Steady periodic water waves bifurcating for fixed-depth rotational flows with discontinuous vorticity

arXiv:2409.08730v1 [math.AP] 13 Sep 2024 Steady periodic water waves bifurcating for fixed-depth rotational flows with discontinuous vorticity David Henry and Silvia Sastre-Gomez School of Mathematical Sciences, University College Cork, Cork, Ireland. Abstract In this article we apply local bifurcation theory to prove the existence of small-amplitude steady periodic water waves, which propagate over a flat bed with a specified fixed mean-depth, and where the underlying flow has a discontinuous vorticity distribution. 1 Introduction In this paper we prove the existence of small-amplitude steady periodic water waves which propagate over a flat bed with a specified fixed mean-depth, and where the underlying flow has a discontinuous vorticity distribution. This existence result is achieved by way of an implementation of the Crandall-Rabinowitz local bifurcation theory applied to a generalised version of a modified-height function formulation of the governing equations. This novel reformulation, introduced in [20, 21], enables us to maintain the mean fluid depth as a fixed parameter in the bifurcation analysis, as opposed to the (by now) standard approach initiated in [8] whereby, following the Dubreil-Jacotin [12] transformation, the mean depth does not feature in the bifurcation setting. Instead, in [8, 9] the mass-flux is maintained as a fixed-parameter in the standard height function formulation. As in many complex bifurcation problems the role of the involved parameters is by no means canonical, and this is particularly true for the water wave problem. Fixing the mean depth of the wave is heuristically and physically quite natural, and furthermore since numerical studies [27] suggest that for fixed mass-flux the mean depth is not a constant of motion, the present contribution represents in a sense a completion of the analysis of the mathematical problem. The seminal work [8] was the first mathematically rigorous work which established the existence of water waves with general vorticity distributions, prior to this much of the rigorous mathematical analysis of water waves focussed on irrotational flows— cf. [37] for a nice survey of the rigorous mathematical analysis of irrotational waves, while surveys of recent progress in the mathematical analysis of rotational flows can be found in [2, 4]. Following [8], mathematically rigorous existence theory for rotational flows underlying steady periodic 1 water waves has thrived, with a proliferation of results modelling scenarios with increasing levels of physical complexity, such as incorporating fluid stratification, the capillary effects of surface tension, the presence of weak stagnation and critical points, and even accommodating the possibility of overhanging wave-profiles, cf. [9, 10, 13, 22, 24, 31, 34, 33, 38, 39, 40, 41, 42]. The practical relevance of flows with vorticity has long been recognised, particularly as they serve as models for wave-current interactions, among other intricate and physically important phenomena [2, 26, 36]. The extension of the existence result in [21] to allow for discontinuous vorticity is both mathematically interesting and challenging. Physically, it is known that wave-current interaction is most dramatically observed where there is a rapid change in the current strength [26], a scenario which may be crudely modelled by a discontinuous vorticity distribution. Examples of such processes include wind-generated waves, and near-bed current layers which account for sediment transport. This paper is laid out as follows. In Section 2 we recast the governing equations in terms of a generalised version of the modified