An existence theory for small-amplitude doubly periodic water waves with vorticity

We prove the existence of three-dimensional steady gravity-capillary waves with vorticity on water of finite depth. The waves are periodic with respect to a given two-dimensional lattice and the relative velocity field is a Beltrami field, meaning that the vorticity is proportional to the velocity. The existence theory is based on multi-parameter bifurcation theory.


Statement of the problem
This paper is concerned with three-dimensional steady water waves driven by gravity and surface tension. An inviscid fluid of constant, unit density occupies the domain for some function η : R 2 → R and constant d > 0, where x = (x, y). Let u : Ω η → R 3 be the (relative) velocity field and p : Ω η → R the pressure. In a moving frame of reference, the fluid motion is governed by the stationary Euler equations (u · ∇)u = −∇p − ge3 in Ω η , with kinematic boundary condition on the top and bottom boundaries u · n = 0 on ∂Ω η , and dynamic boundary condition on the free surface p = patm − 2σKM on z = η(x ).
Here e3 = (0, 0, 1), KM is the mean curvature of the free surface, given by while σ > 0 is the coefficient of surface tension and patm the constant atmospheric pressure. In the original physical frame of reference, the travelling wave is given by z = η(x − νt), where ν = (ν1, ν2) is the (arbitrary) velocity vector. The corresponding velocity field is given by u(x − νt, z) + (ν, 0). Almost all previous studies of three-dimensional steady water waves have been restricted to the irrotational setting, where ∇ × u = 0. It is desirable to relax this condition in order to model interactions of surface waves with non-uniform currents. In the present paper we consider the special case when the velocity and vorticity fields are collinear, that is, for some constant α. In other words, we assume that u is a (strong) Beltrami field. Such fields are well-known in solar and plasma physics (see e.g. Boulmezaoud, Maday & Amari [5], Freidberg [16] and Priest [30]) and are also called (linear) force-free fields. The adjectives 'strong' and 'linear' refer to the fact that α is assumed to be constant. The more complicated case when α is variable has been investigated by several authors (see e.g. Boulmezaoud & Amari [4], Enciso & Peralta-Salas [15] and Kaiser, Neudert & von Wahl [26]), but will not be considered herein. Any divergence-free Beltrami field generates a solution to the stationary Euler equations with pressure The governing equations are thus replaced by

1a)
∇ · u = 0 in Ω η , (1.1b) u · n = 0 on ∂Ω η , (1.1c) where Q is the Bernoulli constant. Condition (1.1b) is actually redundant for α = 0, but we retain it since we want to allow α = 0. For a given η there can be more than one solution to the above equations and we will therefore later append integral conditions in order to enforce uniqueness. The choice of Beltrami flows is mainly motivated by mathematical considerations, since it gives rise to an elliptic free boundary problem. From a physical point of view, the choice is quite specific and it would be desirable to treat more general flows. One interesting feature of Beltrami flows is that they include laminar flows whose direction varies with depth (see Section 1.2.1 and Figure 1). This could potentially be of interest when considering a wind-induced surface current interacting with a subsurface current in a different direction.

Laminar flows
Let us consider a fluid domain with a flat boundary, that is η ≡ 0: In this case we find a two-parameter family of 'trivial' solutions given by laminar flows U [c1, c2] = c1U (1) + c2U (2) , c1, c2 ∈ R, The laminar flow U is constant in every horizontal section of the fluid domain but the direction of the flow depends on the vertical coordinate (see Figure 1). The constants c1, c2 will be used later as bifurcation parameters.

