In the following, we prove new regularity results for two-dimensional steady periodic capillary water waves with vorticity, in the absence of stagnation points. Firstly, we prove that if the vorticity function has a Hölder-continuous first derivative, then the free surface is a smooth curve and the streamlines beneath the surface will be real analytic. Furthermore, once we assume that the vorticity function is real analytic, it will follow that the wave surface profile is itself also analytic. A particular case of this result includes irrotational fluid flow where the vorticity is zero. The property of the streamlines being analytic allows us to gain physical insight into small-amplitude waves by justifying a power-series approach.
In this paper, we prove results for the regularity of the streamlines, including the surface profile, for steady periodic capillary waves allowing for vorticity. Surface tension is the dominant restoration force for capillary waves. We make the important assumption that there are no stagnation points, which means that the wave speed is larger than the horizontal speed of the individual particles—this assumption is physically plausible for many waves, especially those not near breaking [1,2]. Work analysing the a priori regularity of the free surface in the case of irrotational flow has been a subject of considerable historic interest, particularly when one considers Stokes' long-standing conjecture concerning the existence of the wave of greatest height containing cusps, which was proven quite recently, see the discussion in the study of Toland . In 1952, a famous result of Lewy  for irrotational gravity water waves showed that a free surface that is initially C1 near a point will automatically be analytic near the point. In the following, we study flows that are rotational, a situation that greatly complicates the mathematical approach, and for which surface tensions is the sole restoration force on the fluid (cf. [5,6]).
Results concerning whether the solutions of equations that arise in mathematical physics are analytic or not in a certain region also have great practical importance. For if we know whether a function is analytic in some region, then we can approximate it there to any level of accuracy by using power series, e.g. . Recently, results were obtained by Constantin & Escher concerning the real analyticity of the streamlines and of the wave profile in the case of gravity water waves with vorticity in the absence of stagnation points . The results of Constantin & Escher  reduce to Lewy's result in the case where the vorticity is zero. Subsequently, it was shown  that the streamlines of these gravity water waves are real-analytic curves when the vorticity is merely bounded and measurable. Furthermore, such analyticity properties were proven for fluids of infinite depth in Matioc . In Henry , it was shown that for rotational water waves, with no stagnation points and under the additional influence of surface tension, the streamlines are real analytic once the vorticity has a Hölder-continuous first derivative. Here, the presence of surface tension is a significant complicating factor (cf. [11–14] and below) in the analysis and it is perhaps a miracle of the inherent structure of the water wave equations that such results carry over to this case. The analyticity of the free surface for capillary–gravity waves when the vorticity function is analytic was established in Henry .
In the initial stages of wind-generated waves, capillary water waves are generated. The effects of surface tension are extremely important for these waves of small amplitude, and they wrinkle the sea much more than larger waves . In the case where the wind has a large enough fetch and blows both long enough and strong enough, these smaller waves grow larger and subsequently large amplitude water waves are generated, such as swell in the ocean. In this sense, capillary water waves are vital in the study of wave motion. Mathematically, surface tension acts as a significant complicating factor when compared with gravity water waves. However, interestingly Crapper  showed the existence of explicit capillary solutions for irrotational water waves of infinite depth. Kinnersley  then showed a similar existence of waves in the case of irrotational flow over a flat bed. Both of these types of waves are symmetric regular waves, and in both cases, the streamlines are analytic. We note that the only explicit solutions for free-surface water flows with vorticity are for gravity waves with infinite depth having very special vorticity distributions. These flows are Gerstner's celebrated wave (re-discovered by Rankine ), cf. Gerstner  (see also the modern presentations [20,21]), and also a related recently found wave pattern in Constantin . In particular, no explicit solutions are known for rotational capillary water waves with a non-flat free surface, and this fact enhances the importance of qualitative studies of the type pursued in the present paper.
The main novel feature of this work is that we allow for vorticity in the underlying fluid flow. It is well known that the motion of waves over an underlying current causes rotational flow, see the discussion in Constantin & Strauss . Furthermore, the influence of the wind in creating first capillary, and then gravity waves, is thought to introduce vorticity into the upper layer of the fluid region. It is therefore imperative that we allow for vorticity in the study of capillary waves. For a Hölder continuously differentiable vorticity function γ∈C1,α([−m,0]), the existence of a continuum of solutions was recently proven by Walsh .