height function formulation, in the process establishing the equivalence of this new formulation with the weak versions of the standard Euler equation and stream function formulations, and additionally we present strong a priori regularity results for the streamlines (including the unknown free-boundary) of the resulting flow. The main result of our paper, which establishes the existence of small-amplitude weak solutions, is presented in Theorem 2.3, and Section 3 is then dedicated to proving this result. Finally, in Section 4 we present explicit, physically-motivated examples of various flows for which our bifurcation result applies. 2 Preliminaries 2.1 Formulations of the governing equations We consider two-dimensional steady periodic travelling surface waves propagating over water of mean depth d > 0where the predominant external restoration force is gravity. The (x, y)−coordinate system is fixed by locating the mean–water level at y= 0, which implies that the mean of the waves’ free surface over a wavelength must equal zero: if η(x, t)denotes the unknown free surface at some given fixed-time t, then Zη(x, t)dx = 0,(2.1) where the integral is taken over a wavelength. The impermeable flat bed is given by y= −d. Aside from the physically–enforced restriction (2.1), the free surface ηis a priori undetermined and is therefore an unknown in the mathematical formulation of the water wave problem. We seek steady travelling waves propagating with a constant wave speed c > 0in the positive x−direction. Accordingly, all functions must have a x, t relationship of the form x−ct, and the change of coordinates (x−ct, y)7→ (x, y)transforms to a steady reference frame moving alongside the wave where the flow is time independent. We denote the closure of the fluid domain by Dη={(x, y)∈R2:−d≤y≤η(x)}. The governing 2 equations for the motion of the perfect (inviscid and incompressible) fluid take the form of the Euler equations together with boundary conditions given by ux+vy= 0 in Dη,(2.2a) (u−c)ux+vuy=−Pxin Dη,(2.2b) (u−c)vx+vvy=−Py−gin Dη,(2.2c) v= (u−c)ηxon y=η(x),(2.2d) P=Patm on y=η(x),(2.2e) v= 0 on y=−d, (2.2f) where (u, v)is the velocity field, P(x, y)is the pressure distribution function, Patm is the constant atmospheric pressure and gis the gravitational constant of acceleration. We now make the additional assumption that there are no weak-stagnation points, that is, u < c (2.2g) throughout the fluid. This is a physically reasonable assumption for water waves, without underlying currents containing strong non-uniformities, and which are not near breaking [25, 28]. These flows do not contain any stagnation points, and the individual fluid particles move with a horizontal velocity which is less than the speed with which the surface wave propagates. The non-stagnation condition is also essential mathematically in our reformulations of the water wave equations, as we see below. For two-dimensional motion the vorticity is given by ω=uy−vx.(2.3) We work with periodic waves and note that we may effectively choose the period to be equal to 2πwithout loss of generality in subsequent considerations. This is due to the fact that if we are dealing with water waves of wavelength Lin the governing equations (2.2a)–(2.2f) then performing the scaling of variables (x, y, t, g, ω, η, u, v, P, c)7→ (κx, κy, κt, κ−1g, κ−1ω, κη, u, v, P, c), where κ=2π Lis the wavenumber, we end up with a 2π−periodic system in the new variables identical to (2.2) except g, ω are replaced by κ−1g, κ−1ω. We can write the Euler equations (2.2b)-(2.2c) coupled with the mass conservation equation (2.2a) and the boundary conditions (2.2d)-(2.2f) in the (weak) divergence form as −cux+u2x+ (uv)y=−Pxin Dη,(2.4a) −cvx+ (uv)x+v2y=−Py−gin Dη,(2.4b) ux+vy= 0 in Dη,(2.4c) v= 0 on y=−d, (2.4d) v= (u−c)ηxon y=η(x),(2.4e) P=Patm on y=η(x).