Two-and-a-half-dimensional waves
There is a connection between problem (1.1) and the two-dimensional steady water wave problem with affine linear vorticity function. Indeed, let η(x) be the surface and ψ(x, z) be the stream function for a two-dimensional wave with vorticity function α 2 ψ + αβ traveling in thē e-direction, where α and β are real constants,ē is a horizontal unit vector andx = x ·ē. Then where m1, m2, Q0 are constants, while is the two-dimensional fluid region. The corresponding velocity field is given by We can turn this into a solution of the stationary Euler equations in the three-dimensional domain by letting η and u equal the two-dimensional solution for everyȳ = x · e ⊥ , where e ⊥ = e3 ×ē. However, this solution is clearly still two-dimensional in the sense that it is independent ofȳ and the velocity vector is collinear with the direction of propagation, and hence there is no fluid motion in the perpendicular horizontal direction e ⊥ . On the other hand we can put One verifies that u solves (1.1) in The flow generated by u is called 2 1 /2-dimensional, since u only depends on the two variables x and z but has a non-zeroȳ-component; see Majda & Bertozzi [28,Sect. 2.3]. Note that every laminar solution U [c1, c2] can be written in the form (1.3) for some stream function Ψ(z). Note also that one can get rid of the constant β when α = 0 by introducing the new stream function ψ + α −1 β. Conversely, any 2 1 /2-dimensional Beltrami flow arises from a solution to the two-dimensional steady water wave problem with affine linear vorticity function. Indeed, assume that we have a solution (u, η) to (1.1) depending only on one horizontal variablex. Then (ū, u3) is divergence free with respect to the variables (x, z), whereū = u ·ē, and hence there exists a stream function ψ(x, z) such that u3 = ψx andū = −ψz. Now, equation (1.1a) gives u ⊥ = αψ + β for some constant β, as well as ψxx + ψzz + α 2 ψ + αβ = 0.
On the other hand, u is subject to (1.1c), which implies that ψ is constant on the boundaries.  [14]. The gravity-capillary problem has been considered for a general class of vorticity functions (including affine) but restricted to flows without stagnation points by several authors; see e.g. Wahlén [33] and Walsh [35,36]. These two-dimensional existence results immediately yield the existence of 2 1 /2-dimensional waves on Beltrami flows. In this paper we will instead take the opposite approach. As a part of the analysis we will directly obtain the existence of 2 1 /2-dimensional waves, which generate solutions of problem (1.2) by the above correspondence; see Remark 4.7.

Previous results
The theory of three-dimensional steady waves with vorticity is a relatively new subject of studies. In contrast to the two-dimensional case (see e.g. Constantin [9]) one cannot, in general, reformulate the problem as an elliptic free boundary problem. At the moment there are no existence results, save for some explicit Gerstner-type solutions for edge waves along a sloping beach and equatorially trapped waves; see Constantin [8,10] and Henry [23] and references therein. Wahlén [34] showed that the assumption of constant vorticity prevents the existence of genuinely three-dimensional traveling gravity waves on water of finite depth. A variational principle for doubly periodic waves whose relative velocity is given by a Beltrami vector field was obtained by Lokharu & Wahlén [27].
The irrotational theory is on the other hand much more developed. The first existence proofs for doubly periodic, irrotational, gravity-capillary waves in the 'non-resonant' case are due to Reeder & Shinbrot [31] and Sun [32]. These papers consider periodic lattices for which the fundamental domain is a 'symmetric diamond'. The resonant case was investigated by Craig & Nicholls [11] using a combination of topological and variational methods. They proved the existence of small-amplitude periodic waves for an arbitrary fundamental domain. A different approach known as spatial dynamics was developed by Groves & Mielke [20]. The idea is to choose one spatial variable for the role of time and think of the problem as a Hamiltonian system with an infinite dimensional phase space. Using this approach, Groves & Mielke constructed symmetric doubly periodic waves. The asymmetric case was later investigated by Groves & Haragus [18]; see also Nilsson [29]. One of the strengths of spatial dynamics is that is not restricted to the doubly periodic setting. It can also be used to construct waves which e.g. are solitary in one direction and periodic or quasi-periodic in another; see Groves [17] for a survey of different results. One type of solutions which have so far alluded the spatial dynamics method is fully localised solitary waves, that is, solutions which decay in all horizontal directions. Such solutions have however been constructed using variational methods; see Buffoni, Groves, Sun & Wahlén [6], Buffoni, Groves & Wahlén [7] and Groves & Sun [21]. Finally, note that the doubly periodic problem with zero surface tension is considerably harder since one has to deal with small divisors. Nevertheless, Iooss & Plotnikov [24,25] proved existence results for symmetric and asymmetric waves using Nash-Moser techniques. It might be possible to deal with the zero surface tension version of problem (1.1) in a similar way.

