STEADY PERIODIC HYDROELASTIC WAVES IN POLAR REGIONS

. We construct two-dimensional steady periodic hydroelastic waves with vorticity that propagate on water of ﬁnite depth under a deformable ﬂoating elastic plate which is modeled by using the special Cosserat theory of hyperelastic shells satisfying Kirch-hoﬀ’s hypothesis. This is achieved by providing necessary and suﬃcient condition for local bifurcation from the trivial branch of laminar ﬂow solutions.


Introduction
Hydroelastic waves propagate in polar regions at the surface of water covered by a deformable ice sheet.The water is modeled as an inviscid and incompressible fluid with constant density (set to be 1).Assuming that the flow is two-dimensional, the equations of motion are the Euler equations u t + uu x + vu y = −P x , v t + uv x + vv y = −P y − g, where 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 a flat impermeable bed located at y = −d, where d > 0 is a constant, and the free wave surface {y = η(t, x)} is assumed to be a thin ice sheet which is modeled as a thin elastic plate by using the Cosserat theory of hyperelastic shells satisfying Kirchhoff's hypothesis [19,27].The inertia of this thin elastic plate is neglected, we assume that the plate is not pre-stressed, and consider only the effect of bending, neglecting the stretching of the plate.Therefore we impose the following boundary conditions where α > 0 is a constant, and ω(η) := (1 + η 2 x ) 1/2 . (1.1c) We investigate herein the existence of periodic steady water wave solutions to (1.1a)-(1.1c)for which the unknowns u, v, P, η satisfy (u, v, P )(t, x, y) = (u, v, P )(x − ct, y) and η(t, x) = η(x − ct), where c > 0 is the speed of the wave.Moreover, letting λ > 0 denote the period of the wave, we restrict to solutions which fulfill λ 0 η(x) dx = 0. (1.1d) Setting we thus look for functions u, v, P : Ω η → R and η : R → R which are λ-periodic in x and solve (after replacing u − c by u) the coupled system of equations We also exclude the presence of stagnation pointy by requiring that u < 0 in Ω η . (1.2b) A similar setting has been considered in [18] where, using a variational approach, the authors establish the existence of hydroelastic solitary waves for sufficiently large values of the dimensionless parameter α under the assumption that the vorticity is zero.Within the same irrotational scenario, the authors of [2] establish the existence of symmetric envelope hydroelastic solitary waves by using spatial dynamics techniques.Moreover, in [3,4], the existence of periodic hydroelastic waves which may posses a multi-valued height between two superposed irrotational fluid layers with positive densities separated by an elastic plate was shown via global bifurcation theorem, the analysis being based on the reformulation of the problem as a vortex sheet problem.
We also mention the paper [9] where, in the rotational setting, weak periodic solutions to a related problem which acconts also for surface tension effects at the free boundary are constructed via a variational approach, see also [8,28] for hydroelastic wave models which allow for both bending and stretching of the elastic wave surface.
The initial-value problem for flexural-gravity waves has been investigated in [6] by developing a well-posedness theory based on a vortex sheet formulation.
For numerical studies of hydroelastic waves, in the setting of constant vorticity, we refer to the recent works [15,16,31].