(2.4f) 3 With this weak formulation we may consider solutions with weaker regularity than those of (2.2). The equations (2.4a)–(2.4c) will be understood in the sense of distributions, whereas the boundary conditions (2.4d)–(2.4f) will be understood in the classical sense. The type of solutions of (2.4) we are interested in are Hölder continuously differentiable functions u, v, P ∈W1,r per(Dη)⊂C0,α per (Dη), with η∈C1,α per (R), for some Hölder exponent α∈(0,1) and r= 2/(1 −α), where ris chosen to ensure the embedding W1,r per(Dη)⊂C0,α per (Dη). The per subscript indicates that our solutions are 2π−periodic, and furthermore we assume that solutions have a single crest located at x= 0 and troughs located at x=±π, and the condition (2.2g) on uand cwill hold throughout the fluid domain. The governing equations (2.2) have a second and, as we mention below, equivalent formulation when recast in terms of the stream-function. The stream function ψis defined up to a constant by the relations ψy=u−c, ψx=−v. (2.5) The constant may be fixed by setting ψ= 0 on y=η(x), and we observe by expressing ψ(x, y) = −p0+Zy −d (u(x, s)−c)ds, that ψis also periodic with period 2π. We note that one consequence of the definition (2.5) is that the level sets of the stream function ψ(x, y)are streamlines of the fluid motion, hence the name. The condition (2.2g) ensuring the absence of stagnation points is recast in terms of the streamfunction ψas ψy<0.(2.6) The governing equations (2.2) can then be reformulated (cf. [2, 8, 9, 20, 21] for details) in terms of the stream function as follows: ∆ψ=γ(ψ)in −d < y < η(x),(2.7a) |∇ψ|2+ 2g(y+d) = Qon y=η(x),(2.7b) ψ= 0 on y=η(x),(2.7c) ψ=−p0on y=−d, (2.7d) where γ(ψ) = ωis the vorticity function, Qis a physical constant known as the total hydraulic head, and p0is a constant of motion known as the relative mass-flux, defined by p0=Zη(x) −d (u(x, y)−c)dy < 0. The appropriate weak formulation of the stream function governing equations (2.7) is then: (ψxψy)x−1 2(ψ2 x−ψ2 y)y−˜ Γ(ψ/p0)y= 0 in Dη,(2.8a) ψ=−p0on y=−d, (2.8b) ψ= 0 on y=η(x),(2.8c) |∇ψ|2+ 2g(y+d) = Qon y=η(x),(2.8d) 4 where ˜ Γ(p) = Rp 0p0γ(s)ds. In this paper we are interested in solutions of (2.8) with ψ∈ W2,r per(Dη)⊂C1,α per (Dη),˜ Γ∈C0,α([−1,0]), and η∈C1,α per (R), for some α∈(0,1) and r= 2/(1−α), where the boundary conditions (2.8b)–(2.8d) are satisfied in the classical sense, and the equation (2.8a) is satisfied in the sense of distributions, and ψx, ψyare also understood in the classical sense. The third governing equation formulation we introduce is the form most suitable for local bifurcation theory to be implemented in order to prove the existence of water waves on a fluid domain of fixed mean-depth with a discontinuous vorticity distribution. It was first introduced in [20, 21] to prove the existence of classical solutions representing small and large amplitude water waves propagating on a fluid of fixed mean-depth, where it was shown (cf. [20, 21]) that the governing equations (2.7) transform to 1 d2+h2 qhpp −2hq(hp+ 1)hpq + (hp+ 1)2hqq +γ(p) p0 (hp+ 1)3= 0 in −1< p < 0,(2.9a) 1 d2+h2 q+(hp+ 1)2 p2 0 [2gd(h+ 1) −Q] = 0 on p= 0,(2.9b) h= 0 on p=−1.(2.9c) Here the variables (q, p)are defined by the semi-hodograph transformation of variables (x, y)7→ (q, p) := (x, ψ(x, y)/p0),(2.10) which has the effect of transforming the fluid domain with unknown free boundary ηinto the fixed semi-infinite rectangular strip R×[−1,0]. It is now apparent that the non-stagnation condition (2.2g) is vital in order to ensure that the change of variables (2.10) represents an isomorphism. The function h, defined by h(q, p) = y d−p, (2.11) is a modified-height function where it is understood that y=y(q, p)is a function of the new (q, p)−variables. In this paper we seek solutions hof (2.9) which are even and 2π−periodic in q, and the analogous condition to (2.1) which the modified-height function should satisfy is given by Zπ −π h(q, 0)dq = 0.