Stratified periodic water waves with singular density gradients

We consider Euler’s equations for free surface waves traveling on a body of density stratified water in the scenario when gravity and surface tension act as restoring forces. The flow is continuously stratified, and the water layer is bounded from below by an impermeable horizontal bed. For this problem we establish three equivalent classical formulations in a suitable setting of strong solutions which may describe nevertheless waves with singular density gradients. Based upon this equivalence we then construct two-dimensional symmetric periodic traveling waves that are monotone between each crest and trough. Our analysis uses, to a large extent, the availability of a weak formulation of the water wave problem, the regularity properties of the corresponding weak solutions, and methods from nonlinear functional analysis.


Introduction
Stratification is a phenomenon that is common in ocean flows where in the presence of salinity and under the influence of the gravitational force a heterogeneity in the fluid is produced. Stratification corresponds to the formation of fluid layers, normally arranged horizontally with the less dense layers being located on top of the denser ones. This phenomenon may be caused by many other factors including temperature, pressure, topography and oxygenation. Because of the plethora of effects resulting from stratification, such flows have received much attention, especially in geophysical fluid dynamics. In the setting of traveling stratified waves the problem is modeled by the stationary Euler equations for incompressible fluids, subject to natural boundary conditions, cf. (2.2). The study of two-dimensional stratified flows dates back to the pioneering work of Dubreil-Jacotin. In 1937 Dubreil-Jacotin [23] constructed small-amplitude stratified traveling gravity waves by using power series expansions. Previously in [22] she showed that Gerstner's explicit solution [10,27] can be accommodated to describe exact traveling gravity waves with an arbitrary stratification. Furthermore, related to Gerstner's solution, there is a further exact solution describing an edge wave propagating along a sloping beach [9,43,48] allowing even for an arbitrary stratification. Recently also other exact and explicit solutions of stratified flows in different geophysical regimes have been found, cf. [12,16,17,[29][30][31]36].
Many of the papers dedicated to the stratified water wave problem consider the vertical stratification to be fairly smooth. Small-amplitude periodic gravity water waves possessing a linear stratification have been constructed in [25]. These waves may contain critical layers and stagnation points and the authors in [25] provide also the qualitative picture of the flow beneath the constructed waves. Small-amplitude periodic capillary-gravity waves with sufficiently regular density which may still contain critical layers have been found in [34] by means of local bifurcation. The local bifurcation branches of solutions to the stratified water wave problem have been extended by using global bifurcation theory to global branches in [32]. The papers [25,32,34] use the Long-Yih formulation [37,56] (see (2.6)) of the problem whose availability is facilitated by the fact that the density is sufficiently regular. When excluding critical layers and stagnation points the stratified wave problem can be considered by using Dubreil-Jacotin's formulation (see (2.10)- (2.11)). This approach has been followed in [52][53][54] where-by means of local and global bifurcation theory-small-and large-amplitude stratified periodic water waves of finite depth are constructed both in the presence and absence of surface tension. The existence of solitary free surface water waves with general regular density distribution, together with a qualitative study of such flows, has been provided only recently [8], using again Dubreil-Jacotin's formulation in their treatise. Qualitative properties of stratified water waves with regular density, such as symmetry, regularity, and the unique determination of the wave when knowing the pressure on the bed and the fluid stratification, have been addressed in [7,33,51,55].