In the present paper we extend the existence theory for (1.2) by allowing for a general vorticity The vorticity is a very important aspect of ocean flows also in polar regions, a non-zero vorticity characterizing waves that interact with non-uniform currents such as the Antarctic Circumpolar Current or near-surface currents in the Arctic Ocean, see e.g.[1,11,24,29].An essential tool in our analysis is the availability of two equivalent formulations of (1.2), the stream function formulation (2.2) and the height function formulation (2.3), see Proposition 2.1.In particular, the condition (1.2b) enables us to introduce the so-called vorticity function γ : [p 0 , 0] → R, where p 0 < 0 is a constant, which determines, via the stream function formulation (2.2), the vorticity ω of the flow, see Section 2. While we consider a general Hölder continuous vorticity function γ, in order to establish the existence (and uniqueness) of a laminar flow solution to (1.2) (with a flat wave profile located at y = 0 and x-independent velocity and pressure) the restriction (1.3) is required on γ and the physical parameters.This laminar flow solution is a solution to (1.2) for each value of the wavelength λ.We will then use λ as a bifurcation parameter in order to determine other nonlaminar symmetric (with respect to the horizontal line x = 0) hydroelastic waves.It turns out that bifurcation can occur if and only if a second condition, see (1.4), is satisfied.In the setting of irrotational waves these conditions are explicit, see Remark 1.3.In order to prove our main result in Theorem 1.1, we cannot directly use the aforementioned formulations (2.2) or (2.3) of the problem because the boundary condition in these formulations that corresponds to the dynamic boundary condition (1.2a) 4 involves fourth order derivatives of the unknown, whereas the elliptic equation posed in the (fluid) domain is of second order.However, inspired by an idea used also in other steady water wave problems, see [7,14,21,23,25,26,30], we may reformulate (2.3), after rescaling the horizontal variable by λ, as a quasilinear elliptic equation subject to a boundary condition which may be viewed as a compact, but at the same time nonlocal and nonlinear, perturbation of the trace operator, see (3.8).The wavelength λ appears as a free parameter in (3.8) and we show that the local bifurcation theorem of Crandall and Rabinowitz, cf.[13,Theorem 1.7], can be applied in the context of (3.8) to prove our main result and establish in this way the existence of solutions to (1.2) within the regularity class introduced in Proposition 2.1.
Concerning Theorem 1.1, we add the following remarks.
(i) If 0 = |s| < ε, then (u(s), v(s), P (s), η(s)) is also of solution to (1.2) having (not minimal) period kλ(s) for all 1 ≤ k ∈ N. In particular, for each k ≥ 1, (kλ * , u * , v * , P * , 0) is also a local bifurcation point of the trivial branch of solutions to (1.2).In Theorem 1.1 we prove that these are the only points on the trivial branch of solutions from where other symmetric solutions bifurcate.(ii) Our analysis discloses, under the assumption (1.4), that bifurcation from double (actually multiple) eigenvalues of symmetric waves is excluded along the trivial branch of laminar solutions to the hydroelastic waves problem (1.2).
We now illustrate the conditions for bifurcation from Theorem 1.1 in the particular case of irrotational waves (with γ = 0).Moreover, the wavelength λ * > 0 can be determined as the unique solution to the equation

Equivalent formulations of the hydroelastic waves problem
In this section we introduce two further equivalent formulations of the hydroelastic waves problem (1.2) which have been useful also when constructing rotational water waves in other physical scenarios, cf.e.g.[10,20,22].
As a consequence of (1.2b), the function H is a bijection.For smooth solutions to (1.2) we then compute Hence, there exists a function γ : [p 0 , 0] → R, the so-called vorticity function, with the property that ω • H −1 (q, p) = γ(p) for all (q, p) ∈ Ω, or equivalently This relation together with (1.2a) 1 -(1.2a) 2 implies that the energy Evaluating this expression at the wave surface, we deduce together with the relation (1.2a) 4 , that where Q is a constant.Integration by parts further leads to and, since also η has zero integral mean, we infer from (2.1), after integrating over one period, that Consequently, ψ solves the boundary value problem and satisfies The height function formulation.We define the height function h : Ω → R by h(q, p) = y, which associates to a point (q, p) ∈ Ω the vertical coordinate of the fluid particle located at (x, y) = H −1 (q, p) ∈ Ω η .Then, since η = h(•, 0), the function h solves the following boundary value problem together with Proposition 2.1 (Equivalence of formulations).Let β ∈ (0, 1).Then, the following formulations are equivalent: Proof.The proof is similar to that of [12, Lemma 2.1].
