Monatsh MathDOI 10.1007/s00605-013-0583-1

Global bifurcation for water waves with capillaryeffects and constant vorticity

Bogdan-Vasile Matioc

Received: 10 April 2013 / Accepted: 4 October 2013© Springer-Verlag Wien 2013

Abstract We study periodic capillary and capillary-gravity waves traveling over awater layer of constant vorticity and finite depth. Inverting the curvature operator, weformulate the mathematical model as an operator equation for a compact perturbationof the identity. By means of global bifurcation theory, we then construct global continuaconsisting of solutions of the water wave problem which may feature stagnation points.A characterization of these continua is also included.

Keywords Global bifurcation · Constant vorticity · Stagnation points

Mathematics Subject Classification (1991) 35C07 · 35R35 · 47J15 · 76B03 ·76B45

1 Introduction

Surface tension plays an important role in the dynamics of water waves in manyphysical situations. For example, when wind starts blowing over a still fluid surface onecan observe two-dimensional small amplitude wave trains appearing, the capillaritybeing the predominant restoring force in this regime [25]. When the waves grow larger,the gravity starts to influence the motion of these capillary-gravity waves. We shallfocus herein on two-dimensional capillary and capillary-gravity water waves travelingover an inviscid fluid layer of finite depth and bounded from below by an horizontalbed. While there is no known explicit solution describing capillary-gravity waves, in

Communicated by J. Escher.

B.-V. Matioc (B)Institut für Mathematik, Universität Wien, Nordbergstraße 15, 1090 Vienna, Austriae-mail: bogdan-vasile.matioc@univie.ac.at

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B.-V. Matioc

the case of pure capillary waves there is an explicit solution which is due to Kinnersely[24]. This solution, which is obtained by adapting Crapper’s solution [9] for deep waterwaves, describes waves over an irrotational fluid and, when the amplitude is large, thewaves have an overhanging profile.

Wilton [43] observed for capillary-gravity irrotational waves that sometimes a modemay interact with another one of half his size, the resulting waves, called Wilton ripples,possessing two crests within a period. A rigorous existence theory for these type ofwaves was obtained, in the irrotational regime, in [33], the complete picture of thelocal bifurcation problem for capillary-gravity waves being described in [20,21] fordeep water waves and in [19] for waves over a fluid layer of finite depth. Excludingthe presence of stagnation points, that is fluid particles which travel horizontally withthe same speed as the wave, the methods used in [6] in the context of gravity wavesare further developed in [38,39] to prove existence of small-amplitude solutions foran arbitrary vorticity distribution in the capillary-gravity and pure capillary context.More recently, the analysis has been extended to the case of flows with a discontinuousvorticity distribution [31], and to startified flows with a general density distributions[18,41].

In this paper we study the global bifurcation problem for capillary and capillary-gravity waves with constant vorticity, that is traveling over a linearly sheared current,when stagnation points are not excluded. The study of such flows is physically moti-vated by the fact that the tides, which are regular, two-dimensional currents of constantvorticity, are the most significant currents on the coastal shore [5,34]. The correspond-ing local bifurcation problem has been investigated in [27,28] by using the very recentformulation obtained in [7] in the context of pure gravity waves. The study of waveswith stagnation points, and critical layers consisting of closed streamlines, dates backto Lord Kelvin’s pioneering work [35]. While Stokes waves of greatest height pos-sess stagnation points only at their crests where they form upward cusps, cf. [1,36],irrotational waves do not possess stagnation points inside the fluid layer (see e.g. [2]).Exact gravity water waves with stagnation points were shown to exist only recentlyin [7,40] in the case of flows with constant vorticity and in [10] for flows with linearvorticity (in this case the waves may possess arbitrarily many critical layers). The flowpattern of small stratified flows with stagnation points and a linear density distributionis described in [11].