In ocean flows, however, the density varies strongly in thin layers called pycnoclines which exhibit sharp density gradients, cf., e.g., [21,46,47]. For this reason some of the research [3,4,14,15,21,39,40,42,49] is restricted to so-called layered models which consider the flow as consisting of a finite number of vertical layers each of them having uniform density. These layers are separated by internal waves which are mainly driven by the density difference between the layers (some models also consider surface tension effects). In this paper we consider a general continuous stratification, but allow for solutions with a density gradient which is merely L r -integrable with r ∈ (1, ∞) arbitrarily close to 1. Furthermore, Bernoulli's function, called vorticity function in the constant density case, is also a general function and is assumed to be L r -integrable too. A similar setting has been studied in [6] but in the absence of surface tension forces. The authors of [6] deal with a layered model with the density in each layer varying continuously in such a way that the density gradient is L r -integrable, but with r > 2 . The choice r > 2 is related to the Sobolev embedding W 1 r (ℝ 2 ) ↪ C 1−2∕r (ℝ 2 ). In this regime the different formulations of the water wave problem mentioned above are equivalent in the setting of periodic Sobolev solutions. Our first main result is an equivalence result for the three formulations of the problem in a suitable setting of strong solutions, cf. Theorem 2.1 (see Theorem 2.5 for the case when surface tension is neglected). The equivalence in these theorems holds for r ∈ [1, ∞) . For r ∈ [1, 2] the Sobolev regularity is too weak for the equations to be realized in L r -spaces, and therefore our notion of strong solutions involves some complementary Hölder regularity. Here the Hölder exponent = 1 − 1∕r ∈ [0, ∞) results from the embedding W 1 r (ℝ) ↪ C 1−1∕r (ℝ). The main result Theorem 2.3, which relies on the equivalence in Theorem 2.1, establishes the existence of infinitely many periodic solutions to the stratified water wave problem having merely a L r -integrable density gradient. Moreover, the wave profiles are symmetric with respect to crest and trough lines and strictly monotone in between them. The proof of Theorem 2.3 uses the Crandall-Rabinowitz theorem [20, Theorem 1.7] on bifurcation from simple eigenvalues in the context of a weak interpretation of Dubreil-Jacotin's formulation. For traveling water waves this idea was first used by Constantin and Strauss 19] to construct homogeneous periodic gravity water waves with discontinuous vorticity. The situation of heterogeneous water waves is slightly different as the equations in the bulk cannot be recast in divergence form, see also [6]. Due to the presence of surface tension we need to deal with a second-order nonlinear equation on the surface boundary corresponding to the dynamic boundary condition at the waves surface. Here we use a recent trick employed first in [41,44] (a similar idea appears also in [3,32,45]) to transform this equation into a Dirichlet boundary condition perturbed by a nonlinear and nonlocal part of order −1 . A particular feature of our analysis is that we fix both the fluid bed and the mean depth of the fluid (within a period). This fact in combination with the weak regularity of the density gradient reduces the number of possible bifurcation parameters. For this reason the best (probably only) choice for a bifurcation parameter is the wavelength (the wavelength has also been used in [24] as one of the bifurcation parameters). It is worth pointing out that this choice provides a remarkable identity in (see (4.32)) that leads us to a very simple and elegant dispersion relation, cf. Lemma 4.9.
The paper is organized as follows: In Sect. 2 we introduce the three formulations of the problem and we establish their equivalence in Theorems 2.1 (and Theorem 2.5). Moreover, we state our main result in Theorem 2.3 on the existence of laminar and nonlaminar flow solutions and some qualitative properties. In Sect. 3 we first introduce the notion of a weak solution to Dubreil-Jacotin's formulation and establish, by means of a shooting method, the existence of at least one laminar flow solution to this latter formulation. This solution does not depend on the horizontal variable, having thus parallel and flat streamlines, and it solves the problem for each value of . This set of laminar solutions (we have a solution for each > 0 ) is then seen as the trivial branch of solutions to the problem. Merely the existence of the laminar solution imposes some restriction on the physical properties of the flows, cf. (3.10) and Example 3.4. In Sect. 4 we reformulate the equations as an abstract bifurcation problem and identify, by using methods from nonlinear functional analysis, a particular value * of the wavelength parameter where a local branch of nonlaminar weak solutions arises from the set of laminar solutions. For this we need to impose a further, quite explicit restriction in (4.12). The proof of Theorem 2.3 is then completed by showing that the weak solutions that were found are in fact strong solutions, cf. Proposition 4.17. This gain of regularity relies on the regularity result in Theorem 4.14, which is inspired by ideas presented in [13] and [26].