3. An equivalent formulation of (2.3) In our analysis we will take advantage of the height function formulation (2.3) to establish the existence of steady periodic hydroelastic waves.The main tool used to achieve this goal is the local bifurcation theorem of Crandall and Rabinowitz, cf.[13,Theorem 1.7].The appropriate parameter for bifurcation is the wavelength λ > 0. Since h is λ-periodic with respect to q it is therefore suitable to rescale h according to h(q, p) := h(λq, p), (q, p) ∈ Ω. (3.1) The function h is 1-periodic and solves (after dropping tildes) the equations In the following tr 0 is the trace operator with respect to the boundary component {p = 0} of Ω, that is, given f : Ω → R, the function tr 0 f : R → R is defined by tr 0 f (q) = f (q, 0) for q ∈ R.
Let β ∈ (0, 1) be fixed.We will assume that h ∈ X, where the Banach spaces X is defined as follows X := h ∈ C 2+β (Ω) : h is even and 1-periodic with repect to q and 1 0 tr 0 h dq = 0 .
Similarly, given k ∈ N, the space C k+β e (R) consists of the even and 1-periodic functions with uniformly β-Hölder continuous kth derivative.Moreover, C k+β e,0 (R) is the subspace of C k+β e (R) which contains only functions with zero integral mean.We note that the boundary condition (3.2a) 3 is not well-defined for h ∈ X as fourth order derivatives of h appear in this equation.However, since of H(h/λ) involves only derivatives of h with respect to the horizontal variable q, we may reformulate the boundary condition (3.2a) 3 as a nonlocal and nonlinear compact perturbation of the trace operator tr 0 .
To this end we set ζ := tr 0 h/λ and note that, if h ∈ X satisfies tr 0 h ∈ C 4+β e (R), (3.2b), and the boundary condition (3.2a) 3 , then e,0 (R) is the smooth mapping defined by In view of the fact that tr 0 h is an even function we obtain that where ω(•) is the nonlinear operator defined in (1.1c), hence Integrating the last relation once more we arrive at for all q ∈ R, where for q ∈ R, the previous equality identifies, since ζ ′′ has zero integral mean, the constant C as and therefore we have This proves that tr 0 h satisfies (3.7) and therewith the first implication in Lemma 3.1 below.
Lemma 3.1.Let h ∈ X satisfy (3.2b).Then the following are equivalent: (3.7) Proof.It remains to prove that (ii) implies (i).Let thus Lemma 3.1 (ii) be satisfied.Then, since the argument of In view of Lemma 3.1 we have formulated the problem (3.2) as the following system where R) is the smooth mapping given by (3.9)

Local bifurcation analysis
In this section we consider the equivalent formulation (3.8) of the hydroelastic waves problem (1.2) and study its solutions set.In a first step we investigate in Section 4.1 the existence of laminar flow solutions to (3.8).Then, in Section 4.2 we formulate (3.8) as a bifurcation problem, see (4.5), and determine a sufficient and necessary condition, see (4.16), for bifurcation from the set of laminar flow solutions.We conclude this section with the proof of the main result.

4.1.
Laminar flow solutions for (3.8).We next investigate the existence of laminar flow solutions to (3.8), that is, given λ > 0, we look for solutions H = H(λ) ∈ X to (3.8) that depend only on the variable p.Then, H ∈ C 2+β ([p 0 , 0]) solves the Sturm-Liouville problem in (p 0 , 0), together with the inequality that H ′ > 0 in [p 0 , 0].The next result shows that (1.3) is a sufficient and necessary condition for the existence of a (unique) laminar flow solution.
Lemma 4.1.The boundary value problem (4.1) has a solution 3) is satisfied.In this case the solution is unique and it is given by where ϑ > 2 max [p 0 ,0] Γ is the unique solution to Proof.Since (4.1) 1 is equivalent to where the constant ϑ needs to satisfy ϑ > 2 max [p 0 ,0] Γ.From the latter relation we infer that H is given by (4.2) and solves (4.1) iff ϑ is the solution to (4.3).In view of the monotonicity of the integrand in (4.3) with respect to ϑ, the existence of the (unique) solution to (4.3) is equivalent to (1.3).

