Surface water waves as saddle points of the energy

By applying the mountain-pass lemma to an energy functional, we establish the existence of two-dimensional water waves on the surface of an infinitely deep ocean in a constant gravity field. The formulation used, which is due to K. I. Babenko [3, 4] (and later to others, independently), has as its independent variable an amplitude function which gives the surface elevation. Its nonlinear term is purely quadratic but it is nonlocal because it involves the Hilbert transform. Moreover the energy functional from which it is derived is rather degenerate and offers an important challenge in the calculus of variations. In the present treatment the first step is to truncate the integrand, and then to penalize and regularize it. The mountain-pass lemma gives the existence of critical points of the resulting problem. To check that, in the limit of vanishing regularization, the critical points converge to a non-trivial water wave, we need a priori estimates and information on their Morse index in the spirit of the work by Amann and Zehnder [1] (see also [14]). The amplitudes of the waves so obtained ∗Supported by a grant of the Swiss National Science Foundation. †Membre de l’Institut Universitaire de France. ‡Supported by an EPSRC Senior Fellowship. All three authors enjoyed the possibility of collaborating during the “Clay Mathematics Institute Symposium and EuroWorkshop on Hamiltonian Systems” at Edinburgh in May 2001.


Introduction
The water-wave problem is the determination of wave profiles compatible with Bernoulli's theorem which implies that the pressure at the surface of a two-dimensional, infinitely deep, irrotational, incompressible flow under gravity is constant (surface tension is not considered in this paper). The physical parameters are the wave speed, the wavelength, and gravity (acting vertically downwards). After some rescaling, they can be chosen equal to be, respectively, 1, 2π and λ. (Note that λ is now the only (dimensionless) parameter in the problem.) Consider such a wave in a reference frame propagating with the speed of the wave. The constant-pressure condition then has the form where η is a smooth 2π-periodic function and v is a smooth two-dimensional velocity field that is divergence free and irrotational on the domain {(x, y) : y ≤ η(x)}. The asymptotic velocity at infinite depth is lim y→−∞ v(x, y) = (−1, 0) uniformly in x ∈ R.
If such a wave exists then the energy, kinetic plus potential, in one wavelength is where 1 is subtracted from | v| 2 so that the integral is finite (this is related to the choice of the constant 1/2 on the right-hand side of (1)).
P. R. Garabedian [15] argued that variational methods applied to E should yield the existence of water waves. However Turner [25] pointed out some technical difficulties with the idea and introduced a different variational approach which led to the existence of periodic and solitary waves on the surface of a fluid layer of finite depth. (In the absence of surface tension, it is known that solitary waves do not exist if the depth is infinite [13]).
More recently Babenko [3,4] introduced a new variational formulation of the periodic water-wave problem, but used non-variational methods to study it. (See also [18].) Plotnikov [19] discovered the analogous formulation for solitary waves. (See [23] for references to other independent discoveries of Babenko's formulation.) The advantage of this formulation is that it involves a quadratic equation for a function of a single real variable; the difficulty is that it involves the Hilbert transform which is a non-local operator.
Underlying our approach is the idea of adapting Turner's ideas and the mountain-pass lemma to the Babenko formulation. Like Turner, we get the existence of small-amplitude water waves. However our method works globally if, roughly speaking, the gravitational force field λ in the Bernoulli boundary condition vanishes a little below the maximal height of a Stokes' wave. Only when precise estimates are needed to deal with constant gravity fields is the argument forced to become local. A bifurcation-theoretic variational approach to the Babenko formulation was given in [8] where a Lyapunov-Schmidt reduction, followed by a study of the reduced problem in terms of finite-dimensional constrained optimization, was used. However this is a strictly local method which avoids the essential difficulties of the infinite-dimensional formulation and it gives no quantitative information about the solutions found.
In the end we prove the existence of a nonzero symmetric wave for some λ * ≤ 0.99 . This result is not a consequence of the local existence theorem obtained by Babenko [3]. Indeed, in Babenko's paper, solutions are found for λ * = 1/(1 + ε 2 ) with ε ∈ (0, 1/25], i.e. λ * ∈ [0.9984, 1). Note that λ * = 0.99 corresponds to 2 = 0.2, which, for a small wave, is a good approximation of the trough-to-crest height (for the extreme wave, this height is approximatively 0.8868 [7,18]). Much more is known from global continuation methods [7,8,9], from numerical investigation [12,18] and from computer assisted proofs [5,6]. However the present work may be a small step towards a better understanding of large-amplitude water waves from a variational viewpoint.
Note added in proof. Since this paper was completed we have developed a general theory of quasi-linear problems in an abstract Hilbert-space setting (see [10]) which is sufficiently general to cover, for example, the the existence question for periodic capillarygravity waves with its awkard curvature term that represents surface-tension effects. In the abstract version of the theory, we stay closer to the spirit of Turner's work and seek minimizers, rather than mountain pass solutions.