Mathematical formulations and the main results
We now present three classical formulations of the steady water wave problem for stratified fluids. We start with the classical Euler formulation. The motion of an inviscid, incompressible, and stratified fluid is described by the Euler equations where is the fluid's density, u is the horizontal velocity, v is the vertical velocity, P is the pressure, and g is the gravitational acceleration. The fluid domain is bounded from below by the impermeable flat bed y = −d , where d is a fixed positive constant, and y = (t, x) denotes the wave surface. In addition to the conservation of momentum which is expressed by the first two equations of the system, the fluid is assumed to be incompressible and mass conserving (these properties correspond to the third and fourth equations, respectively). Our analysis is restricted to the physically relevant case of positive density, that is we assume throughout this paper that there exists a constant 0 > 0 such that The equations in the fluid domain are subject to the following boundary conditions where the atmospheric pressure is set to zero and ≥ 0 denotes the surface tension coefficient. As we are interested in periodic waves we introduce the positive constant to denote the associated (minimal) wavelength. Moreover, we require that at each time t. This condition implies in particular that also the mean depth of the fluid is fixed. Traveling periodic waves correspond to solutions of the previously introduced equations that exhibit a (t, x)-dependence of the form where c > 0 is the wave speed, and which are -periodic in x. Observed from a frame that moves with the wave speed c, traveling waves appear to be steady and we are left with the free boundary value problem where We point out that since u and c appear only in terms of the difference u − c in (2.2), we may view the quintuplet (u − c, v, P, , ) as being the unknown, each of these functions additionally being -periodic with respect to the horizontal variable x.
In order to study problem (2.2) analytically it is useful to consider equivalent formulations. To this end we define the so-called stream function by the relations One may observe that in the moving frame the streamlines of the flow coincide with the level curves of the stream function. Moreover, the density and the total hydraulic head are both constant along the streamlines. In particular, if we require that a condition which is a priori satisfied for homogeneous irrotational water waves, the hodograph transformation H ∶ Ω → Ω defined by is a bijection. Here Ω ∶= ℝ × (p 0 , 0) and p 0 ∶= − | |y=−d is a negative constant. Using this property one can find two functions , ∶ [p 0 , 0] → ℝ , the so-called streamline density function and the Bernoulli function, respectively, such that In particular (x, y) = (− (x, y)) in Ω . As the density usually increases with depth we restrict our considerations to the stably stratified regime defined by the inequality 1 (2.5) ′ ≤ 0.
These considerations lead one to the Long-Yih [37,56] formulation of the hydrodynamical problem (2.2): Under the assumption (2.1) of positive density the condition (2.3) is equivalent to The constant Q in Eq. (2.6) 4 is related to the energy E. Equation (2.6) 4 identifies Q for waves with zero integral mean as follows: Using the (partial) hodograph transformation H , a further equivalent formulation of (2.2)-(2.3) may be derived in terms of the height function h ∶ Ω → ℝ that is defined by The system (2.6) can then be recast in the fixed rectangular domain Ω in the following form: the relation (2.3) taking the form In virtue of (2.11) the quasilinear equation (2.10) 1 is uniformly elliptic. This equation is complemented by a nonlinear and nonlocal boundary condition on p = 0 and a homogeneous Dirichlet condition on p = p 0 . This formulation gives an insight into the flow as the streamlines in the moving frame are parameterized by the mappings In the setting of classical solutions it is not difficult to show that the three formulations (2.2)-(2.3), (2.6)-(2.8), and (2.10)-(2.11) are equivalent, cf., e.g., [11,[52][53][54]. This feature remains true in the more general framework described below.