(2.12) The condition (2.2g) excluding stagnation points becomes hp+ 1 >0,(2.13) and consequently the system (2.9) is a uniformly elliptic quasilinear partial differential equation with oblique nonlinear boundary conditions. The appropriate weak form of the 5 modified-height function formulation (2.9) is given in divergence form by (−1 + d2h2 q 2d2(1 + hp)2+Γ(p) 2d2)p +hq 1 + hpq = 0 in −1< p < 0,(2.14a) −1 + d2h2 q 2d2(1 + hp)2−gd(h+ 1) p2 0 +Q 2p2 0 = 0 on p= 0,(2.14b) h= 0 on p=−1,(2.14c) where Γ(p) = 2 Zp 0 d2γ(s) p0 ds, −1≤p≤0, and in the following Γmin := min p∈[−1,0] Γ(p). We understand by a solution of (2.14) a function h∈W2,r per(R)⊂C1,α per (R), where r > 2/(1 −α)for α∈(0,1) and R= [−π, π]×[−1,0]. Here the per subscript indicates that our solutions are even and 2π−periodic in the q−variable, and the inclusion relation (along with the choice of r) are derived from Morrey’s inequality. 2.2 Equivalence of formulations In [21] it was proven that the Euler equation (2.2), the stream function (2.7) and the modified-height function (2.9) formulations of the governing equations are equivalent in the sense of classical solutions. This approach can be modified (in the same vein as in [9] for the standard height function formulation) to prove the equivalence of the weak versions (2.4), (2.8) and (2.14) of these formulations that we are interested in. Furthermore, we note that results concerning the equivalence of the standard height formulation of the governing equations were recently strengthened in [43], and this approach was suitably adapted in [35] for the modified height function form of our equations giving the following result: Theorem 2.1. [35] Let 1/3< α < 1. Then the following formulations are equivalent: (i) the weak velocity formulation (2.4) with (2.2g), for η∈C1,α per (R), and u, v, P ∈C0,α per (Dη); (ii) the weak stream function formulation (2.8) with (2.6), for ˜ Γ∈C0,α([−1,0]),η∈ C1,α per (R)and ψ∈C1,α per (Dη); (iii) the weak height formulation (2.14) together with (2.13), for Γ∈C0,α([−1,0]) and h∈C1,α per R. 2.2.1 Regularity properties of solutions Recently, instigated by the paper [7], a wide body of research has flourished in establishing strong a priori regularity properties for periodic water waves with vorticity in a variety of physical settings, such as flows with stratification [23], free-boundary problems incorporating the capillarity effects of surface tension [17], and flows with very general vorticity 6 distributions [14, 33]. We note in particular that the approach and results of [14], which apply to vorticity functions which are merely integrable, may be adapted for our modified height function formulation (2.14), giving the following result: Theorem 2.2 (Regularity of streamlines).Given a solution h∈C1,α per Rof (2.14) such that (2.13) holds, then: •∂m qh∈C1,α per Rfor all m∈N. •Furthermore, there exists a constant L > 1such that, for all integers M≥3, we have k∂m qhk1,α ≤Lm−2(m−3)!. •Accordingly, all the streamlines for solutions of the Euler equation formulation (2.4), including the free-surface wave profile η(x), are a priori real analytic curves. 