Preliminaries
Let L p 2π , 1 ≤ p < ∞, denote the linear space of 2π-periodic, real-valued, measurable 'functions' on R with and essentially bounded if p = ∞. The space of functions u ∈ L 2 2π with u also in L 2 2π is denoted by W 1,2 2π . Denote by C ∞ 2π the space of 2π-periodic functions u which are infinitely differentiable.
Let the Fourier coefficients of u ∈ L 1 2π with respect to the orthonormal basis {(2π) − 1 2 e ikt : k ∈ Z} be denoted byû k , k ∈ Z. Thenû −n =û n , since u is real, and L 2 2π is a real Hilbert space with inner product u, v = n∈Zû nvn , In this notation the fractional order Sobolev space H 1 2 is the Hilbert space of functions u ∈ L 1 2π with norm given by u 2 The periodic Hilbert transform of an L 1 2π -function u is defined for almost all x ∈ R by the Cauchy principle value integral It is well known [26] that C : L p 2π → L p 2π is a bounded linear operator if 1 < p < ∞, C does not map L p 2π into itself when p = 1 and ∞, and the Hilbert transform of an infinitely differentiable function is infinitely differentiable. Moreover C is a Fourier multiplication operator on L p 2π , p > 1, in the sense that equivalently, C(cos nt) = sin nt for n ≥ 0 and C(sin nt) = − cos nt for n ≥ 1.
From this it is clear that u → Cu is symmetric in the sense that The Hilbert transform has a geometric interpretation in the complex plane. Suppose that w ∈ W 1,2 2π and that w(t) = ∞ n=0 a n cos nt + b n sin nt , a n , b n ∈ R, b 0 = 0.
Then the complex-valued function i(a n + ib n )e −in(φ+iψ) = ∞ n=1 a n sin nφ − b n cos nφ e nψ + i ∞ n=0 a n cos nφ + b n sin nφ e nψ is holomorphic on the open half plane {φ + iψ ∈ C : ψ < 0}, its trace on the boundary and it converges to ia 0 as ψ → −∞. Since {a n } and {b n } are square-summable, it is well known [26] that Cw(φ) + iw(φ) = 0 on a set of positive measure if and only if {a n } and {b n } are both zero sequences.
Finally we mention an alternative way of interpreting the Hilbert transform in the framework of complex analysis. The function w, given by (6), and Cw can be extended to the open unit disc D = {z ∈ C : |z| < 1} by writing z = re it with 0 ≤ r < 1 and t ∈ R in such a way that w(t) + iCw(t) is the trace on the unit circle (r = 1) of the holomorphic function (a n − ib n )z n , z ∈ D.
Observe that the value of this extension at z = 0 is a 0 = [w] and the trace on the circle is non-zero almost everywhere if w ≡ 0.

Variational Formulation
Our approach is based on the following form of the water-wave problem: to find (λ, w) such that 1 2 Given a solution w of (7) in the form (6), we obtain a solution of (1) heuristically as follows. For φ ∈ R and ψ ≤ 0, define The function y is harmonic and is an extension of w in the sense that Equivalently, x and y are harmonic conjugates and satisfy the Cauchy-Riemann equations: New variables (x, y) can now be defined by writing We find that the half-plane where η is given implicitly by and the velocity field is defined by Hence, as functions of x and y, φ can be regarded as a velocity potential and ψ as a stream function. Clearly, when ψ = 0 and φ = t ∈ R, . This intuitively explains the equivalence between (1) and (7).
Note that (7) is not a variational equation as it stands. However Babenko [4] found that the first equation of (7) is satisfied by any w ∈ C ∞ 2π that satisfies which is the Euler equation of the functional Recently Toland [23] (improving on [21]) observed that all smooth, non-constant solutions of (8) automatically satisfy the remaining condition in (7). The energy (2)  The simple form of J, by comparison with (2), leads one naturally to enquire if water waves can be found using the direct method of the calculus of variations on (9). The present work shows that this is indeed possible. The waves which emerge are small, but not too small. In the next three sections we study a functional more general than J, and in the final section we specialize to J.

