As a function of the vorticity, the amplitudes of all the waves tend to zero uniformly as the vorticity becomes infinite:

 limΥ→∞sups∈[0,∞){maxS(s)Y−minS(s)Y}→0.

(iv) For any , as one moves along the bifurcation curve, the waves approach their maximum possible amplitude :

 lims→+∞{maxS(s)Y−Q(s)2g}=0,

while and remain uniformly bounded as , where is the wave flux, is the total head and is the wave profile.

In Section 2 of the present paper we discuss the governing equations and this new formulation for both upstream and downstream waves. In particular, the scalar constraint is conductive to an elegant characterization of the downstream waves. In Section 3 we prove that the wave profile of each downstream wave on the solution curve is a graph; that is, the waves do not overhang. In Section 4 we derive the key a priori bounds for the downstream waves. These bounds use explicit detailed analysis of the Dirichlet-Neumann operator acting on the nonlinear terms. They are novel, surprising and extremely delicate. In Section 5 we discuss the physical interpretation of these mathematical results. Some background material is collected in the Appendix (Section 6).

As already mentioned, Theorem 1 is sufficient to ensure, by a very slight adaptation of the arguments in [45] that let along a subsequence, the existence of a limiting wave with stagnation points at its crests. An in-depth study of the properties of this limiting wave remains an important open problem. Another open problem is the determination of a priori bounds and geometric properties of upstream waves of large amplitude.

## 2. Preliminaries

We consider the problem of two-dimensional spatially periodic travelling free surface gravity water waves in a flow of constant vorticity over a flat bed. In a frame of reference moving with the speed of the wave, the fluid is in steady flow and occupies a laterally unbounded region of the -plane, whose boundary is made up of two parts: a lower part consisting of the real axis and representing the flat impermeable water bed, and an upper part that consists of a curve representing the free surface between the fluid and the atmosphere (see Fig. 1). The steady flow in may be described by means of a stream function , so that the velocity field is , and satisfies the following equations and boundary conditions:

 (2.1a) Δψ =Υin Ω, (2.1b) ψ =0on S, (2.1c) ψ =−mon B, (2.1d) |∇ψ|2+2gY =Qon S.

Here is the gravitational constant of acceleration, the constant is the relative mass flux, the constant is the total head, and the vorticity of the flow is assumed to take the constant value . In addition, both the domain and the stream function will be assumed to be -periodic in the horizontal direction, for some . This is a free-boundary problem, in which both the fluid domain and the function satisfying (2.1) need to be found as part of the solution.

A stagnation point of the flow is a point where . Stagnation points below the surface occur in the case of flow-reversal, even for waves that are small perturbations of a flat surface (see [17]). On the other hand, a stagnation point on the surface is the hallmark of the limiting ‘wave of greatest height’, which is on the verge of breaking and whose profile may have a corner singularity at the wave crest, see [45, 46, 47]. Such waves present special features that stand apart from those of regular waves and, in order to avoid technicalities that are of little relevance for the purposes of this paper, will not be investigated directly. Instead, we look for smooth (regular) solutions of (2.1) that satisfy

 (2.2) Q>2gYon S,

bearing in mind the possibility that singular waves may potentially arise as limits of such regular waves.

In order to obtain existence results for (2.1), it is usually necessary to consider an equivalent reformulation of the problem over some fixed domain. In the present setting, for waves of large amplitude the most comprehensive results available are due to [16] and are based on the following alternative formulation of the governing equations (2.1):

 (2.3a) Ckh((Q−2gv−Υ2v2)v′) +(Q−2gv−Υ2v2)(1k+Ckh(v′)) −2Υv(mkh−Υ2kh[v2]−ΥCkh(vv′)) −Q−2Υm−2ghk+2g[vCkh(v′)]=0, (2.3b) [(mkh−Υ2kh[v2]−ΥCkh(vv′)+Υv(1k+Ckh(v′)))2] −[(Q−2gv)((v′)2+(1k+Ckh(v′))2)]=0.