Our approach to the global bifurcation problem is based on the new formulation ofthe capillary and capillary-gravity water wave problems obtained herein, that allows usto make use of Rabinowitz’s global bifurcation theorem. This formulation is obtainedby observing that if the mean fluid depth is fixed, then we can invert the curvatureoperator in Bernoulli’s condition and reduce the problem to an equation for a compact,but nonlinear and nonlocal, perturbation of the identity. We then establish the existenceof global continua consisting of solutions of this equation, some of them featuringstagnation points. Moreover, starting with a laminar flow with stagnation points locatedon one of the continua we show, by using estimates for weak solutions of ellipticequations, that along a subcontinuum: (i) the peak-deviation approaches the meandepth; (i i) the limiting wave has very low regularity, situation related to the possibilitythat it may not be represented as a graph; or (i i i) the flow loses the property ofcontaining stagnation points. We emphasize that the numerical simulations performed

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Global bifurcation for waves with capillary effects

in the irrotational case (thus for flows without stagnation points) indicate that thereare two kinds of limiting waves: waves with self-intersection or waves with contactpoints between the surface and the bottom, cf. [32].

The outline of the paper is as follows: in Sect. 2 we introduce the mathematicalmodel and present our main result Theorem 2.2. The problem is then formulated asan abstract operator equation in Sect. 3 where, employing Crandall and Rabinowitz’stheorem on bifurcation from simple eigenvalues, local bifurcation branches consistingof solutions of the problem are found. Using global bifurcation theory, the branchesare continued in Sect. 4 to global continua, the analysis culminating with the proof ofthe main result.

2 The free boundary problem and the main result

In this paper we consider the free boundary value problem

⎧⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎩

�ψ = ω in �η,

|∇ψ |2 + 2gη − 2ση′′

(1 + η′2)3/2= Q on y = η,

ψ = 0 on y = η,

ψ = m on y = −1,

(2.1)

describing periodic capillary-gravity (only capillary when g = 0) water waves trav-eling over an horizontal bed when we observe them from a reference frame movingin the same direction and with the same speed as the waves. Hereby, g is the gravityconstant, m is the relative mass flux, and Q/2 is the hydraulic head (see [3]). Further-more, we denote by ω the constant vorticity of the underlying current and σ is thesurface tension coefficient at the water surface y = η, with η assumed to belong tothe set

V := {η ∈ C2+α0,e (S) : η > −1}.

The constant α ∈ (0, 1) is fixed in the remaining of the paper. Given p ∈ N, the spaceC p+α

0,e (S) consists of even, 2π -periodic functions with zero integral mean and havinguniformly Hölder continuous p-th order derivative. The fluid domain is then the set

�η := {(x, y) : −1 < y < η(x)}.

Note that because the horizontal bed is located at y = −1, the average mean depth ofthe fluid is equal to 1. Moreover, ψ is the stream function and it satisfies additionallythe relation ∇ψ = (−v, u − c), with c being the constant speed of the wave and(u − c, v) the velocity field of the steady flow. We recall that stagnation points arewater particles for which ∇ψ = 0 and that the vorticity constant is related to thevelocity vector through ω = uy − vx .

Let us observe that considering the mean depth fixed, we can in fact identify the con-stant Q from the second equation of (2.1) by simply integrating Bernoulli’s condition(the second relation of (2.1)) over a period

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B.-V. Matioc

Q = 1

2π

2π∫

0

|∇ψ |2(x, η(x)) dx . (2.2)

This relation is a consequence of the fact that the curvature may be written in divergenceform

κ(η) := η′′

(1 + η′2)3/2=

(η′

(1 + η′2)1/2

)′for η ∈ V.

Note also that κ is a real-analytic map κ ∈ Cω(V,Cα0,e(S)).

Finally, we introduce the parameter

λ := −m + ω/2 (2.3)

into the problem. As we shall see later on, λ is the speed at the wave surface forthe laminar flow solutions of (2.1) when Q and m are chosen as in (2.2) and (2.3),respectively. In this paper, we reduce the problem (2.1)–(2.3) to an operator equationfor (λ, η) and we shall seek for solutions by using λ as a bifurcation parameter.

We shall study the capillary and the capillary-gravity water wave problems at once.When g = 0 then all our results refer to capillary waves and when g > 0 they arestated for capillary-gravity waves.