Equivalent formulations In Theorem 2.1 we present our first main result which establishes for capillary-gravity stratified water waves, that is for > 0 , the equivalence of the three formulations in a suitable setting of strong solutions. The case = 0 is treated in Theorem 2.5. A strong solution of any of the three formulations possesses weak derivatives up to highest order (the order is required by the equations) that are L r -integrable. Moreover the lower order derivatives enjoy some additional Hölder regularity to ensure that all equations are satisfied in L r -spaces (in particular pointwise a.e.). (2.10) Theorem 2.1 (Equivalence for > 0 ) Let , > 0, and assume that (2.1) holds true. Given r ∈ [1, ∞) , set ∶= (r − 1)∕r ∈ [0, 1). Then, the following formulations are equivalent: r ((p 0 , 0)) , and ∈ L r ((p 0 , 0)). (iii) The height function formulation (2.10)-(2.11) for h ∈ W 2 r (Ω) ∩ C 1+ (Ω) with tr 0 h ∈ W 2 r (ℝ) , ∈ W 1 r ((p 0 , 0)) , and ∈ L r ((p 0 , 0)).
The proof of Theorem 2.1 and the corresponding result for = 0 are presented at the end of this section. It is worthwhile to add the following remarks.

Remark 2.2
(a) Given r ≥ 1 , the Hölder coefficient ∶= (r − 1)∕r ∈ [0, 1) corresponds to the onedimensional Sobolev embedding W 1 r (ℝ) ↪ C (ℝ). It is worthwhile to note that W 1 r (ℝ 2 ) is not embedded in a space of continuous functions if r ∈ [1, 2] . Therefore, the Hölder regularity required above is not implied by the Sobolev regularity. (b) All function spaces in Theorem 2.1 consist only of functions that are -periodic with respect to x and q, respectively. (c) The symbol tr 0 stands for the trace operator with respect to the boundary component In the proof of Theorem 2.1 (and also later on) we make use of the following properties The properties (2.12) and (2.14) are classical results, while (2.13) is a direct consequence of (2.12).

Local bifurcation
The main issue of this paper is the local bifurcation result stated below. Under the natural assumptions (2.1) and (2.5) on the fluid density and the following restrictions on the physical quantities 2 (2.12) is an algebra; (2.14) where x * ≈ 1.9368 is the positive solution to e x − x = 5, we prove that the water wave problem (2.2)-(2.3) possesses, for each > 0 , at least one laminar flow solution with flat streamlines. Besides, a critical wavelength * > 0 is identified such that (2.2)-(2.3) has also other solutions with nonflat wave surface and with wavelength close to * . More precisely, the following result holds true.

Then there exists a local bifurcation curve
where > 0 is small, having the following properties: flat streamlines, streamline density , and Bernoulli function .
3) with minimal period (s), streamline density , and Bernoulli function . Moreover, the wave profile has precisely one crest and one trough per period, is symmetric with respect to crest and trough lines, and is strictly monotone between crest and trough. (iv) The wave profile and all other streamlines are real-analytic graphs.

Remark 2.4
(a) We point out that we do not impose any restrictions on the value of > 0, cf. (2.15).
Nevertheless, the critical wavelength * depends in an intricate way on . We conclude this section by proving the equivalence of the three formulations in the setting of strong solutions introduced above. For (x, y) ∈ Ω we now define where p 0 < 0 is a constant to be fixed below. It is obvious that is continuously differentiable with respect to y with y = U. Moreover, making use of Fubini's theorem, the generalized Gauß theorem in [1, Appendix A 8.8], and the relations (2.17) and (2.2) 7 , we find for ∈ C ∞ 0 (Ω ) that where ∈ C 1 (Ω ) is defined by the formula Thus, ∇ = (−V, U) and since these weak derivatives belong to W 1 r (Ω ) , we conclude that ∈ W 2 r (Ω ). Moreover, (2.14) implies that ∈ C 1+ (Ω ). The relation (2.7) is clearly satisfied in view of (2.3). Since is constant on the fluid bed and by (2.2) 6 also on the free surface y = (x) , we infer from (2.7) that we may chose the negative constant p 0 such that = 0 on the free surface.