The square bracket will be used throughout the paper to denote, for a periodic function, its average over a period. Thus (2.3b) is merely a scalar equation. In (2.3), is a suitably smooth -periodic function of a real variable that represents the wave elevation in a parametrization of the free surface related to a conformal mapping from a horizontal strip onto the fluid domain (see Fig. 1). We denote , the average of over a period, which is a positive constant that may be called the conformal mean depth of the fluid domain . The constant is the wave number corresponding to the wave period . The operator denotes the periodic Hilbert transform for a strip of height (see the Appendix). As in (2.1), , and are real numbers that represent, respectively, the constant vorticity, the relative mass flux, and the total head.

Although any (smooth) solution of (2.1) gives rise to a solution of (2.3), a (smooth) solution of (2.3) gives rise to a solution of (2.1) if and only if the parametrization is regular, that is,

 (2.4) v(x)>0for all x∈R, (2.5) the mapping x↦(xk+(Ckh(v−h))(x),v(x)) is injective on R, (2.6) (v′(x))2+(1k+Ckh(v′)(x))2≠0for all x∈R.

If (2.4)-(2.6) hold for a solution of (2.3), then a solution of (2.1) can be constructed as described in detail in the Appendix, with the fluid domain being the image of the strip

 Rkh={(x,y):x∈R,−kh

through a conformal mapping obtained easily from . Note also that any solution of (2.3) also satisfies (see the Appendix)

 (2.7) (mkh−Υ2kh[v2]−ΥCkh(vv′)+Υv(1k+Ckh(v′)))2 −(Q−2gv)((v′)2+(1k+Ckh(v′))2)=0.

In view of this, if (2.2) holds, then we see that (2.6) yields

 (2.8) mkh−Υ2kh[v2]−ΥCkh(vv′)+Υv(1k+Ckh(v′))≠0onR.

The existence of solutions of (2.3) such that was studied in [16] in the space , where

 C2,α2π,e(R)={f∈C2,α2π(R): f(x)=f(−x) for all x∈R},

for any fixed Hölder exponent . The requirement that be an even function reflects the symmetry of the corresponding wave profile about the crest line that corresponds to .

Furthermore, seeking wave profiles whose vertical coordinate strictly decreases between each of its consecutive global maxima and minima, which are unique per principal period, we consider the following properties of the pair :

 (2.9) v(x)>0for all x∈R, (2.10) v′(x)<0for all x∈(0,π), (2.11) v′′(0)<0,v′′(π)>0, (2.12) 00,1k+(Ckh(v′))(π)>0, (2.14) ±(mkh−Υ2kh[v2]−ΥCkh(vv′)+Υv(1k+Ckh(v′)))>0on R.

The condition (2.14) comes from merely (2.8). We define the open sets

 (2.15) V±={(m,Q,v)∈R×R×C2,α2π,e(R):(???)−(???) hold},

where the choice of sign in is the same as that in (2.14). We emphasize that the sets can accommodate waves with overhanging profiles since no claim is being made that the horizontal coordinate of the wave profile is also a strictly monotone function of the parameter .

The constraints (2.9)-(2.14) ensure, in particular, that the associated free surface is not flat. Consequently, they exclude the family of trivial (laminar) solutions of (2.3) for which , and for which and are related by

 (2.16) Q=2gh+(mh+Υh2)2,

where is arbitrary. This family represents a curve

 Ktriv={(m,2gh+(mh+Υh2)2,h):m∈R}

in the space . These trivial solutions correspond to parallel shear flows in the fluid domain bounded below by the rigid bed and above by the free surface , with stream function

and velocity field

 (2.17)

Returning to the general case, it is explained in the Appendix how a solution of (2.1) can be constructed from a solution of (2.3) by means of a conformal map from onto , an important role being played by a function on that satisfies the system (6.5) and is related to by

 (2.18) ζ(x,y)=ψ(U(x,y),V(x,y))+m−12ΥV2(x,y),(x,y)∈Rkh.

It then follows that the fluid velocity at the location with , is given by

 (2.19) (ψY,−ψX)=(Vxζx+VyζyV2x+V2y+ΥV,Vxζy−VyζxV2x+V2y).

The main existence result for waves of large amplitude, which is based on an application of global real-analytic bifurcation theory, is as follows (see Theorem 5 in [16]).