In the following, we prove results regarding the regularity of the streamlines and of the free surface for capillary water waves, in the absence of stagnation points, but allowing for rotational flow. In particular, we show that the streamlines are analytic when the vorticity function has a Hölder-continuous first, while in this case, the free surface will be smooth (). The proof of this result on the streamlines uses a novel weighted translation, introduced in Constantin & Escher , which fits the water wave problem perfectly. The idea of studying the parameter dependence of nonlinear PDEs goes back to the studies of Masuda  and Angenent , and was further developed in Escher & Simonett , for example, yet the typical parameter translation classically used in these works was not sufficient to prove regularity in the water wave problem. We then show that if the vorticity function is real analytic, then so is the wave surface profile. This setting encompasses irrotational flow, where the zero vorticity is trivially real analytic. The proof of this result depends on us showing that our system satisfies a certain complementing condition, and upon showing this, we then use some powerful regularity results for elliptic equations [26–28]. The results contained in this paper were recently announced in Henry .
2. Governing equations
The waves we deal with are two-dimensional periodic travelling capillary water waves, where we imagine the wave motion to occur in a vertical plane perpendicular to the wave crest line. Capillary water waves are waves where surface tension is the dominant restoring force, and so we can ignore the effects of gravity. We let the x-coordinate denote the horizontal direction and then the y-coordinate corresponds to the vertical direction, and we assume without loss of generality that the wave is periodic with period 2π in the x-variable. We suppose that the water moves over a flat bed where we take y=0 to represent the location of the undisturbed water surface, and then the flat bed is located at y=−d for d>0. Then, periodicity considerations imply that for any fixed time t0, where η(x,t) is the wave surface profile. Let c>0 denote the constant speed of the travelling waves, then the velocity field for travelling waves takes the form (u(x−ct,y),v(x−ct,y)). Furthermore, the wave profile η is a free surface since it is a priori undetermined and thus represents an unknown in the problem, it is given by η(x−ct).
By using the change of coordinates (x−ct,y)↦(x,y), we move to a reference frame travelling alongside the wave with constant speed c>0. Here, the fluid flow is steady—we now work with a time-independent problem.
We denote the closure of the fluid domain by . The governing equations for the capillary water wave problem are embodied by the nonlinear free boundary problem (2.1)–(2.6) [2,6], where P=P(x,y) is the pressure distribution function, P0 is the constant atmospheric pressure, the parameter σ(>0) is the coefficient of surface tension and is the mean curvature in the x-direction, 2.1 2.2 2.3 2.4 2.5 2.6 For this two-dimensional flow, the vorticity is prescribed by the equation 2.7 We furthermore make the assumption that there are no stagnation points throughout the fluid by insisting that 2.8 This assumption (2.8) is physically plausible since it holds for most flows that are not near breaking . Note that there are recent results about steady waves (with vorticity) in flows with stagnation points (cf. [30,31]).
Equation (2.1) allows us to define the stream function ψ up to a constant by 2.9 and we fix the constant by setting ψ=0 on y=η(x). We can integrate (2.9) using (2.8) to get ψ=m on y=−d, for where −m is the relative mass flux, and by writing we can see that ψ is also periodic, with period 2L. The level sets of ψ(x,y) are the streamlines of the fluid motion. We can use (2.8) to show that the vorticity can be expressed as a function of the streamline, 2.10 with the vorticity function γ∈C1,α([−m,0]) if ω∈C1,α(Dη) [5,8,11], where C1,α is the space of functions with a Hölder-continuous first derivative, with Hölder exponent α∈(0,1).
We can see by direct calculation that and from integrating (2.2) and performing various manipulations, we derive Bernoulli's law, which states that the expression is constant throughout the fluid domain . We can reformulate the governing equations in the moving frame in terms of the stream function as follows: 2.11 2.12 2.13 2.14
The assumption (2.8) is vital now because it enables us to apply the following partial hodograph transformation, using the fact that the stream function ψ is constant on both boundaries of the fluid domain ((2.4) and (2.6)), and it is also a strictly decreasing function of y ((2.8) and (2.9)). We make the change of variables 2.15 The partial hodograph transformation (2.15) has the advantage of transforming the fluid domain Dη, with the a priori unknown free boundary η, into the fixed semi-infinite rectangular strip .
For the height function in the new (q,p)-variables, the change of variables (2.15) transforms the system of equations (2.11)–(2.14) on an unknown domain into the following system for the function h(q,p) in the fixed rectangular domain R: 2.16 2.17 2.18 The equivalency of the systems of governing equations (2.1)–(2.7) and (2.16)–(2.18) follows from arguments presented in Constantin & Strauss  and Walsh .
Let us now discuss what we mean by solutions of the three equivalent water wave problems. We wish to consider solutions (u,v,P,η) of (2.1)–(2.7) in the class of Hölder continuously differentiable functions, with Hölder exponent α∈(0,1), and where the ‘per’ subscript indicates that our solutions are 2π-periodic. Scaling arguments then allow us to recover solutions of any general period 2L.