The present contribution
The main contribution of our paper is an existence result for small-amplitude doubly periodic solutions of problem (1.1). Given two linearly independent vectors λ1, λ2 ∈ R 2 we define the two-dimensional lattice We assume that for λ ∈ Λ, so that the fluid domain Ω η and velocity field are periodic with respect to the lattice Λ (see Figure 2). In addition, we impose the symmetry conditions For later use it is convenient to define and x ∈ B lj }, l, j ∈ Z, which splits the domain Ω η into simple periodic cells. We denote the top and bottom boundaries of Ω η lj by ∂Ω η,s lj and ∂Ω η,b lj respectively. We study solutions bifurcating from laminar flows U [c1, c2], where c1 and c2 act as bifurcation parameters. We look for solutions satisfying (1.1d) with the same constant Q = Q(c1, c2) as the underlying laminar flow. Therefore, the Bernoulli constant Q will vary along the bifurcation set. For purposes of uniqueness we also impose integral conditions on u1 and u2. This results in the system In Section 2 we introduce a suitable functional-analytic framework. Since we are dealing with a free boundary problem it is convenient to transform the problem to a fixed domain. After a sequence of changes of variables we derive an equivalent version, problem (2.5), which is amenable to further analysis. We also show that it is possible to reduce the governing equations to a single nonlinear pseudodifferential equation for the free surface (cf. Theorem 2.1 and equation (2.7)). In Section 3 we study the linearisation of the problem and identify bifurcation points (c 1 , c 2 ) at which the solution space is two-dimensional. In doing so, we derive a kind of dispersion relation ρ(c, k) = 0 (see equation (3.4)) which shows how the parameters c = (c1, c2) are related to the wave vector k of a solution to the linearised problem. The bifurcation points are those for which ρ(c , k1) = ρ(c , k2) = 0 for two linearly independent wave vectors k1, k2.
The conclusions can be found in Propositions 3.1 and 3.3. In Section 4 we finally give a precise formulation and proof of the main result, Theorem 4.1. While the main interest of this paper Figure 2: A sketch of a doubly periodic wave.
lies in the case α = 0, the existence result also covers the case α = 0 and therefore yields another existence proof for doubly periodic irrotational gravity-capillary waves. In the irrotational case there is an additional symmetry which allows one to treat the case of symmetric fundamental domains using classical one-dimensional bifurcation theory. The lack of this symmetry is one of the reasons for using a multi-parameter bifurcation approach. We have formulated the main result in terms of the relative velocity field. It is worth keeping in mind that in the original physical frame the solution is a small perturbation of the laminar flow U + (ν, 0), where ν is the velocity vector.

Function spaces and notation
If Ω is an open subset of Euclidean space, we let C k,γ (Ω), with k ∈ N0 := {0, 1, 2, . . .} and γ ∈ (0, 1), denote the class of k times continuously differentiable functions whose partial derivatives of order less than or equal to k are bounded and uniformly γ-Hölder continuous in Ω. This is a Banach space when equipped with the corresponding norm · C k,γ (Ω) . If u ∈ C k,γ (Ω), then u and its partial derivatives of order up to k extend continuously to the boundary. We will tacitly assume that this extension has been made. We will consider surface profiles in the space C k,γ per,e (R 2 ), consisting of η ∈ C k,γ (R 2 ) which satisfy the periodicity condition (1.4a) and the evenness condition (1.5a), and velocity fields in the space We let denote the lattice dual to Λ. Note that any η ∈ C k,γ per,e (R 2 ) can be expanded in a Fourier series with Fourier coefficientsη Functions u ∈ C k,γ per,e (Ω 0 ) have a similar expansion, with Fourier coefficients depending on z.
If X and Y are normed vector spaces and G : X → Y is a Fréchet differentiable mapping, we will denote its Fréchet derivative at x ∈ X by DG[x].