###### Theorem 2.

Let and be given. Set

 (2.20) m∗±=−Υh22+Υhtanh(kh)2k±h√Υ2tanh2(kh)4k2+gtanh(kh)k,
 (2.21) Q∗±=2gh+(m∗±h+Υh2)2.

Then for any real , there exists a neighborhood in of the point on , where is related to by (2.16), within which there are no solutions of (2.3) in . On the other hand, for either choice of sign in , there exists in the space a continuous curve

 (2.22) K±={(m(s),Q(s),vs): s∈(0,∞)}

of solutions of (2.3), such that the following properties (i)-(vi) hold:
Local behavior:

• ;

• in as ;

• there exist a neighbourhood of in and sufficiently small such that

 {(m,Q,v)∈W±: v≢h,{\rm(???)} holds}={(m(s),Q(s),vs): 0<|s|<ε},

where, for any , we have defined

 (2.23) m(s):=m(−s),Q(s):=Q(−s),vs(⋅):=v−s(⋅+π).

Global behavior:

• for all and ;

• has a real-analytic reparametrization locally around each of its points;

• one of the following alternatives occurs:

• either and

 (2.24) min{11+||(m(s),Q(s),vs)||R×R×C2,α2π,e(R),minx∈R{Q(s)−2gvs(x)}}→0as s↑∞;
• or there exists some such that:

 (m(s),Q(s),vs)∈V±for all s∈(0,s∗),

whereas satisfies (2.9)(2.11), (2.13) and (2.14), while instead of (2.12) it satisfies

 (2.25a) 0

In this result, (2.20) and (2.21) identify the local bifurcation points along the trivial solution curve , (i)–(iii) describe the local behavior of the curve, and (iv)–(vi) describe the global behavior. The alternative () means that either the curve is unbounded in the function space or it approaches stagnation at the crest (and thus, may have waves of greatest height as limit points). The alternative () means that solutions on that correspond to physical water waves with qualitative properties as described by (2.9)-(2.14) do exist until a limiting configuration whose profile self-intersects on the line strictly above the trough is reached at .

Note that for the laminar flows given by (2.17), the horizontal velocity at the free surface is . Introducing the parameter

 (2.26) λ=mh+Υh2,

it is observed in [16] that, for a flow with a flat free surface at which nonlinear small-amplitude waves bifurcate, the horizontal velocity at the surface is given by

 (2.27) λ∗±=Υtanh(kh)2k±√Υ2tanh2(kh)4k2+gtanh(kh)k.

Note that and , so that there are no stagnation points on the free surface for the waves of small amplitude whose existence is guaranteed by local bifurcation. This property holds for all the genuine waves provided by Theorem 2, even for the waves of large amplitude that are not merely small perturbations of a laminar flow. It is also noted in [16] that, for any solution of (2.3), the function is necessarily smooth (of class ) on .

Let us also mention that [16] contains in fact a more comprehensive global bifurcation theory for (2.3) than that in Theorem 2, where we have restricted attention only to nontrivial waves with the nodal properties expressed by .

## 3. On the favorable branch the waves do not overhang.

Numerical simulations (for instance [24]) clearly indicate that the behavior of the solutions on the branches can be quite different, depending on the choice of the branch and on the sign of . The main result of this section partly confirms the numerical observations, ruling out alternative on the favorable bifurcating branch. Note that the equations (2.3) are invariant under the change of parity , . This simply reverses the vorticity and the direction of the flow in the moving frame. The favorable case is represented either by the curve with or by the curve with . Thus we may consider just the former case. The following theorem establishes that, all along the branch, the free surface of the wave is the graph of a function and no flow reversal occurs within the corresponding fluid domain.

###### Theorem 3.

Let . Then, in the notation of Theorem 2, the bifurcating curve of solutions of (2.3) satisfies alternative . Moreover, any solution on satisfies

 (3.1) 1k+(Ckh(v′))(x)>0for all % x∈R,

and, if denotes the corresponding solution of (2.1), then

 (3.2) ψY<0in ¯¯¯¯Ω.
###### Proof.