It is easy to see now that the mapping H defined in Consequently, there exists ∈ L r ((p 0 , 0)) with •H −1 = . Moreover, it actually holds that ∈ W 1 r ((p 0 , 0)) with weak derivative We now consider the expression which defines a function in W 1 Appealing to (2.2) 1 -(2.2) 3 , it follows that q (E•H −1 ) = 0. This relation has at least two implications. Firstly, E is constant at the wave surface, which implies the existence of a constant Q such that which is the semilinear elliptic equation in (2.6). This completes this first step of the proof.

A weak setting for Dubreil-Jacotin's formulation
In this section we seek solutions to problem (2.10)-(2.11) under the general assumptions that where and are arbitrary but fixed. Moreover, we restrict to the setting of stably stratified flows defined by (2.1) and (2.5). The reason for studying the height function formulation is twofold. Firstly, the equations have a single unknown, the height function h, and secondly, the Bernoulli function and the streamline density appear as coefficients in the equations.
Since we aim to formulate (2.10) as a bifurcation problem and to use the wavelength as bifurcation parameter, we let Then h is 1-periodic 3 and (2.10) may be rewritten (after dropping tildes) as while (2.11) remains unchanged. Now not only h is unknown in (3.3) but also the wavelength .
In order to determine strong solutions to (3.3) and (2.11) as defined in Theorem 2.1(iii), we shall first find weak solutions to this problem and then improve their regularity. We now introduce a proper notion of weak solutions. ∈ W 1 r ((p 0 , 0)), and ∈ L r ((p 0 , 0)), We now show that T 1 is a self-map. Indeed, recalling that ′ ≤ 0 , it holds that and consequently T 1 H ∈ C([p 0 , p 1 ], [0, ∞)).
provided that Hence, T 2 possesses a fixed point H 2 ∈ W 2 r ((p 1, , p 2, )) . Thus, we may extend H 1 to a solution to (3.5) which lies in W 2 r ((0, p 2, )) and which equals H 2 on (p 1, , p 2, ) . Arguing in this way, if necessary, we may extend (in a finite number of steps) H 1 onto the whole interval [p 0 , 0]. The uniqueness claim is obvious. ◻ We next show that the solution found in Proposition 3.2 depends smoothly on the parameter . One can easily verify that K actually maps continuously into W 2 r ((p 0 , 0)) . Since the embedding of W 2 r ((p 0 , 0)) into C([p 0 , 0]) is compact, it follows that K is a compact operator. Hence, H F(H, ) is a compact perturbation of the identity. Using the Riesz-Schauder theorem, we can conclude that H F(H, ) is a Fredholm operator of index zero.
Let us now return to the more involved setting of stratified waves addressed in this paper. Given > * , we define Then recalling (3.9), we observe that p = 0 and H(0; ) < d if is sufficiently large. The size condition that we require reads Since H(p; ) ∈ [0, d] for p ∈ [p 0 , p ] and ′ ≤ 0 , it follows that In view of (3.10) we conclude (by a contradiction argument) that p < 0 for all which are sufficiently close to * . Hence Together with Lemma 3.3 we conclude the following result establishing the existence of at least one strong laminar flow solution to (3.3) and (2.11).

Proposition 3.5 Assume that (3.10) is satisfied. Then there is at least one solution
H ∈ W 2 r ((p 0 , 0)) to (3.4). Moreover, H ′ is a positive function.
Proof The proof follows from Proposition 3.2, Lemma 3.3, and the discussion preceding Proposition 3.5. ◻

Local bifurcation
In the following we assume that the density and the Bernoulli function satisfy (2.1), (2.5), and (3.1), i.e., and that (3.10) holds true. This guarantees, in particular, the existence of a laminar flow solution to (3.3) and (2.11). The first goal of this section is to recast the weak formulation of (3.3) and (2.11) as an abstract bifurcation problem. For this goal only the Hölder regularity of the strong solutions to (3.3) and (2.11) is needed. To proceed, we define the Banach space H(0; ) > d for sufficiently close to * .
which is endowed with the norm and we set We recall that these Banach spaces consist only of periodic distributions of period 1. Furthermore, we fix a laminar flow solution H ∈ W 2 r ((p 0 , 0)) ↪ (as found in

4), it holds that
We note that F is smooth with respect to its variables, that is The aim is to apply the Crandall-Rabinowitz theorem [20, Theorem 1.7] on bifurcation from simple eigenvalues to (4.1) to determine other solutions to (4.1) that depend on the variable q. To this end we need to determine * > 0 such that the partial Fréchet derivative h F( * , 0) is a Fredholm operator of index zero with a one-dimensional kernel. A certain transversality condition also needs to be satisfied, cf. Proposition 4.13.