2.3 Main Result The principle aim of this paper is to use local bifurcation theory to prove the following: Theorem 2.3. Let 0< α < 1and Γ∈C0,α ([−1,0]), and suppose we are given the wave speed c > 0and a fixed mean-water depth d > 0. Suppose for λ∗>−Γmin there exists a weak solution 06≡ M(p)∈C1,α([−1,0]) to the Sturm-Liouville problem        (a3Mp)p=d2aM in −1< p < 0 a3Mp=gd3 p2 0 Mon p= 0 M= 0 on p=−1, (2.15) where a(λ, p) = pΓ(p) + λfor p∈[−1,0]. Then for sufficiently small ε > 0there exists aC1−curve Cloc ={(λs, hs)∈R×C1,α per (R) : |s|< ε}of solutions to the water wave problem (2.14), subject to hp+1 >0. The solution curve Cloc contains precisely one function independent of q, namely (λs(0), hs(0)) = (λ∗, H(p, λ∗)) where the laminar flow solution H(p, λ∗)is defined by (3.1) below. For |s|< ε we have hs(q, p) = H(p, λs) + sM(p) cos(q) + o(s)in C1,α(R),(2.16) where Mis the solution to (2.15) with M(0) = 1, and Qis specified in terms of λby (3.3). In Section 3 below we demonstrate how the existence of non-trivial water wave solutions of the form (2.16) is determined by system (2.15). In the process, it will be clear that the existence of solutions to (2.15) depends on the nature of the vorticity distribution, something we expand further upon in Section 4. In Theorem 4.2 it is shown that solutions of (2.15) exist for a wide range of general vorticity distributions, yet at the same time there also exist vorticity distributions for which solutions of (2.15) do not exist. In particular, we present 7 necessary and sufficient conditions for solutions of (2.15) to exist for constant, and some piecewise constant, vorticity distributions. We further note that the “physical” solutions u, v, η of the Euler equations formulation (2.4) which correspond to (2.16) are periodic in the x−variable, with a single crest and trough per period, with u, η being symmetric (and vanti-symmetric) functions about the crestline. The inherent symmetry of these waves is not particularly prohibitive since its has been rigorously proven under quite general conditions that water waves with vorticity are intrinsically symmetric, cf. [2, 5, 6, 32]. 3 Local bifurcation 3.1 The Crandall-Rabinowitz local bifurcation theorem The primary tool we employ to prove existence of small amplitude water waves for the system (2.14) is the Crandall-Rabinowitz [11] local bifurcation theorem, which may be stated as follows (cf. [1] for a detailed proof and discussion regarding the theorem and its hypotheses): Theorem 3.1 (Crandall-Rabinowitz).Let X, Y be Banach spaces and let F ∈ Ck(R×X, Y ) with k≥2satisfy: 1. F(λ, 0) = 0 for all λ∈R; 2. The Fréchet derivative Fx(λ∗,0) is a Fredholm operator of index zero with a onedimensional kernel: ker(Fx(λ∗,0)) = {sx0:s∈R,06=x0∈X}; 3. The tranversality condition holds: Fλx(λ∗,0)(1, x0)6∈ R(Fx(λ∗,0)). Then λ∗is a bifurcation point in the sense that there exists ǫ0>0and a branch of solutions {(λ, x) = {(Λ(s), sχ(s)) : s∈R,|s|< ǫ0} ⊂ X×R,}, with F(λ, x) = 0,Λ(0) = λ∗, χ(0) = x0, and the maps s7→ Λ(s)∈R, s 7→ sχ(s)∈X, are of class Ck−1on (−ǫ0, ǫ0). Furthermore there exists an open set U0∈R×Xwith (λ∗,0) ∈U0and {(λ, x)∈U0:F(λ, x) = 0, x 6= 0}={(Λ(s), sχ(s)) : 0 <|s|< ǫ0}. 8 3.2 Operator reformulation of system (2.14) The first step in our implementation of Crandall-Rabinowitz’ theory is to determine solutions H(p)of system (2.14) which have no q−dependence, and which therefore describe parallel shear flows with a flat horizontal surface and streamlines. In the process we will elicit a suitable bifurcation parameter, we will establish a reformulation of system (2.14) as an operator between appropriate Banach spaces, and we will achieve a characterisation of the “trivial” solutions which fulfil the first condition of Theorem 3.1. We solve (2.14) along similar lines as in [21] to get, for −Γmin < λ < Q, a q−independent solution of the form H(p;λ) = Zp 0 ds pλ+ Γ(s)+1 2gd Q−p2 0 d2λ−(p+ 1),−1< p ≤0, =Zp −1 1 pλ+ Γ(s)ds −(p+ 1) ∈C1,α per ([−1,0]) .