Flattening transformation
Since (1.6) is a free boundary problem, it is convenient to perform a change of variables which fixes the domain. Under the 'flattening' transformation Note that the vectors fj(ẋ) are tangent to the coordinate curves. In the new variables, the vector field u is given byu where the coordinate functions are determined bẏ Note that under this change of variables the space The divergence and curl take the following forms in the new coordinates: and Thus, in view of (2.1) and (2.2) problem (1.6) transforms into where the divergence and curl are now with respect to the dotted variables and the nonlinearities B and N = (N1, N2, N3) are given by Note that N is linear inu and that N (u, 0) = 0. From now on we drop the dots on x, y and z to simplify the notation.
Since we are interested in solutions that are close to a laminar flow U [c1, c2], we writė u = U +ṽ.
This is a linear system of equations forc η 1 ,c η 2 , which is uniquely solvable if and only if αd ∈ 2πZ \ {0}. We shall make this assumption from now on. The transformation then gives Note that G is affine linear in its first argument. We have G(v, 0) = 0 and DG[0, 0] = 0 as well as R(0, 0) = 0 and DR[0, 0] = 0.

Reduction to the surface
We now go on to show that problem (1.6) (or, equivalently, problem (2.5)) can be reduced to a nonlinear pseudodifferential equation for the surface elevation η in a neighbourhood of a laminar flow. To do this, we eliminate the vector field v from equation (2.5f) by solving (2.5a)-(2.5e) for v. In order for the procedure to work, we need to impose the non-resonance condition Theorem 2.1. Let d > 0, α ∈ R and Λ be given and assume that condition (2.6) holds. There exists a constant r0 > 0 such that for any η ∈ C 2,γ per,e (R 2 ) and any c ∈ The constant r0 depends only on α and d. Furthermore, the vector field v depends analytically on η and c.
Using Theorem 2.1, we can solve equations (2.5a)-(2.5e) for v and express the solution as an analytic operator of η and c, with v(0, c) = 0. We can thus write problem (2.5) as the single equation (2.7) We will approach problem (2.5a)-(2.5e) using a perturbative approach. We begin by defining suitable Banach spaces and then prove a technical lemma for the unperturbed problem. In the analysis of (2.5a)- The corresponding range space for the operator is given by Proof. Let us prove the first claim. We start with the injectivity. For a given v ∈ Y such that Thus, the components v1 and v2 must be constant throughout Ω 0 as they solve the homogeneous Neumann problem. But the integral assumptions in the definition of the space Y require these constants to be zero and so v vanishes everywhere in Ω 0 and we obtain the injectivity.
Next we turn to the surjectivity. We need to solve the equations (note that integral of w3 over Ω 0 00 vanishes due to oddness). Furthermore, it's easily seen that A1 and A2 are even in x and that A3 can be chosen odd. We put and note that A ∈ C 1,γ per (Ω 0 ) solves The unique solvability of the Dirichlet problem implies ∇ · A = 0 everywhere in Ω 0 . Moreover, we let ϕ ∈ C 2,γ per (Ω 0 ) be the unique odd solution to It is straightforward to verify that v satisfies (2.9a). Furthermore, the boundary conditions (2.9b) and v3 = 0 on ∂Ω 0,b follow from the relation v3 = −∂xA2 + ∂yA1 + ∂zϕ.
Finally, the formulas also reveal that v1 and v2 are even in x , while v3 is odd. Thus, v ∈ Y and the surjectivity is verified. To complete our proof of the first statement we use the inverse mapping theorem which ensures that C0 : Y → Z is an isomorphism. The second claim follows from an observation that the composition is a sum of the identity and a compact operator. Thus it is Fredholm of index zero and so is Cα.
The last statement (iii) follows from (ii) since under the given assumption the kernel of Cα is trivial. Indeed, if Cα(v) = 0 for some v ∈ X, then the Fourier coefficients solve for all k ∈ Λ and j = 1, 2, 3. The componentv (k) 3 also satisfies homogeneous Dirichlet boundary conditions both at z = −d and z = 0. According to the assumption of the claim, we see that α 2 − |k| 2 is not an eigenvalue and hencev (k) 3 must be zero everywhere in Ω 0 . Using this fact, we can compute the third coordinate of ∇ × v − αv to find Together this shows that |k| 2v(k) j = 0, j = 1, 2.
Thus, v = 0 identically and the kernel is trivial. This finishes the proof of the lemma.
We are now ready to treat the perturbed problem.
Proof of Theorem 2.1. Note that Cα : Y → Z is an isomorphism by the hypotheses of the theorem and Lemma 2.2. We can therefore rewrite problem (2.5a)-(2.5e) as where the left-hand side is a bounded linear operator on Y (since v → G(v, η) is affine linear), which is analytic in η ∈ U := {η ∈ C 2,γ per (R 2 ) : min η > −d}, and the right hand side is an ) v Y it follows by a Neumann series argument that the equation has a unique solution v = v(η, c) which is analytic in η and c for η C 2,γ (R 2 ) < r0 with r0 sufficiently small.