, 2 are even, 1 is odd in q  Proof According to [28,Theorem 8.34], the mapping is an isomorphism (we recall that H ′ is positive). Since (1 − 2 q ) −1 ∈ L(C(ℝ), C 2 (ℝ)) , it follows that the operator is compact. Therefore, the desired claim follows. Hence, for k = 0 , we find that w 0 ∈ W 2 r ((p 0 , 0)) solves the system where Furthermore, given k ≥ 1, the function w k ∈ W 2 r ((p 0 , 0)) solves the system with ∶= (2k ) 2 . The remainder of this section is organized as follows. In the first paragraph we specify a condition under which (4.4) has only the trivial solution w 0 = 0 . The objective of the second paragraph is twofold. On the one hand we determine special values * of such that (4.5) has a one-dimensional space of solutions for k = 1 , respectively = (2 ) 2 . On the other hand we prove that (4.5) admits only the trivial solution w k = 0 for ∈ {(2k ) 2 ∶ k ≥ 2} . The third paragraph treats the above mentioned transversality condition and the last paragraph is devoted to the proof of Theorem 2.3.
Conditions such that w 0 = 0 . We now show that, under some additional restrictions (cf. (4.7)), the system (4.4) has only the trivial zero solution. Equivalently formulated, we show that the elliptic operator [u ↦ (a 3 u � ) � − g � u] , which is supplemented by homogeneous Dirichlet boundary conditions, does not have zero as an eigenvalue. We point out that the coefficient of u is positive and it may be unbounded, while the coefficient a 3 is not explicitly determined. This is where the assumption (2.15) 2 becomes important, as it constitutes an explicit relation in terms of d, p 0 , , and which ensures that (4.7) is satisfied, see Example 4.3. We also emphasize that in the constant density case the assertion of Lemma 4.2 follows via maximum principles, hence (4.7) (or (4.12)) is then not needed. Moreover, the restriction (4.7) (or its weaker version (4.12)) is also used at several places in the paragraphs below in order to establish the existence of a bifurcation point. In particular, for homogeneous irrotational waves (that is = ∈ ℝ and = 0 ), see Remark 4.10, the explicit condition (4.12) ensures, due to the fact that x * < 2 , that the dispersion relation (4.40) has at least a solution. Since x * ≈ 1.9368 , this shows that (4.12) is close to being optimal with respect to this issue.
To prove that z 0 is positive in [p 0 , 0] , we assume there exists p 1 ∈ (p 0 , 0] with z 0 > 0 in [p 0 , p 1 ) and z 0 (p 1 ) = 0 . The relation (4.9) implies that w 0 > 0 in (p 0 , p 1 ] . Invoking (4.8) 2 it holds that Since A ′ is positive, we find that z 0 e A is decreasing in [p 0 , p 1 ] . Consequently Since z 0 (p 1 ) = 0, the relation (4.10) yields and using (4.11) we arrive at Since (s) ≤ (r) for p 0 ≤ r ≤ s ≤ 0 and recalling the definition of A we get in view of (4.7) that which is a contradiction. Our assumption is thus false and the proof complete. ◻ We now provide a quantitative condition which ensures that (4.7) is satisfied.