Truncation, Penalization and Regularization
In the case when R = ∞, ρ ≡ 0 and f (w) = λw 2 − w, the functional J coincides with J.
The idea is to replace the function λw 2 − w in J with its truncation f in J so that inf(−f ) > 0 . This property will be very important when getting a priori estimates and when studying the regularity of the critical points. Note that, as observed in [11], much of the structure of the problem persists for general f . The role of the penalization term ρ( π −π wCw dt) comes in proving the estimates needed to show that when w is a mountainpass critical point of the regularized J (see (20)), the function λw 2 −w is unaffected by the truncation. In fact it allows us to work in the domain {w ∈ W 1,2 2π : π −π wCw dt < πR 2 } by preventing a mountain-pass critical point from approaching the boundary. Alternatively, we could have used the Hampwile theorem [20] It follows that critical points satisfy For φ ∈ W 1,2 2π , the quadratic form J (w)(φ, φ) is given by In the light of relation, E = −2J, between (2) and (9), smooth solutions of (11) should correspond to water waves when ρ ≡ 0 and f (w) = λw 2 − w. The proof of the general result, which is new even for the special case, has an immediate generalization to the regularized functional introduced in (20).
In particular, if ρ ≡ 0, then f (w(t)) < 0 and Then as we saw in Section 2, W * is the trace on the circle of a function W holomorphic on the unit disc and |W * | is non-zero almost everywhere on [−π, π]. Now suppose that v ∈ C ∞ 2π , so that v + iCv is the trace on the boundary of some function V holomorphic on the disc with V (0) = [v]. Let Φ = iW V and note from (14) that Then Φ(0) = −[v] ∈ R and its trace satisfies We conclude that, for any w ∈ C ∞ 2π the range of the mapping . Let z and z denote the real and imaginary parts of z ∈ C and suppose that w ∈ C ∞ 2π satisfies (11). If φ ∈ C ∞ 2π then clearly Since φ is arbitrary, the first assertion follows. Now suppose that ρ ≡ 0. Then Since f, w and Cw are smooth, it follows that f (w) is nowhere zero and the result follows.
When f (w) = λw 2 − w, equation (13) can be written in the form When (17) is compared with (7) we see that the parameter λ has been replaced by the a priori unknown value λ However the introduction of ρ will in the end yield an a priori bound on the norm of w.
In other words, ρ plays the same role as a constraint in the calculus of variations.
For general f , (13) still has a geometric interpretation, because it can be written formally as 1 Thus a Bernoulli condition still holds but the gravitational force field has now been replaced by a potential field of force pointing vertically downwards with intensity As before, y is the vertical coordinate (the height) of the surface above some horizontal axis.
Observe that, when λ = 0 and therefore f (w) = λw 2 − w = −w, every constant w is a solution of (11). This family of "trivial" solutions makes the problem unnecessarily complicated. To eliminate these solutions and to remove the translation invariance of the problem, we work in the subspace H of W 1,2 2π consisting of even functions of zero mean with norm given by A critical point w of J restricted to H satisfies almost everywhere for some constant c. It can also give rise to a water wave, as the next result shows. (18),

Proposition 2. Let w be a smooth critical point of J restricted to H. Then for the constant c in
Proof. The proof is the same as that of Proposition 1.
When f (w) = λw 2 − w, (19) can be written and the value of c can be found by integrating (18): This leads us to impose a new condition on ρ : If ρ has this property, then 1 + ρ π −π wCw dt − c ≥ 1, and so the coefficient is less than λ. The functionw := w − c/(2λ) is then a solution of the Bernoulli equation (7) in which λ is replaced byλ.
Now, in addition to restricting J defined by (10) to H, we add a regularizing term. Consider the regularized functional J defined by where Then critical points w ∈ H of J such that π −π wCw dt < πR 2 have w ∈ W 1,2 2π and satisfy almost everywhere for some constant c. We will see that that w = 0 is a non-degenerate minimizer of J when > 0. When = 0, the linear operator J (0) = J (0) defined on H is not invertible and present knowledge of the regularity of the critical points of J in larger (i.e. weaker) spaces than H is rather limited [24]. On the other hand, a simple bootstrap argument shows that a solution w ∈ W 1,2 2π of (21) has w ∈ C ∞ 2π .
In what follows, we do not need the generalization of the pointwise identity (19) to the case > 0; an integral identity that is much easier to prove, namely (22) suffices. (This integral identity also holds for = 0 and w ∈ H, as it is seen by multiplying (18) by Cw and integrating.) However, for completeness, we show that if w ∈ H satisfies π −π wCw dt < πR 2 and is a (necessarily smooth) solution of (21) with > 0, then The proof is the same as that for (19), the only new ingredient being the following lemma.
Lemma 3. For all w ∈ C ∞ 2π ∩ H and W * defined by (14), Proof. We have C w w − Cw Cw = w Cw + w Cw since (φ + iCφ)/W * has a holomorphic extension to the unit disc in the complex plane which is zero at 0. Now This completes the proof.