Dispersion equation
In this section we analyse the linearisation of (2.5). We choose to work with this problem rather than the reduced equation (2.7) since it is slightly more general (see condition (2.6)) and since it also gives direct information about the velocity field. However, it is of course straightforward to draw conclusions concerning the linearisation of problem (2.7) from this analysis. We aim to show that for a broad range of parameters the kernel of the linearised problem is exactly four-dimensional (and therefore two-dimensional when restricting to solutions satisfying the symmetry conditions (1.5)). For the convenience of the reader, the results are summarised in Proposition 3.1 at the end of the section.
The kernel of the linearisation of problem (2.5) is described by the system By Fourier analysis, it is enough to consider the Ansatz with k = (k, l) ∈ Λ in order to find periodic solutions of these equations. We split the analysis into four cases, in which √ · denotes the principal branch of the square root.

The first two equations (3.1a) and (3.1b) imply that
Taking into account the boundary conditions (3.1c) and (3.1d), we find v3(z) = ληφ(z), where λ = i(c · k) and φ = φ(z; |k|) is a solution to Note that (3.1e) is satisfied automatically since k = (0, 0). Substituting v into (3.1f) and assumingη = 0 (otherwise v vanishes), we arrive at the dispersion equation This is an equation for k which will be analysed below.
This corresponds to the constant function η =η. The vector field v solving (3.1a)-(3.1d) with no x -dependence must coincide with a laminar flow U [c1,c2] for some constantsc1,c2 ∈ R, but the condition (3.1e) forces them to be zero. Finally, condition (3.1f) forcesη = 0. Thus, in this case we find no non-trivial solutions to the linearised problem.
As in Case II, we get that η =η is constant and v = U [c1,c2] for some constantsc1,c2 ∈ R. If |α| is an odd multiple of π/d, (3.1e) forcesc1 =c2 = 0 and then (3.1f) leads toη = 0. We will avoid the last two cases by assuming the non-resonance condition (2.6). A complete analysis of equation (3.4) is a complicated problem. Our aim here is to find sufficient conditions on the problem parameters that guarantee that there exists some c = c such that the dispersion equation ρ(c , k) = 0 has exactly four different roots in the dual lattice Λ . Note that the roots come in pairs ±k, so that the dimension of the solution space is halved when we consider solutions with the symmetries (1.5). We use a geometric approach and restrict ourselves to the case when the roots are generators of Λ . We will also restrict attention to the case α = 0 in the main part of the analysis and leave the irrotational case to Remarks 3.2, 3.4 and 4.2 (see also the references mentioned in the introduction). Let us assume that the constants α, σ and d are fixed. Then for a given k = 0 we want to describe the set of all c ∈ R 2 such that (3.4) holds true. In other words, we are going to analyse the zero level set ρ(c, k) = 0, (3.6) where the vector k is fixed. For this purpose we put and write equation (3.6) in the form κ(|k|)x 2 = a(|k|) + αxy, a(|k|) := g + σ|k| 2 .