The solutions we find have the property that η ∈ C2+α0,e (S). However, exploiting

the fact that the curvature operator is a quasilinear elliptic operator together with theinvariance of (2.1) with respect to horizontal translations, it is shown in [30, Remark2.1] that the surface profile of any solution of (2.1)–(2.3) is a real-analytic curve. Theresult is true even if the flow has a general vorticity ω = γ (−ψ) provided the vorticityfunction γ is real-analytic and non-increasing.

In the case of a regular vorticity distribution, but in the absence of stagnation pointsit was shown in [14–16], by further developing some of the technics from [4], and in[26] that all the streamlines, including the wave surface, are real-analytic curves. This istrue even when the voticity is merely integrable [31]. The regularity of the streamlinesfor waves over stratified flows with capillarity has been addressed in [17,42]. Suchregularity though, cannot be expected for flows with stagnation points, see e.g. Fig. 1.We summarize with the following regularity result.

Proposition 2.1 Let η ∈ V be a solution of problem (2.1)–(2.3). Then, the wave profileis a real-analytic graph, that is η ∈ Cω(S).Moreover, if there are no stagnation pointsin the flow, then all the streamlines are real-analytic graphs.

The main result of this paper is the following global bifurcation theorem.

Theorem 2.2 (Global bifurcation) Let ω ∈ R, α ∈ (0, 1), and σ ∈ (0,∞) be fixed,and let λ0 be a real constant. Then, either:

(a) there exists a small neighborhood of (λ0, 0) in R×V with the property that if (λ, η)belongs to this neighborhood and solves the problem (2.1)–(2.3), then η = 0;

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Global bifurcation for waves with capillary effects

(b) (λ0, 0) is a bifurcation point which belongs to a continuum C of solutions of theproblem (2.1)–(2.3) in R × C2+α

0,e (S). Moreover, C contains at most countablymany solutions (λ, η) with η �= 0, the map η is real-analytic for all (λ, η) ∈ C,and

(1) the continuum C contains a bifurcation point (λ1, 0) with λ1 �= λ0, or(2)

sup(λ,η)∈C

1

min[0,2π ] |η + 1| + infβ∈(0,1) sup

(λ,η)∈C‖η′‖β + sup

(λ,η)∈C|λ| = ∞. (2.4)

Moreover, if

sup(λ,η)∈C

1

min[0,2π ] |η + 1| + infβ∈(0,1) sup

(λ,η)∈C‖η′‖β < ∞ (2.5)

and the flow corresponding to (λ0, 0) contains stagnation points, then, along asub-continuum of C, the waves will lose this property.

We enhance that if the first term of (2.4) becomes unbounded, then the peak devi-ation will approach the mean depth of the flow and the waves will touch in the limitthe fluid bed. On the other hand, when the second term becomes unbounded thereis not enough information to deduce that the sequence (ηp)p possesses a convergentsubsequence in C1(S) even if (ηp)p is bounded in C1(S).1 We relate this case to thepossibility that the limiting wave is not described by a graph, as suggested by [24] andmany numerical studies, see e.g. [22,37]. In the remaining cases, that is when (2.5)holds, the flow determined by (λ0, 0) may contain stagnation points, cf. (3.14), butthis property will be lost along a sub-continuum of C.

3 Local bifurcation for the new mathematical formulation

In this section we first reformulate the problem (2.1)–(2.3) as a nonlinear and nonlocalequation for a compact perturbation of the identity. To this end, we transform theproblem on the fixed domain� := S× (−1, 0), by flattening the free boundary y = η

with the help of the mapping �η : � → �η given by

�η(x, y) := (x, (y + 1)η(x)+ y), (x, y) ∈ �.Assuming that η ∈ V, it is not difficult to see that �η is a diffeomorphism. While theflattening transformation used in [7,27,28] is a conformal map,�η does not have thisproperty, but it is more suitable when considering the global bifurcation problem.

The diffeomorphism�η transforms all the operators appearing in the system (2.1).Letting C p+α

e (�), p ∈ N, denote the subspace of C p+α(�) consisting of functionsthat are even in the x-variable, we define

1 For example, the sequence (ηp)p defined by ηp(x) := cos(px)/p for x ∈ R and p ≥ 1 is uniformlybounded in C1(S) and infβ supp ‖η′

p‖β = ∞ for all β ∈ (0, 1). Though ηp is a real-analytic map for all

p ≥ 1, the sequence (ηp)p does not have a convergence subsequence in C1(S).