(3.1) The parameter λis related to the fluid velocity at the flat surface by the relation √λ=1 Hp+ 1p=0 =d(u−c) p0on the flat surface ,(3.2) and it is implicitly related to Qby the following formula Q= 2gd Z0 −1 ds pλ+ Γ(s)+p2 0 d2λ > 0.(3.3) We see that Q(λ)is a positive convex function of λwith a minimum occurring at the unique value λ0>0satisfying p0 gd3=Z0 −1 1 λ0+ Γ(s)3 2 ds, (3.4) and Q(λ)is monotonically decreasing for −Γmin < λ < λ0and monotonically increasing for λ > λ0. We infer that the appropriate bifurcation parameter for system (2.14) is λ > −Γmin. To re-express system (2.14) as an operator F(λ, x) : R×X→Y, where X, Y are Banach spaces, we work as follows. Let us consider first the operators G1h=(−1 + d2h2 q 2d2(1 + hp)2+Γ(p) 2d2)p +hq 1 + hpq , G2h= −1 + d2h2 q 2d2(1 + hp)2−gd(h+ 1) p2 0 +Q 2p2 0!T where T={(q, 0) : q∈[−π, π]}. With α∈(0,1) fixed we define the Banach spaces X=h∈C1,α per (R) : ZT h(q, 0)dq = 0, h = 0 on −1, h is even in q, 9 where the constant Cdepends only on kHp+ 1kC0,α per (R). We will conclude our proof of the existence of a solution to the system (3.15) by proving that there exists a subsequence {ǫn} → 0whose associated solutions {v(εn)}n≥1of (3.16) converge to some function v. First we show that for any εn→0, the sequence kv(εn)kC1,α per (R)n≥1is bounded. Arguing by contradiction, if kv(εn)kC1,α per (R)→ ∞ as εn→0then (3.18) yields kv(εn)kL∞(R)→ ∞, while the functions un=v(εn) kv(εn)kL∞(R)are bounded in C1,α per (R). Accordingly, due to the compact embedding C1,α per (R)⊂⊂ C1 per(R)we can extract a subsequence {unk}that converges in C1 per(R)to some uwith kukL∞(R)= 1 as nk→ ∞. Upon dividing (3.16) by kv(εnk)kL∞(R) and taking limits we see that umust solve (weakly)            0 = up d2(1 + Hp)3p +uq (1 + Hp)q in R, 0 = up d2(1 + Hp)3−gd p2 0 uon p= 0, u= 0 on p=−1, (3.19) and due to the Schauder estimate (3.18) we have an immediate gain of regularity, namely u∈C1,α(R). Comparing (3.19) with (3.9) we infer that u∈ker Fw(λ∗,0), and Lemma 3.5 implies that u=δϕ∗for some δ∈R. Choosing ϕ=ϕ∗in (3.17) and bearing in mind (3.10) we obtain (1+εn)ZZRϕ∗ q 1 + Hp vεn q+ϕ∗ p d2(1 + Hp)3v(εn) pdq dp+εnZZR v(εn)ϕ∗dqdp =gd p2 0ZT v(εn)ϕ∗dq, (3.20) which following integration by parts and using the fact that ϕ∗solves (3.19) gives εnZZR v(εn)ϕ∗dqdp +εn gd p2 0ZT v(εn)ϕ∗dq = 0.(3.21) Dividing by εnkv(εn)kL∞(R), and taking limits gives δZZRϕ∗2dqdp +δgd p2 0ZTϕ∗2dq = 0, since un=v(εn)/kv(εn)kL∞(R)→δϕ∗, which implies that δ= 0 and hence u=δϕ∗= 0. This contradiction of the identity kukL∞(R)= 1 leads us to conclude that the sequence kv(εn)kC1,α per (R)n≥1is indeed bounded as ǫn→0, which together with the compact embedding C1,α per (R)⊂⊂ C1 per(R)enables us to infer the existence of a subsequence of {v(εn)}n≥1 converging in C1 per(R)to some v∈C1 per(R)which is a solution to (3.15). By standard elliptic regularity [16] we have that v∈C1,α per (R). We have therefore proven that, when (3.10) holds, there exists unique solutions Φ0and v of (3.12) and (3.15) respectively, and Φ=Φ0+vthen is a unique solution of (3.11). This proves the sufficiency of relation (3.10). The considerations of this section, coupled with those of subsection 3.4, prove that when µ(λ∗) = −1the second hypothesis of Theorem 3.1 is satisfied. 16 3.6 Transversality condition To complete the proof that local bifurcation occurs for the simple eigenvalue λ∗, thereby proving our main result Theorem 2.3, we must prove that the transversality condition, hypothesis three of Theorem 3.1, holds whenever µ(λ∗) = −1. We recall that the transversality condition states that Fw,λ(λ∗,0)ϕ∗6∈ R(Fw(λ∗,0)) , where Fw,λ(λ∗,0) is given by Fw,λ(λ∗,0)w=3 2d2(awp)p+1 2(a−1wq)q,3 2d2awpT, for a=pΓ(p) + λ. As a result of Proposition 3.6 we only need to check that 1 2ZZRa−1ϕ∗ qq+3 d2aϕ∗ ppϕ∗dq dp −3 2d2ZT aϕ∗ pϕ∗dq 6= 0.