We can solve for y and get a curve of solutions in the xy-plane given by The curve can be recognised as a hyperbola with the y-axis as one asymptote. We denote by the angle between the asymptotes of one branch of the hyperbola (Figure 3). It is clear that γ is obtuse if κ is positive and acute if κ is negative. We call the shaded open set in Figure 3c the set between the asymptotes. To express this curve in (c1, c2) coordinates we note that x = c1 cos(θ) + c2 sin(θ) and y = −c1 sin(θ) + c2 cos(θ), where θ is the angle that k makes with e1. Hence x y , so going from (x, y) to (c1, c2) is a counterclockwise rotation by the angle θ. We denote this curve of solutions by C(k). Now we want to find linearly independent vectors k1 and k2 so that there is a point of intersection of C(k1) and C(k2). It is not hard to see that a sufficient condition is that the sets between the asymptotes of C(k1) and C(k2) have nonempty intersection, but one is not contained in the other. (3.7) We now analyse this in more detail in the special case when α, κ1 := κ(|k1|) and κ2 := κ(|k2|) are all positive. In that case we get two hyperbolas as in Figure 3a. Note that if α = 0 we we can always assume that it is positive by replacing u by −u. Without loss of generality we can assume that k1 is parallel with e1 and that k2 makes an angle θ with k1. Moreover we can always choose the generators in such a way that 0 < θ < π by changing k2 to −k2 if necessary. By possibly relabelling, we may assume that γ1 ≥ γ2 where γi is the angle between the asymptotes of C(ki). We see in Figure 4a that if 0 < θ ≤ γ1 − γ2 then the set between the asymptotes of C(k2) is completely contained in the set between the asymptotes of C(k1); in Figures 4b and 4c we see that if γ1 − γ2 < θ < π then condition (3.7) is satisfied. In summary we necessarily get intersection between C(k1) and C(k2) if γ1 − γ2 < θ < π. (3.8) In the subsequent analysis it will be important to make sure that the only solutions to the dispersion equation (3.6) for a fixed c = c are the generators ±k1 and ±k2. This property is expected to hold for generic values of the parameters since three hyperbolas in the plane generally don't have a common point of intersection. However, verifying it analytically is nontrivial. We content ourselves with analysing the case of a symmetric lattice, |k1| = |k2| = k > 0. In that case γ1 = γ2 and as before we assume that the angles are obtuse (meaning that κ := κ1 = κ2 > 0) and that α > 0. Condition (3.8) is clearly satisfied. It's convenient to assume that the generators have the form k1 = k(cos ω, sin ω), k2 = k(cos ω, − sin ω), with ω ∈ (0, π/2), which can always be achieved after rotating and relabelling. Note that θ = 2ω. Similarly, we write c = c(cos ϕ, sin ϕ).
Since the left hand side of the dispersion equation is proportional to g + σk 2 , we either get equality for all σ or for at most one σ > 0. However, the former can happen only if qn 1 n 2 = 1 and the proportionality constant is 1. We have with equality only if either n1 or n2 vanishes or if both are non-zero and ω is an integer multiple of π/2. It follows that the only way to get qn 1 n 2 = 1 is if (n1, n2) = (±1, 0) or (0, ±1), or if n1 = ±n2. The former is the trivial case when k = ±kj, j = 1, 2. In the latter case, we get q 2 n 1 n 2 = q 2 n 1 n 1 = 4n 2 1 cos 2 ω = 1 or q 2 n 1 n 2 = q 2 n 1 (−n 1 ) = 4n 2 1 sin 2 ω = 1.
(i) Assume that the non-resonance condition (2.6) is satisfied for the dual lattice Λ . Then the dimension of the space of solutions (v, η) ∈ C 1,γ per,e (Ω 0 ; R 3 ) × C 2,γ per,e (R 2 ) to the linearised problem (3.1) is equal to half the number of solutions k ∈ Λ of the dispersion equation (3.4).
(ii) If, in addition, α = 0 and the sufficient condition (3.7) holds for the generators k1 and k2 of Λ , then there exists a constant c such that the solution space is at least twodimensional when c = c .
(iv) If, in addition to the conditions in (i) and (iii), the lattice is symmetric, that is, |k1| = |k2|, then the solution space is exactly four-dimensional for all but countably many values of σ.