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B.-V. Matioc

A : V → L(C2+αe (�),Cα

e (�)) and B : V × C2+αe (�) → C1+α

e (S)

by the expressions

A(η)v := �(v ◦�−1η ) ◦�η and B(η, v) := tr|∇(v ◦�−1

η )|2 ◦�η

for (η, v) ∈ V × C2+α0,e (�). Hereby, tr denotes the trace operator with respect to the

unit circle S = S × {0}. The following explicit expressions

A(η) = ∂11 − 2(1 + y)η′

1 + η∂12 + 1 + (1 + y)2η′2

(1 + η)2∂22 − (1 + y)

(1 + η)η′′ − 2η′2

(1 + η)2∂2,

B(η, v) = trv21 − 2η′

1 + ηtrv1trv2 + 1 + η′2

(1 + η)2trv2

2 for (η, v) ∈ V × C2+α0,e (�)

(3.1)are used later on when studying the bifurcation problem (3.7).

With this notation, the problem (2.1)–(2.3) may be recast as the following nonlocaland nonlinear equation

�(λ, η) = κ(η) (3.2)

where � : R × V → Cα0,e(S) is the operator defined by

�(λ, η) := 1

2σ

⎛

⎝B(η, T (λ, η))+ 2gη − 1

2π

2π∫

0

B(η, T (λ, η)) dx

⎞

⎠ (3.3)

for (λ, η) ∈ R × V. Here T : R × V → C2+αe (�) is the solution operator associated

to the Dirichlet problem⎧⎨

⎩

A(η)v = ω in �,

v = 0 on y = 0,v = −λ+ ω/2 on y = −1,

(3.4)

that is for all (λ, η) ∈ R×V , the map T (λ, η) is the unique solution of (3.4). In orderto write the problem in a more accessible form, we now invert the curvature operatoras shown in the following lemma.

Lemma 3.1 Let R denote the open subset of Cα0,e(S) defined by

R :=⎧⎨

⎩ξ ∈ Cα

0,e(S) : supx∈[0,2π ]

∣∣∣∣∣∣

x∫

0

ξ(s) ds

∣∣∣∣∣∣< 1

⎫⎬

⎭.

Then, the operator κ : V → R is a real-analytic diffeomorphism.

Proof The fact that curvature operator is one-to-one follows readily from its definition.Let now ξ ∈ R be given. If κ(η) = ξ, then we obtain by integrating this relation that

η′(x)(1 + η′2(x))1/2

= �(x) :=x∫

0

ξ(s) ds for all x ∈ R.

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Global bifurcation for waves with capillary effects

Hence, η′ and � have the same sign and therefore

η′ = �

(1 −�2)1/2.

Integrating once more and using the fact that η has integral mean zero, we concludethat curvature operator κ : V → R is invertible and its inverse is given by

κ−1(ξ)(x) :=x∫

0

�(s)

(1 −�2(s))1/2ds − 1

2π

2π∫

0

x∫

0

�(s)

(1 −�2(s))1/2dsdx, x ∈ R,

(3.5)for ξ ∈ R, whereby � is the odd antiderivative of ξ. ��

In virtue of Lemma 3.1, we conclude that if (λ, η) is a solution of the problem(3.2), then (λ, η) ∈ U := �−1(R). With this notation, the problem (3.2) can bereformulated as the following problem

F(λ, η) := η + f (λ, η) = 0 in C2+α0,e (S), (3.6)

whereby f : U ⊂ R × C2+α0,e (S) → C2+α

0,e (S) is the mapping defined by

f (λ, η) := −κ−1(�(λ, η)) for (λ, η) ∈ U . (3.7)

Note that the relation (3.1) together with Lemma 3.1 ensure that f is a real-analyticmap and that U is an open subset of R × C2+α

0,e (S).