Bifurcation analysis for (3.8).
In the following we assume that (1.3) is satisfied and we denote by H the laminar flow solution identified in Lemma 4.1.We next define the Banach spaces Y and Z 1 ×Z 2 consisting of 1-periodic functions with respect to the variable q by setting We further introduce the operator ) Hence, the problem (3.8) is equivalent to the nonlinear and nonlocal equation where ) has the property that F(λ, 0) = 0 for all λ > 0. (4.7)Our goal is to apply the Crandall-Rabinowitz theorem [13, Theorem 1.7] on bifurcation from simple eigenvalues in the context of (4.5) in order to determine new solutions to (4.5) which are also q-dependent.For this reason we shall determine λ * > 0 with the property that the partial Fréchet derivative ∂ h F(λ * , 0) is a Fredholm operator of index zero with a one-dimensional kernel.
Given λ > 0, the partial Fréchet derivative ∂ h F(λ, 0) := (L, T ) is given by for h ∈ Y, where Proof.In view of [17,Theorem 6.14], the operator 2 is an isomorphism.We may infer from this property that (λ is a compact operator, the desired claim for ∂ h F(λ, 0) follows at once.
The next lemma characterizes the functions that belong to the kernel of ∂ h F(λ, 0).
Lemma 4.3.Assume that (1.3) is satisfied and set where H is the unique solution to (4.1).Then, given The latter identity is obtained by multiplying the equation L[h] = 0 by cos(2kπq), followed by integration on [p 0 , 0].Since a is positive and γ = −aa ′ , cf. (4.1) 1 , we may reformulate the latter equation as Furthermore, since h ∈ Y we have h 0 (0) = 0, while, arguing similarly as above, the relation T [h] = 0 is equivalent to Finally, since each h ∈ Y vanishes on the boundary p = p 0 , it holds that Noticing that the function h 0 ∈ C 2+β ([p 0 , 0]) solves the boundary value problem it is straightforward to conclude that actually h 0 = 0.This proves the claim.
In Lemma 4.3 we have shown that a function h ∈ Y with h 0 = 0 solves ∂ h F(λ, 0)[h] = 0 iff for all k ≥ 1 the function h k defined in (4.10) is a solution to the Sturm-Liouville problem with µ := (2kπ) 2 .We next determine λ * > 0 such that (4.11) has a nontrivial solution for µ = (2π) 2 and only the trivial solution f = 0 when µ > (2π) 2 .As a first step we show that the solutions to (4.11) build a vector space of dimension less or equal to 1 for each choice of the parameters λ > 0 and µ ∈ R. To this end we define the Sturm-Liouville type operator R λ,µ : C 2+β 0 where and Lemma 4.4.Given λ > 0 and µ ∈ R, the operator R λ,µ is a Fredholm operator of index zero with dim ker R λ,µ ≤ 1.Additionally, dim ker R λ,µ = 1 iff the functions f 1 and f 2 are linearly dependent.In this case we have ker R λ,µ = span{f 1 }.
, is a constant function, the initial conditions in (4.13)-(4.14)ensure that this function is in fact identically zero, hence f 1 and f 2 are linearly dependent.Viceversa, if f 1 and f 2 are linearly dependent, then they both belong to ker R λ,µ , and this completes the proof.
In view of Lemma 4.4 it remains to look for a value λ * > 0 with the property that the Wronskian 0] and a = 0, the Wronskian vanishes in [p 0 , 0] iff it vanishes at the point p = 0. Therefore we consider the function W : (0, ∞) × R → R given by Observing that (4.13) 1 depends smoothly on µ and λ, this property is inherited also by the solution to (4.13), cf.e.g.[5], and therefore we have For the special value µ = 0 we have ).In view of (4.13) we compute, in the particular case µ = 0, that which leads to Hence, if then W (λ, 0) < 0 for all λ > 0. (4.17) We next investigate the behavior of W (λ, µ) when µ → ∞ .