A priori estimates
The mountain-pass lemma will be applied to J . To understand the convergence of the critical points as → 0, we need a priori estimates that are independent of . First we recall a useful lemma from [22].
Here is our first result giving an a priori estimate: and ρ ≥ 0 (this also holds for = 0 and w ∈ H).
It follows from this and Hölder's inequality that and the first assertion follows. That follows immediately from (23). We then apply Lemma 4 with h (s) = s sup f − f (s).
Let H denote the completion of H with respect to the metric defined by the inner product in which u k , v k denote the Fourier coefficients of u, v ∈ H with respect to the orthonormal basis {π − 1 2 cos nt : n ∈ N}. Thus H is a subspace of the fractional Sobolev space defined in (3). Clearly H is a Hilbert subspace of even functions with zero mean in L 2 2π and Now for > 0 define a new inner product on H by Let w ∈ C ∞ 2π be a solution of (21) (with π −π wCw dt < πR 2 ). It follows that there exists a linear operator L : H → H such that Assume that µ is a non-positive eigenvalue of L : Proposition 6. If inf{−f (w(t))} > 0, ρ ≥ 0 and ρ ≥ 0, then Remark.The right-hand side can be further estimated thanks to Proposition 5.

Application of the Mountain-Pass Lemma
In this section we prove the existence of a saddle point of J restricted to H if J has the right mountain-pass structure. For a correct choice of the parameters, the critical point then gives a smooth solution of (7) (see next section). It is convenient to distinguish the linear and nonlinear parts of f . To do so we write where λ ∈ (0, 1) is fixed, g : R → R is a smooth function with Note that where the first inequality follows from Lemma 4 applied to h (w) = −f (w) = w − λg(w).
The functional J is of class C 2 on H, but it is not of class C 2 on H (see (24) for the definition of H). However, for all w ∈ H, the second derivative J (w) : H × H → R has a continuous extension J (w) : H × H → R. Indeed, for all w ∈ H, (12) implies that for some constant K > 0 and all φ ∈ H, because φCφ − C(φφ ) ≥ 0 almost everywhere by Lemma 4. Applying the Cauchy-Schwartz inequality to the positive symmetric bilinear for all φ 1 , φ 2 ∈ H. Therefore J (w)(φ 1 , φ 2 ) and J (w)(φ, φ) can be extended by continuity to all φ 1 , φ 2 , φ ∈ H.
Then there exists a solution w ∈ H of (18) such that π −π it follows that {w k } is bounded in H. Extracting a subsequence we may assume that This shows that J satisfies the Palais-Smale condition.
Since the functional J on H has mountain-pass geometry at 0 ∈ H and satisfies the Palais-Smale condition, there exists a critical point w of J on H with the property that where Γ denotes the set of continuous paths in H joining 0 and u * [2].
The next step is to let → 0 and for this we need an a priori bound. Since w is a critical point of J , Proposition 5 gives and therefore w is bounded in H uniformly in . By taking a sequence of → 0, we may suppose that w w as → 0. Since J (w )v = 0, taking the limit as → 0 in (28) yields that for all v ∈ H, −λg (w)v+(1−λg (w))vCw +(w−λg(w))Cv dt+2ρ π −π wCw dt π −π vCw dt and hence that w ∈ H satisfies (18). Moreover Recall space H in (24), the inner product on H for > 0 in (25), and the linear operator L : H → H such that Since w is a mountain-pass critical point of J , the bilinear form J (w )(v, v) is nonpositive for v in a non-trivial subspace of H (see [17], [16,Ch. 6]). Therefore the selfadjoint operator L has a non-positive eigenvalue µ : L u = µ u for some u ∈ H\{0}. Hence, by Proposition 6, where the constant is independent of . Now normalize u such that u 1 2 = 1. Then, since H is continuously embedded in L 4 (−π, π), there is a constant K 1 such that u 2 Suppose that u u in H and µ →μ ∈ (−∞, 0] as → 0. Since u 2 converges strongly in L 2 2π to u 2 , it follows that u = 0. Since, by assumption, w w in H, it follows from (26)  where 0 < r < R < ∞. Moreover we assume that M < 1/2 (which implies M < 1/(2λ) uniformly in λ ∈ (0, 1)). We easily get the following particular case of Theorem 7.