Example 4.3
Let * be as defined in (2.15) and assume that (3.10) holds. If Hence (4.7) holds true. ◻ Relation (4.12) provides an explicit condition which ensures that the system (4.4) has only the trivial solution w 0 = 0 . Consequently, for all > 0 , the kernel of h F( , 0) does not contain functions that depend only on the variable p (except for the zero function). We now address the second issue of determining * > 0 such that h F( * , 0) has a onedimensional kernel spanned by a function of the form w 1 (p) cos(2 q) , with w 1 being (up to a multiplicative constant) the only nontrivial solution to (4.5) when = * and ∈ {(2k ) 2 ∶ k ∈ ℕ}.
The system (4.5) with as a variable. We seek > 0 such that (4.5) has a one-dimensional space of solutions for = (2 ) 2 and only the trivial solution for > (2 ) 2 . To this end we first determine the dimension of the space of solutions to (4.5). Given ∈ (0, ∞) (4.12) 14) and ∈ ℝ , we let R , ∶ W 2 r,0 ((p 0 , 0)) → L r ((p 0 , 0)) × ℝ denote the Sturm-Liouville-type operator where we set We associate with (4.5) the initial value problems and Lemma 4.4 Given > 0 and ∈ ℝ , the operator R , ∶ W 2 r,0 ((p 0 , 0)) → L r ((p 0 , 0)) × ℝ defined in (4.15) is a Fredholm operator of index zero and dim ker R , ≤ 1 . Furthermore, dim ker R , = 1 if and only if the solutions w 1 and w 2 to (4.16) and (4.17) are linearly dependent. In this case it holds that Proof We first decompose R , = R I + R c , where It is clear that R c is a compact operator. Furthermore, R I is an isomorphism. Hence, R , is a Fredholm operator of index zero.
colinear. Invoking (4.15) 2 and (4.17) 2 , (4.19) shows that also w and w 2 are colinear. Finally, if w 1 and w 2 are colinear, it is easy to see that they both belong to ker R , . ◻ We can now reformulate our task as the problem of determining * > 0 such that the Wronskian We next prove that for each > 0 there exists at least one solution to W(0; , ) = 0 . As a first step we show that This is a direct consequence of the following more general statement. where C 4 denotes a nonnegative constant that is independent of . Hence, recalling (4.25), we observe that the right-hand side tends to infinity as → ∞ . In particular, the assertion (4.30) directly follows. ◻ The relations (4.21), (4.22), and (4.28) ensure that the equation W(0; , ) = 0 has at least one solution > 0 for any fixed > 0 . The next result provides a remarkable identity, cf. (4.32), that will enable us later to identify the largest solution ( ) to the above equation in a quite explicit way. (3.10) and (4.12) hold and that ( , ) ∈ (0, ∞) 2 satisfies W(0; , ) = 0 . Then, it holds � w 1 (0)W (0; , ) < 0 and Proof Let w 1 =w 1 ( ⋅ ; , ) denote the solution of (4.16) corresponding to and . We first consider the derivative W (0; , ). Using the algebra property of W 1 r ((p 0 , 0)) we conclude that the partial derivative w 1, = w 1 ( ⋅ , , ) belongs to W 2 r ((p 0 , 0)) and solves the problem We multiply (4.16 1 by w 1, ) and (4.33) 1 by w 1 and subtract the resulting equations. This yields Integrating with respect to p from p 0 to 0 and using integration by parts then gives Recall that W(0; , ) = ( 2 g (0) + )w 1 (0) − 2 a 3 (0)w � 1 (0) according to (4.20). Hence, the derivative with respect to is given by We point out that, since W(0; , ) = 0 , the functions w 1 and w 2 are linearly dependent and not identically zero. This implies that w 1 (0) ≠ 0. Multiplying the latter identity by w 1 (0) and using (4.34) and the colinearity of w 1 and w 2 we then obtain
Hence, (4.35) implies that which completes the proof. ◻ Given > 0 , let ( ) denote the largest solution to W(0; , ) = 0. In the following lemma we identify, using Lemma 4.8, the mapping up to a positive multiplicative constant. Hence there exists a constant C D > 0 such that ̃ ( ) = C D 2 for all ∈ ( 0 − , 0 + ). The desired claim follows now at once. ◻