In the following, we will find it expedient to work in a fixed domain, and so the notion of solution given above translates into the following for the system (2.16)–(2.18) in the domain R. Given , by a Hölder continuously differentiable solution of the governing equations, we mean , which satisfies (2.16)–(2.18), with h an even function in the q-variable. Furthermore, we require that 2.19 which is the re-expression of (2.8) in terms of h and the new (q,p)-variables.
3. The operator formulation of the problem
Now, for a given , let be the corresponding solution to (2.16)–(2.18) that also satisfies (2.19). To show that our streamlines are curves with a certain regularity, we will show that the mapping 3.1 possesses this regularity, for fixed p0∈[−m,0]. The regularity of this mapping corresponds to the regularity of the streamlines beneath the surface for p0∈[−m,0), and to the regularity of the free surface for p0=0.
We define the following spaces and operators. Keeping in mind the boundary condition (2.18), we let the Banach space and let the open subset of be which is compatible with the condition (2.19). Also let . The reformulation of the water wave problem (2.16)–(2.18) as an operator is achieved by setting where 3.2 3.3 The operator is real analytic in its input variable h, and we will denote such an operator's real-analytic dependence on the input variables by writing 3.4 By comparing (2.16)–(2.17) and (3.2)–(3.3), we see that h0 represents a solution to the water wave problem if and only if . Furthermore, since is analytic, we can compute its Fréchet derivative at h0, resulting in where (2.19) ensures that is a uniformly elliptic operator, given by 3.5 and is given by 3.6 If the flow is non-trivial, , then it can be seen that is not an isomorphism since . This observation comes from the following. Let represent the usual translation in the q-variable by the constant a, and so . Then, , since this translation commutes with differentiation and then h0 is a solution to the water wave problem. We have , by the linearity of , and so
4. Main results
(a) Streamlines and free surface are smooth
We now show that if γ∈C1,α([−m,0]),(Cω([−m,0])), then the mapping q↦h(q,p) is for all fixed p∈[−m,0]. Here, Cω denotes real analyticity. Therefore, all streamlines beneath the surface, together with the wave surface itself, are smooth functions, respectively, analytic. If we were to take the partial derivative of the system (2.16)–(2.18) with respect to q, and denote w=hq, we get 4.1 4.2 4.3 It is a consequence of (2.19) that the above quasi-linear system is uniformly elliptic, for the system of order 2 with weights s=r=0,t=2,N=1; see the discussion in Kinderlehrer et al. . We now show that the boundary operators satisfy the complementing condition [26–28]. To check this condition, we freeze the coefficients of by fixing (q0,p0), and then we consider the principal parts of these operators, namely the second-order derivatives in both and . Next, we formally replace (∂q,∂p) with (ξ,−i∂s) in the above expression, and we reach the following second-order ODE for an auxiliary function v(s): 4.4 4.5 where we use the ‘0’ superscript to denote frozen coefficients. The complementing condition is now equivalent to the following: the only exponentially decaying solution v(s) of the initial value problem (4.4)–(4.5) as is the null solution. Dividing out non-zero factors, we can rewrite this system as 4.6 4.7 and a general solution will be of the form v(s)=c1eias e−bs+c2eias ebs. We can easily see that the only such solution that decays exponentially and which satisfies the boundary conditions is the null solution. Therefore, our system (4.1)–(4.3) satisfies the complementing condition. In fact, it follows directly from the above arguments that in , any second-order elliptic boundary-value problem, with boundary conditions whose principal part contains tangential derivatives alone, must satisfy the complementing condition.
Let γ∈C1,α([−m,0]) and consider the corresponding solution of the governing equations, representing a periodic travelling capillary water wave such that the wave speed exceeds the horizontal fluid velocity throughout the flow. Then, each streamline beneath the wave profile, along with the wave profile itself, possesses regularity.
We have shown above that the quasi-linear system for w in (4.1)–(4.3) is uniformly elliptic and the boundary condition satisfies the complementing condition, with the coefficients of w in . Therefore, if we also have γ∈C1,α([−m,0]), then the system (4.1)–(4.3), viewed as an operator on q,p,h and the derivatives of h, is sufficiently regular in these variables to apply the regularity results for elliptic systems with boundary conditions satisfying the complementing condition (see , ch. 11). These results tell us that .
The above procedure has shown us that hq has the same regularity as h, that is, they are both in , and we may repeat the procedure indefinitely: we differentiate (4.1)–(4.3) with respect to q, then this system will similarly be uniformly elliptic in w′=wq and the boundary conditions will also satisfy the complementing condition, and so on. From these considerations, it follows that for n≥0, and so for fixed p∈[−m,0], the mapping q↦h(q,p) is in .