Remark 3.2.
In the irrotational case, α = 0, the hyperbolas instead become straight lines, x = ± a(|k|)/κ(|k|). The rotated lines C(k1) and C(k2) will clearly intersect as soon as k1 and k2 are not parallel. The multiplicity analysis for symmetric lattices applies also in the irrotational case, but can then be simplified and expanded; see Section 7 of Reeder & Shinbrot [31].

Transversality condition
The local bifurcation theory that we are going to apply requires the bifurcation point (a laminar flow in our case) to satisfy a transversality condition. This can be stated as the condition that the vectors ∇cρ(c , k1) and ∇cρ(c , k2) are not parallel, (3.12) where ρ(c , k1) = ρ(c , k2) = 0. It turns out that this is automatically satisfied under the conditions discussed above.
Proof. Assume instead that C(k1) and C(k2) intersect tangentially at c . Writing the equations ρ(c, kj) = 0 in the form where Aj are real indefinite symmetric 2 × 2 matrices, we see that (A1 − A2)c * = 0. Hence, either A1 ≥ A2 or vice versa. For definiteness, we shall assume the former since it is consistent with the assumptions in the previous section. But this implies that the hyperbola c T A2c = 1 is contained in the set c T A1c ≥ 1. Moreover, the set {c : c T A2c > 0} between the asymptotes of C(k2) is contained in the set {c : c T A1c > 0} between the asymptotes of C(k1), contradicting (3.7).
Remark 3.4. In the irrotational case, α = 0, the transversality condition is automatically satisfied as soon as k1 and k2 are not parallel since the lines ρ(c , kj) = 0 then intersect transversally.