Remark 4.10
Our analysis shows, under the assumptions (3.10) and (4.12), that the constant C D found in Lemma 4.9 is the largest constant such that the solutions to determined by the initial data are linearly dependent. The constant C D depends only on Earth's gravity g, the mass flux p 0 , the water depth d, the surface tension coefficient , the density function , and on Bernoulli's function . In the homogeneous case = ∈ ℝ we obtain, under the assumption that the flow is irrotational (that is for = 0 ), that C D is the largest positive solution to We cannot exclude the possibility that there exist finitely many (since W(0; , ⋅) ∶ ℝ → ℝ is real-analytic there cannot exist infinitely many) positive constants for which the two solutions defined above are linearly dependent. Each of these constants defines a new function i ∶ (0, ∞) → ℝ with (a 3 w � ) � − g � w − C D aw = 0 in L r ((p 0 , 0)) (w, w � )(p 0 ) = (0, 1) or (w, w � )(0) = (a 3 (0), g (0) + C D ), respectively, which satisfies W(0; , i ( )) = 0 for all > 0 . This complicates the bifurcation analysis for (4.1) a lot as in this situation the dimension of ker h F( , 0) may be larger than 1 for certain . However, this behavior is expected since, even in the case of constant density, phenomena like bifurcation from double eigenvalues or secondary bifurcation may occur when allowing for surface tension effects, cf. [35,53,54]. We now shortly discuss the dispersion relation (4.40) in the homogeneous irrotational case (that is for = ∈ ℝ and = 0 ). Setting √ C D = x , the problem reduces to finding the positive zeros of the function f ∶ ℝ → ℝ with Since f (0) = g d − p 2 0 ∕d 2 < 0 (this inequality is a direct consequence of (4.12)) and f (x) → ∞ as x → ∞ , Eq. (4.41) has at least a positive solution. Note that f is even. Furthermore, direct computations show that Moreover, it can be shown that if f � (x) = 0 for some x > 0 , then f �� (x) > 0 (see, e.g., [38,Lemma 3]). Consequently, regardless of the sign of the equation f (x) = 0 has a unique solution. Indeed, if > g d 2 ∕3 , then f is strictly increasing on (0, ∞) . If = g d 2 ∕3 then f is again strictly increasing on (0, ∞) since then f (4) , then f has a unique global minimizer let's say at x 0 and f is strictly decreasing on (0, x 0 ) and strictly increasing on (x 0 , ∞).
We proceed with the following result.
To this end we need the following regularity result.

Remark 4.15
The estimate (4.49) implies in particular that all the streamlines, including the wave surface, of the corresponding strong solution to (3.3) and (2.11), see Proposition 4.17 and Theorem 2.1, are real-analytic graphs. Similar results for classical solutions to (3.3) and (2.11) have been obtained in [33,55] under the more restrictive assumption that ′ , ∈ C ([p 0 , 0]) . We point out that the study of the a priori regularity of homogeneous but rotational waves has been initiated in [13], see also [26].
The next lemma plays an important role in the proof of Theorem 2.3(iii).   Proof In virtue of Theorem 4.14 we have h q ∈ C 1+ (Ω), hence q h p = p h q ∈ C (Ω). This in turn implies that q h p is the classical derivative of h p with respect to q. Recalling that it follows that and (2.14) in turn yields Consequently, and the repeated use of [28, Lemma 7.5] finally yields h p ∈ W 1 r (Ω) . Since h q ∈ C 1+ (Ω) , it follows that h ∈ W 2 r (Ω) is a strong solution to (3.3) and (2.11) (as tr 0 h is real-analytic, the condition tr 0 h ∈ W 2 r (ℝ) is obvious). ≤ C‖A 1 W + B 1 W q + C 1 W p + w 2 (A 1 − A 2 ) + w 2,q (B 1 − B 2 ) + w 2,p (C 1 − C 2 )‖ C ∕2 (Ω) article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creat iveco mmons .org/licen ses/by/4.0/.