The proof that follows is similar in spirit to that of Theorem 2, being based on a continuation argument that shows that property (3.1) is preserved all along the curve . Since (3.1) represents a strengthening of (2.12) and (2.13), and since alternative involves the failure of (2.12), the global validity of (3.1) prevents the occurrence of .

Though perhaps not impossible, it appears difficult to study the validity of (3.1) in isolation, and thus our approach will be to study it in conjunction with all the other properties that occur in Theorem 2. We consider therefore in what follows the set

 (3.3) U−={(m,Q,v)∈V−:(???) holds},

and define

 (3.4) J={s∈(0,∞):(m(s),Q(s),vs)∈U−}.

Since is an open set in , is an open subset of . We claim that . To that aim, it is necessary to revisit some of the arguments in the proof of Theorem 2 in [16] .

We argue by contradiction, assuming that the open set is not the whole of . Of course it is immediate that there exists such that . Let be the upper endpoint of the largest open interval that contains and is contained in . Then and . In what follows we shall investigate the properties of the solution .

The fact that necessarily follows by the same argument as in [16], which involves the knowledge of all local bifurcation points on , the different nodal patterns of the solutions on the local bifurcating curves (expressed by conditions such as (2.23), (2.14) and (2.10)), and the specific manner of construction of the global curves in the real-analytic theory of Dancer, Buffoni and Toland.

For notational simplicity, we shall denote by . This is a limit of solutions satisfying (2.9)–(2.14) and (3.1). The definition of implies that the following non-strict inequalities hold:

 (3.5) v(x)≥0for all x∈R, (3.6) v′(x)≤0for all x∈[0,π], (3.7) 0≤xk+(Ckh(v−h))(x)≤πkfor all x∈[0,π], (3.8)

as well as

 (3.9) 1k+(Ckh(v′))(x)≥0for % all x∈R.

Now an examination of the arguments in Section 4 of [16] shows that the inequalities (3.5)–(3.8), together with the fact that is the limit of a sequence of solutions that correspond to solutions of (2.1) in the physical plane (which is the case here because of the definition of ), are enough to ensure that the strict inequalities (2.9)–(2.11) and (2.13) are satisfied, while (2.14) holds in the form

 (3.10) mkh−Υ2kh[v2]−ΥCkh(vv′)+Υv(1k+Ckh(v′))<0on R.

On the other hand, if (3.1) were also satisfied, then it would imply the validity of (2.12), and thus would contradict the definition of . Therefore (3.1) necessarily fails. This property, combined with the validity of (2.13) and the periodicity and evenness of , ensure that there exists such that

 (3.11) 1k+(Ckh(v′))(x0)=0.

Note also that the validity of (2.13) and (3.9) implies that (2.12) also holds. As explained in [16], in combination with the evenness and monotonicity of , this ensures the validity of (2.5). Moreover, since (2.4) also holds, and (2.6) holds as a consequence of (3.10) and (2.7), it follows that the solution under discussion, which is in fact , corresponds to a solution of (2.1). We now work in the physical plane and show that such a solution, with all the properties that have been established so far, cannot exist.

For simplicity, we denote

 (3.12) u(x)=xk+(Ckh(v−h))(x)for all x∈R,

so that for all , where is a harmonic conjugate of , the holomorphic function being a conformal mapping from onto . Then , the top boundary of , admits the parametrization

 (3.13) S={(u(x),v(x)):x∈R},

whose properties may be summarized for easy reference as follows:

 (3.14) v is 2π-periodic and even, (3.15) v′(x)<0 for all x∈(0,π), (3.16) u is odd, x↦(u(x)−2πkx) is 2π-periodic, (3.17) (u′(x))2+(v′(x))2>0 for all x∈R, (3.18) u′(0)>0, u′(π)>0, (3.19) u′(x)≥0 for all x∈R, (3.20) there exists x0∈(0,π) such that u′(x0)=0.

It is (3.20) that will lead to the contradiction. Let us examine the sign of in . Note first that, as a consequence of (6.5b), we have on that

 ζx=−ΥVVx,

which, when substituted in the formula (2.19) for the velocity field, and using also the Cauchy–Riemann equations, leads to the following relations, for all :

 (3.21) ψY(u(x),v(x)) =u′(x){ζy(x,0)+ΥV(x,0)Vy(x,0)}(u′(x))2+(v′(x))2, (3.22) −ψX(u(x),v(x)) =v′(x){ζy(x,0)+ΥV(x,0)Vy(x,0)}(u′(x))2+(v′(x))2.