Given (λ, µ) × R ∈ (0, ∞), an application of the chain rule yields that We next multiply (4.19) 1 by f 1 and subtract from this relation the identity (4.13) multiplied with f 1,λ , to obtain, after integration on [p 0 , 0], that If W (λ, µ) = 0, the latter relation together with the identity f 1 = Θf 2 , Θ > 0, leads us to We may now multiply (4.13) 1 by f 1 and integrate over [p 0 , 0] to obtain that hence, on the one hand and, on the other hand Hölder's inequality now yields and together with (4.22) and (4.16) we have which proves (i).
In order to prove (ii), we proceed similarly as above and compute, for given λ, µ ∈ (0, ∞), We next multiply (4.24) 1 by f 1 and subtract from this relation the identity (4.14) multiplied with f 1,µ , to obtain, after integration on [p 0 , 0], that Hence, if W (λ, µ) = 0, the latter identity combined with the relation and (ii) follows in view of (4.21) and (i).
Since W is smooth, (4.17) and Lemma 4.5 imply that µ(λ) is well-defined for all λ > 0 and moreover µ(λ) > 0. In fact, Lemma 4.6 implies that, for each λ > 0, µ(λ) is the unique positive zero of the mapping W (λ, •).Moreover, the implicit function theorem together with Lemma 4.6 ensures that [µ → λ(µ)] : (0, ∞) → (0, ∞) is smooth.We now use the chain rule together with (4.18) to compute that from where we infer that there exists a positive constant C 0 such that It is remarkable that exactly the same expression for µ (with a possibly different constant C 0 ) has been obtained in [14] in the analysis of the bifurcation problem for stratified capillarygravity waves.We arrive at the following result.
Proof.Since µ(λ * ) = (2π) In order to apply [13,Theorem 1.7] in the context of the bifurcation problem (4.5), it remains to prove that the transversality condition with λ * and h * introduced in (4.27) and (4.28), is satisfied.Therefore, we first characterize in Lemma 4.8 below the range of ∂ h F(λ * , 0).
Proof.As a starting point we observe that for ∂ h F(λ * , 0) = (L, T ) we have We now assume that (F, ϕ) ∈ im ∂ h F(λ * , 0) is the image of a function h ∈ Y. Multiplying the identity L[h] = F by a 3 h * and integrating the resulting relation by parts, we find that * a 3 h p h * ,p + ah q h * ,q d(q, p).* a 3 h p h * ,p + ah q h * ,q d(q, p) = 0.
Noticing that (4.30) defines a closed subspace of Z 1 × Z 2 of codimension 1 which contains the range of ∂ h F(λ * , 0), the desired claim follows now from Lemma 4.2.
We are now in a position to verify the transversality condition (4.29).This proves the claim.
We are now in a position to establish Theorem 1.1.
Proof of Theorem 1.1.In view of Lemma 3.1 and of Proposition 2.1, in the framework of waves which are symmetric with respect to the vertical line x = 0, the Euler formulation (1.2) of the steady hydroelastic waves problem is equivalent to the height function formulation (3.8), hence also to the bifurcation problem (4.5).Therefore, the assertion (i) is a straightforward consequence of Lemma 4.1.
R), and we denote by O the open subset of Y defined by
.5) Equation (1.5) is the dispersion relation for irrotational hydroelastic waves.Outline: In Section 2 we present two further equivalent formulations of (1.2): the stream function formulation (2.2) and the height function formulation (2.3).Then, in Section 3, we reformulate (2.3) by reexpressing the boundary condition in (2.3) obtained from the dynamic boundary condition as a compact, but nonlinear and nonlocal, perturbation of a Dirichlet boundary condition, see (3.8).Finally, in Section 4, we recast (3.8) as a bifurcation problem and prove Theorem 1.1.