Main result
Now we formulate our main theorem, providing the existence of three-dimensional steady gravity-capillary waves.
Then there exists a neighbourhood of zero W = B (0; R 2 ) ⊂ R 2 and real-analytic functions δ1, δ2 : W → R satisfying δ1, δ2 = O(|t| 2 ) as |t| → 0, and such that for Furthermore, the solution depends analytically on t ∈ W . In a neighbourhood (0, 0, c ) in C 1,γ per,e (Ω 0 ; R 3 ) × C 2,γ per,e (R 2 ) × R 2 , these are the only non-trivial solutions, except for two twoparameter families of 2 1 /2-dimensional solutions which can be obtained by simple bifurcation from nearby points where the kernel is one-dimensional.
Remark 4.2. Propositions 3.1 and 3.3 show that it is possible to satisfy the assumptions of the theorem. Indeed for any α > 0 and d > 0, we can choose the lengths |k1| and |k2| so that α 2 − n 2 |kj| 2 / ∈ π d Z+ and κ(|kj|) > 0 for j = 1, 2 and n ∈ Z. Since γj depends only on the lengths |kj|, we can then choose θ, the angle between k1 and k2, so that (3.8) is fulfilled. We also choose it so that (2.6) is satisfied, that is for all n1, n2 ∈ Z and n3 ∈ Z+. Note that we only have to avoid finitely many angles and that the case when either n1 or n2 vanishes is already satisfied by the choice of |k1| and |k2|. Then there exists a c = c * such that k1 and k2 are roots of the dispersion equation. Moreover, the transversality condition is satisfied. At least in the case of a symmetric lattice, |k1| = |k2|, we can then choose σ outside some countable set, which depends only on k1, k2, α and d, such that the dispersion equation has exactly the four roots ±k1, ±k2 in the lattice Λ .
In the irrotational case, α = 0, the first and third conditions are always satisfied (see Remark 3.4). As above, the second condition can be verified outside an exceptional set of parameter values in the case of a symmetric lattice; see Remark 3.2 and Reeder & Shinbrot [31]. Remark 4.3. In the proof of this result we will work with the reduced equation (2.7) for the surface profile. We loose a little bit of generality in doing this since we have to impose the non-resonance condition (2.6) which might be a bit stronger than needed if we were to work directly with problem (2.5) (see Cases III and IV in Section 3.1). On the other hand, we gain the elegance of the reduced equation and the bifurcation conditions become simpler to state.
In what follows it will be useful to introduce the notation per,e (R 2 ).
We rewrite ( Let us study the action of the operator DηH[0, c] in terms of the Fourier coefficients of η. Abbreviating the Fourier coefficients of DηH[0, c] to DηH (k) , we find that is the expression from the dispersion equation (3.4), and Similarly, we find that By the assumptions of the theorem, ρ(c , kj) = 0, j = 1, 2 and ρ(c , k) = 0, k = ±k1, k = ±k2. Using this together with standard properties of Fourier multiplier operators on Hölder spaces (see e.g. Bahouri, Chemin & Danchin [3, Prop. 2.78]), one obtains the following lemma.
Lemma 4.4. The operator L : X 2 → X 0 is Fredholm of index 0. Its kernel ker L is twodimensional and spanned by the functions ηj(x ) = cos(kj · x ), j = 1, 2, and the operator L :X 2 →X 0 is invertible, whereX k denotes the orthogonal complement of ker L in X k with respect to the L 2 per inner product. This allows us to use the Lyapunov-Schmidt reduction. For this purpose we split whereη ∈X 2 . Thus the problem is written as Let Pj be the orthogonal projection in L 2 per (R 2 ) on the one-dimensional subspace spanned by ηj and letP = I − 2 j=1 Pj. Taking projections in (4.3) we obtain the 2 × 2 system Applying the implicit function theorem to (4.5), noting that L is an isomorphism fromX 2 tõ X 0 , we obtain the following reduction.
For convenience we writeη(t, c) instead of ψ(t, c) below. We need one more technical lemma before the proof of the main result.
Proof. Considering formula (4.1) for Hr, we see that the first term is quadratic in c − c and linear in η. Therefore, when replacing η by t1η1 + t2η2 +η(t, c) we get a contribution to the remainder of order O(|t||c − c | 2 ). The second term in (4.1) is quadratic in η to lowest order. The quadratic part is obtained by forming products of terms involving e ±ik j ·x and by evenness and realness we obtain precisely the above expression.
Remark 4.7. The last observation in the proof holds more generally. If we have a kernel spanned by cos(x · k0), k0 ∈ Λ , for some value of c, then we can fix one of the parameters and apply local bifurcation theory with a one-dimensional kernel. The resulting family of solutions will be 2 1 /2-dimensional since we can consider the same problem but for functions that depend only onx = x · k0. This explains why one has to consider at least two-dimensional kernels in order to construct genuinely three-dimensional waves.