Observe that, by (6.6), we have on that

 ζy+ΥVVy=mkh−Υ2kh[v2]−ΥCkh(vv′)+Υv(1k+Ckh(v′)),

and this quantity is strictly negative, as (2.14) shows. It follows from (3.21) and (3.19) that

 (3.23) ψY≤0 on S,ψY(u(x0),v(x0))=0andψY≢0 on S.

But as a consequence of (2.1a), the function is harmonic in and satisfies

 (ψY)Y=Υ≥0on the bottom B of the % fluid domain.

Thus, by the strong maximum principle and the Hopf boundary point lemma, the maximum of over the periodic domain can be attained neither in nor on . It is now a consequence of (3.23) that

 (3.24) ψY<0in Ω∪B.

Also, it is a consequence of (3.21), (3.22) and (3.10) that on . In combination with (3.24), this implies that

 |∇ψ|≠0in ¯¯¯¯Ω.

In order to rule out the existence of a solution of (2.1) for which the free boundary has the properties (3.14)–(3.20), we follow an idea that goes back to Spielvogel [32], and consider in the function given by

 (3.25) R:=12|∇ψ|2+gY−12Q−Υψ.

The function is in fact, up to an additive constant, the negative of the fluid pressure. A direct calculation (see [35, 45]) shows that satisfies in

 (3.26) ΔR−2(RX+ΥψX)|∇ψ|2RX−2(RY−2g+ΥψY)|∇ψ|2RY=2g|∇ψ|2(g−ΥψY).

(Note that the definition of vorticity considered in [35, 45] differs by sign from the definition we are considering here.) One can also easily check that

 (3.27) RY=g>0on B.

As a consequence of (3.24) and the assumption , the right-hand side in (3.26) is positive. It follows from (3.26) and (3.27) that the maximum over of can be attained neither in nor on , and therefore is attained all along , since there by (2.1b) and (2.1d). From the Hopf boundary point lemma we infer that the normal derivative of has a strict sign all along , and in particular at the point , at which the tangent is vertical by (3.20) and (3.23). We therefore deduce that at we have

 (3.28) 0≠RX=ψXψXX+ψYψXY−ΥψX=−ψXψYY,

where we have taken into account (2.1a) and the fact from (3.23) that at that point.

On the other hand, twice differentiating

 ψ(u(x),v(x))=0for all x∈R

with respect to , evaluating the result at , and using and , we find that

 (3.29) ψX(u(x0),v(x0))u′′(x0)+ψYY(u(x0),v(x0))(v′(x0))2=0.

Note, however, that (3.19) and (3.20) imply that . Thus taking also into account that , it follows from (3.29) that . This contradicts (3.28).

The source of the contradiction can only be the assumption that , which must therefore be false. We have thus proved that , which means that

 K−⊂U−,

and therefore all solutions on satisfy (3.1), as required.

It remains to prove the validity of (3.2) for all solutions on . Let be an arbitrary such solution, and let be the associated solution of . Then the top boundary of admits the parametrization (3.13), where the function is given by (3.12), and now the properties (3.14)–(3.16) hold, while instead of (3.17)–(3.20) we now have the condition

 (3.30) u′(x)>0for all x∈R.

Application of the maximum principle for in in a way similar to the preceding part of the proof now leads to (3.2) instead of (3.24). This completes the proof of Theorem 3. ∎

The validity of (3.2) and the fact that the free surface profile is a graph place the solutions on within the framework of the considerations made in [10], which yield the folllowing regularity result.

###### Corollary 1.

Let . Then any Hölder continuosly differentiable solution on the curve is real analytic.

## 4. Bound on the amplitudes of the waves

In this section we derive an explicit bound on the wave amplitudes of the favorable waves. We then give the proof of Theorem 1. For brevity we write , for any and all .