(3.22) Integrating (3.22) by parts, and using the fact that ϕ∗= 0 on p=−1, we have −1 2ZZR a−1ϕ∗ q2dq dp −3 d2ZZR aϕ∗ p2dq dp < 0. Thus the transversality condition is satisfied, and this completes the proof of our main result. 4 Applications of Theorem 2.15 In this section we examine the applications of our local existence result Theorem 2.3 with regard to vorticity distributions, both general and specific. In particular, for arbitrary vorticity distributions we present conditions which are sufficient for Theorem 2.3 to apply, whereas for some examples of constant, and piecewise-constant, vorticity distributions we obtain physically explicit conditions which possess the additional value of being necessary for local bifurcation to occur. 4.1 General vorticity distributions We recap that it was shown in [21] for the setting of classical solutions of the water wave problem (where the vorticity distributions are at least continuous) that a sufficient condition for bifurcation to occur is given by the following: Proposition 4.1. [21] Suppose that γis an (otherwise arbitrary) vorticity distribution which satisfies √2 3γ 3 2 ∞|p0|1 2|p1|1 2+2√2 5γ 1 2 ∞|p0|3 2|p1|3 2< g, (4.1) where γ∞=kγkC[−1,0] and p1= max{p∈[−1,0] : Γ(p) = Γmin}. Then local bifurcation occurs and we obtain non-trivial solutions to the classical water wave problem. 17 The proof of the above theorem can be modified to give the following result for discontinuous vorticity distributions, which we present a brief sketch of here for the sake of completeness. Theorem 4.2. For 0< α < 1and Γ∈C0,α ([−1,0]), let θ= sup p6=p1 |Γ(p)−Γ(p1)| |p−p1|α. If g > θ3/2 6αd3|p0|2|p1|3α/2−1+θ1/2 (2 + α/2)d|p0|2|p1|α/2+1,(4.2) then Theorem 2.3 holds and hence local bifurcation occurs. Remark 4.3. We note that γ∈L∞[−1,0] implies Γ∈C0,1[−1,0] is Lipschitz continuous, and if γ(p)≥0then p1= 0 by definition. Hence, a consequence of Theorem 4.2 is that for any positive, bounded, but otherwise arbitrary vorticity distribution, local bifurcation always occurs. We note that this remark applies trivially to irrotational flow, for which γ≡0. Proof. We first note that since 0≤Γ(p)−Γmin = Γ(p)−Γ(p1)≤ |p−p1|α·sup p6=p1 Γ(p)−Γ(p1) |p−p1|α=θ|p−p1|α, we have that a(−Γmin, p) = pΓ(p)−Γmin ≤pθ|p−p1|α, p ∈[−1,0]. From (4.2), we can take k > 1/2so that gd3 p2 0 > θ3/21 2k−1 + 3α/2|p1|3α/2−1+d2θ1/21 2k+α/2 + 1|p1|α/2+1.(4.3) We define pn=1−1 np1−1 n∈[−1, p1], n ≥2(4.4) and ϕn(p) = 0,−1≤p≤pn (p−pn)k, pn≤p≤0.(4.5) for all n≥2. It follows that −1≤p≤pn≤p1≤0, and since k > 1/2we have that ϕn∈H1([−1,0]). From the definition of pnwe have that ϕn(−1) = 0 and ϕn6≡ 0. By way of direct computations we obtain Z0 −1 a3(−Γmin, p)∂pϕn(p)2dp+Z0 −1 d2a(−Γmin, p)ϕ2 n(p)dp ≤θ3/2k2"|pn|2k−1+3α/2 2k−1 + 3α/2+3α/2|p1−pn|2k−1+3α/2 (2k−1)(2k−1 + 3α/2)# +d2θ1/2"|pn|2k+1+α/2 2k+ 1 + α/2+α/2|p1−pn|2k+1+α/2 (2k+ 1)(2k+ 1 + α/2)# =|pn|2kAn+Bn=ϕ2 n(0)An+Bn, 18 where An=θ3/2k2|pn|3α/2−1 2k−1 + 3α/2+d2θ1/2|pn|α/2+1 2k+ 1 + α/2, Bn=θ3/2k23α/2|p1−pn|3α/2−1 (2k−1)(2k−1 + 3α/2) +d2θ1/2α/2|p1−pn|α/2+1 (2k+ 1)(2k+ 1 + α/2). From the definition (4.4) of pnwe have that |p1−pn| ≤ |pn|, and lim n→∞ |p1−pn|= 0. From (4.3) there exists N≥2and ε > 0such that An< gd3/p2 0−εfor all n≥N. With this specified value of ε, we choose n≥Nlarge enough such that Bn< ε. With the choice of this n, the function ϕnsatisfies Z0 −1 a3(−Γmin, p)∂pϕn(p)2dp +Z0 −1 d2a(−Γmin, p)ϕ2 n(p)dp < gd3 p2 0 ϕ2 n(0). Therefore, there exists λ > −Γmin satisfying µ(λ)<−1, hence Corollary 3.3 ensures the existence of a solution of (2.15), which in turn through Theorem 2.3 implies that local bifurcation occurs. 4.2 Constant vorticity We observe that a very convenient implication of Remark 4.3 is that local bifurcation occurs for any positive constant vorticity γ≥0. In [21] it was demonstrated that, for constant, continuous negative vorticity γ < 0, bifurcation occurs if and only if γ2d2<(g+γ2d) tanh(d).