We note that a result similar to the above, but for gravity water waves, was obtained by using a different approach in Constantin & Strauss .
Let γ∈Cω([−m,0]) and consider the corresponding solution of the governing equations, representing a periodic travelling capillary water wave such that the wave speed exceeds the horizontal fluid velocity throughout the flow. Then, each streamline beneath the wave profile, along with the wave profile itself, is an analytic curve.
Similar to the previous theorem, the difference being that if γ∈Cω([−m,0]), then the regularity results of Morrey  imply that . It follows automatically that all streamlines, including the free profile, are real-analytic curves.
(b) Streamlines, in fact, analytic
The aim of this section is to prove a stronger result for the streamlines, namely that if we only have γ∈C1,α, then all streamlines beneath the surface are real-analytic curves. The analysis here does not produce such a result for the free profile itself, as we might expect. To begin, we follow the example of Constantin & Escher  by introducing a weighted translation that particularly suits the water wave problem. We let τa denote translation of the q-variable by the amount ap, and so Then, for any and with |a| small enough to ensure that , we have 4.8 where the operator is given by It is evident that the operator is a real-analytic operator from the space to the space Y ×Z, and so we can write 4.9
Let γ∈C1,α([−m,0]) and consider the corresponding solution of the governing equations, representing a periodic travelling capillary water wave such that the wave speed exceeds the horizontal fluid velocity throughout the flow. Then, each streamline beneath the wave profile is a real-analytic curve.
Consider a Hölder continuously differentiable solution of the governing equations, representing a periodic travelling capillary water wave in a flow with a Hölder continuously differentiable vorticity function and such that the wave speed exceeds the horizontal fluid velocity throughout the flow. Then, each streamline beneath the wave profile is a real-analytic curve. We consider now the operator defined by 4.10 where we will prescribe the condition (4.15) on λ below. It follows immediately from (3.4) and (4.9) that 4.11 If h0 is a solution to (2.16)–(2.18) for a given , then 4.12 Furthermore, using (4.8), we have 4.13 The key to the rest of the proof is the application of the implicit function theorem for real-analytic maps  to Φ(h,a)=0 in a neighbourhood of (h0,0). In order to do this, we must show that the partial Fréchet derivative is an isomorphism, where 4.14 Now, if we make the restriction 4.15 then from (2.19) and (3.6), we have that is a non-degenerate, uniformly oblique Venttsel-type boundary operation [12,13]. It follows directly from the results of §5 in the study of Walsh  that D1Φ(h0,0) is a Fredholm operator of index zero. Therefore, if we can show that D1Φ(h0,0) is injective, then it follows that it is an isomorphism.
The map D1Φ(h0,0) is an isomorphism.
As mentioned above, since D1Φ(h0,0) is a Fredholm operator of index zero, all we need to show is that it is injective. We assume the contrary and obtain a proof by contradiction using maximum principles. Suppose that for some non-zero h∈X, and suppose that h has a positive maximum value (if not, we would simply consider −h). By the weak maximum principle , and the fact that h=0 along the boundary p=−m, we must conclude that h attains its maximum value on the other boundary p=0, at the point (q0,0) say, and so h(q0,0)>0. Since this is a maximum value for h along the line p=0, it must follow that hq(q0,0)=0,hqq≤(q0,0). Applying Hopf's maximum principle  at the point, we must have hp(q0,0)>0. Using the above relations together with (4.15), evaluating at (q0,0), we get which contradicts our initial assumption that . Therefore, for λ>0 satisfying (4.15), we have
We may now apply the implicit function theorem for real-analytic maps , which tells us that there exists an ϵ>0 and a unique real-analytic function such that for a sufficiently small neighbourhood of (h0,0), the solutions of Φ(h,a)=0 are given by (h,a)=(ϕ(a),a) for a∈(−ϵ,ϵ). But, (4.12) and the uniqueness property of ψ imply that τah0=ϕ(a) for a∈(−ϵ,ϵ). Thus, for any , the mapping is real analytic, implying that in fact h0(⋅,p) is real analytic in the first variable for all p∈[−m,0). This demonstrates (3.1) and concludes our proof.
We note that the analyticity of the streamlines provides insight into the particle path pattern beneath the waves, since for small-amplitude waves it permits a power-series approach. Recent results have shown that the trajectories of particles, at first order [36,37] or at a higher order , correspond to the particle trajectories pattern that was found for Stokes waves [38–40].
The author thanks the anonymous referees for their helpful suggestions.
One contribution of 13 to a Theme Issue ‘Nonlinear water waves’.
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