###### Theorem 4.

Let . Then, along the whole global bifurcation curve , we have the estimate

 (4.1a) v(0)−v(π)≤ ⎷36g2Υ4+24πgΥ2kβhk(π2)−6gΥ2<2πkβkh(π2)≤2π2(π−2)kifΥ>0, and (4.1b) v(0)−v(π)<2πkβkh(π2)≤2π2(π−2)kifΥ=0.

In order to prove this theorem, it is convenient to associate, to any solution on the global bifurcation curve , the function

 (4.2) f(x)=k2g(Q−2gv(x))for all x∈R,

which is smooth, even and -periodic on . Note that yields

 (4.3) [f]=k(Q−2gh)2g, (4.4) =Qhg−Q24g2+[f2]k2.

Due to Theorem 2(iv), (2.10), (3.1) and (2.14), respectively, we have

 (4.5) f(x)>0for all x∈R, (4.6) f′(x)>0for all x∈(0,π), (4.7) (Ckh(f′))(x)<1for all x∈R, (4.8) mh+ΥQ28hg2−Υ2k2h[f2]−Υkf+Υk{fCkh(f′)−Ckh(ff′)}<0on R.

Writing (2.3a) in terms of , after multiplication by we obtain the equation

 f+(aA+B)f−A2f2 = {fCkhf′+Ckh(ff′)}+b −A2{f2Ckhf′+Ckh(f2f′)−2fCkh(ff′)}

on , where , , and denote the constants

 (4.10) A = Υ2kg≥0,a=k2g(Q−2gh), (4.11) B = Υg{mh+Υh−Υ[v2]2h}=Υg{mh+Υh−ΥQ2g+ΥQ28hg2−Υ[f2]2k2h}, (4.12) b = kQ2g−Υkmg−kh−[fCkh(f′)]+Υk2Q2g2B+Υ2kQ28g3.

Note that, multiplying (4.8) by , and using the fundamental assumption that , we obtain

 (4.13) aA+B−Af+A{fCkh(f′)−Ckh(ff′)}≤0onR.

Our approach relies on some structure-exploiting integral representations of the cubic and quadratic terms in (4), as an effective tool to obtain estimates. Let us recall from Appendix A in [16] that, for any smooth -periodic function with mean zero over each period, we have

 (4.14) (Ckh(F))(x)=12πPV∫π−πβkh(x−s)F(s)ds,x∈R,

where (with ) the kernel , is given by

 (4.15) βd(s) =−sd+πdcoth(πs2d)+πd∞∑n=12sinh(πsd)cosh(πsd)−cosh(2π2nd) =−sd+πd∞∑n=−∞{coth(π2d(s−2πn))+sgn(n)}.

It is -periodic, odd, and smooth on . The function is continuous at . Although an explicit representation of the kernel in terms of three Jacobi elliptic functions is provided in [1], our series representation has the advantage that its term-by-term differentiation reveals that is strictly decreasing from to on . This is an important property that is not obvious from the explicit closed-form representation. In our proof we will need the following property of .

###### Lemma 1.

For all , is a positive function of strictly decreasing from to . Furthermore,

 (4.16) βd(π/2)≥π−2πfor all d∈(0,∞).
###### Proof.

From (4.15) it is clear that the series converges uniformly in for all , so that is continuous there. Furthermore, term-by-term differentiation shows that is strictly decreasing on . The oddness and -periodicity imply that , while the fact that is immediate.

For convenience in proving (4.16), let us write and consider as a function of . Then

 πβd(π/2)=2xcoth(x/2)−x−4xsinh(x)∞∑n=11cosh(4nx)−cosh(x).

We claim that, for any and any , we have

 0

To that aim, we examine the power series expansions of the numerator and denominator in the left side. Note that

 xsinh(x)=∞∑k=11(2k−1)!x2kfor all x∈R,
 cosh(θx)−cosh(x)=∞∑k=11(2k)!(θ2k−1)x2kfor all x∈R.

The required result is obtained by simply comparing the coefficients of each even power of , using the Bernoulli inequality:

 θ2k−1=(1+(θ2−1))k−1≥k(θ2−1)

for each and .

It therefore follows that

 πβd(π/2)≥2xcoth