(4.6) 4.3 Piecewise-constant vorticity We present two examples here of different physical scenarios which are modelled by piecewise continuous vorticity. The first case, comprises a uniform layer of constant non-zero vorticity located at the surface of the flow, and beneath this layer the flow is purely irrotational. This model has an interesting physical relevance, since is well known that wind-generated water waves exhibit a localised layer of vorticity near the surface. In the initial stages of wind generated-waves on the ocean, the wind “grips” the surface, producing small-amplitude water waves which subsequently develop in size. The second case is, in a sense, the opposite set-up, consisting of a layer of constant non-zero vorticity at the bottom of the fluid, and above this layer there is no vorticity. As a physical motivation for this type of flow, it is well-known that wave-current interaction (and hence vorticity) is most dramatically observed where this is a rapid change in the current strength, and the near-bed region is a location which commonly experiences currents, accounting for sediment transport, for instance. We note that, as a direct consequence of Remark 4.3, local bifurcation occurs in both examples if γ≥0. In [18, 19] the Sturm-Liouville problem (2.15) was analysed for both models with the principle purpose being to obtain dispersion relations for the flow by 19 way of a detailed analysis of relation (3.2). The dispersion relations for small-amplitude waves detail how the relative speed of the wave at the free-surface varies with respect to certain physical flow parameters, such as the fixed mean-depth of the fluid, the wavelength, the vorticity distribution, and— for the discontinuous vorticity distributions — the location dof the isolated layer of vorticity. Analogous results concerning dispersion relations, obtained instead through analysing the standard height function formulation, were derived in [3, 9, 29, 30]. Aside from obtaining dispersion relations, a serendipitous boon of the approach employed in [18, 19] is the derivation, when the region of vorticity is negative, of necessary and sufficient conditions for the existence of solutions to (2.15). As a result of Theorem 2.3, these double–up as necessary and sufficient conditions for local bifurcation to occur, and we outline them now below. 4.3.1 Case (a): Localised vorticity layer at the surface The vorticity distribution takes the form ω=γ, −d< y < 0, 0,−d < y < −d, where d∈[0, d]is some given fixed depth. It is understood by d= 0 in this context that the flow is entirely irrotational, and d=dsignifies that the flow has a constant, continuous vorticity distribution γthroughout the entire domain. It was shown in [18] that, for γ < 0, a necessary and sufficient condition for both solutions of (2.15) exist and, consequent to Theorem 2.3, that local bifurcation occurs, is given by γ2d(d−tanh(d)) < g tanh(d).(4.7) We note that, upon taking the limit d→d, this condition for bifurcation particularises to the equivalent condition (4.6) for flows of constant, continuous negative vorticity. 4.3.2 Case (b): Localised vorticity layer at the bed For this case, the vorticity distribution ω=0,−d< y < 0, γ, −d < y < −d. In this setting, d= 0 represents purely rotational flow while for d=dthe flow is irrotational. For γ < 0, it was shown in [19] that a necessary and sufficient condition that there be a solution of (2.15), and so local bifurcation occurs, takes the form |γ|<pg(sinh (d) (d−d)−sinh (d) sinh (d−d)) (d−d)p(d−d) cosh (d)−γ2cosh (d) sinh (d−d). 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