Local bifurcation. We have showed so far the original problem (2.1)–(2.3) is equiv-alent to the operator equation (3.6) and that the latter problem involves a nonlinearand nonlocal, but compact (see the proof of Theorem 4.1), perturbation of the identity.Using Crandall and Rabinowitz’s theorem on bifurcation from simple eigenvalues [8],we next show that for certain values of the constant λ local branches consisting ofnon-laminar solutions of the problem (3.6) emerge from the branch of trivial solutions(λ, 0). Because we are interested to study the global bifurcation branches we do notstrive to obtain a precise picture of the local bifurcation phenomenon (which is ratherintricate as double and secondary bifurcations may occur [19,29]). We only deter-mine in this section the points where the global continua intersect the trivial branchof laminar flows.

First, let us observe that (λ, 0) ∈ U for all λ ∈ R and that

F(λ, 0) = 0 (3.8)

for all λ ∈ R. Indeed, if η = 0, then T (λ, 0) =: ψ0 is given by the relation

ψ0(x, y) := ωy2

2+ λy for (x, y) ∈ �.

Taking into account that B(0, ψ0) = λ2, we find

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B.-V. Matioc

�(λ, 0) = 1

2σ

⎛

⎝B(0, ψ0)− 1

2π

2π∫

0

B(0, ψ0) dx

⎞

⎠ = 0 = κ(0),

and therefore (λ, 0) ∈ U is a solution of (3.6). The trivial solutions (λ, 0) describehorizontal flows with a flat surface and parallel streamlines. For these laminar flowswe compute ψy(0, 0) = (u − c)(0, 0) = λ, so that the parameter λ can be interpretedas being the horizontal speed at the wave surface for the steady flow.

In order to apply the theorem on bifurcation from simple eigenvalues due to Crandalland Rabinowitz, it is necessary to compute the Fréchet derivative ∂ηF(λ, 0). It is notdifficult to see that

∂ηF(λ, 0)[η] = η − ∂κ−1(0)[∂η�(λ, 0)[η]], η ∈ C2+α0,e (S),

whereby, in virtue of (3.3) and (3.5), we obtain by simply using the chain rule that

∂κ−1(0)[ξ ](x) =x∫

0

�(s) ds − 1

2π

2π∫

0

x∫

0

�(s) dsdx, x ∈ R,

∂η�(λ, 0)[η] = 1

2σ

(

∂ηB(0, ψ0)[η] + ∂vB(0, ψ0)[∂ηT (λ, 0)[η]] + 2gη

− 1

2π

2π∫

0

∂ηB(0, ψ0)[η] + ∂vB(0, ψ0)[∂ηT (λ, 0)[η]] dx

⎞

⎠ ,

with � denoting again the odd antiderivative of ξ ∈ Cα0,e(S). Expanding η by its

Fourier series, we determine the following representation of ∂ηF(λ, 0)

∂ηF(λ, 0)[η] =∞∑

l=1

μl(λ)al cos(lx) for η =∞∑

l=1

al cos(lx),

whereby the symbol μl(λ) is given by

μl(λ) := 1 + 1

σ l2

(

g − λ2l

tanh(l)+ λω

)

. (3.9)

Invoking [12, Theorem 3.4], we see that the spectrum of ∂ηF(λ, 0)) consists onlyof the eigenvalues {μl(λ) : 1 ≤ l ∈ N}, and that μ = 0 is an eigenvalue if and onlyif λ = λ±

k where

λ±k := ω

2

tanh(k)

k±

√

ω2

4

tanh2(k)

k2 + (g + σk2)tanh(k)

k, k ≥ 1. (3.10)

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Global bifurcation for waves with capillary effects

It is well-known that bifurcation occurs only when zero is in the spectrum of∂ηF(λ, 0). Thus, let λ0 := λ+

k or λ0 := λ−k for some integer k ≥ 1. In order to

count the multiplicity of the eigenvalue μk(λ0) = 0, we observe that λ+l > 0 and

λ−l < 0 for all positive integers l ∈ N.Moreover, as shown in [28], zero may be either

a simple or a double eigenvalue of ∂ηF(λ0, 0) depending on the sign of the expressions

�± := �±(g, ω, σ ) := σ −(

g

3+ ω2

6± ω

6

√

ω2 + 4g

)

.

Assume that λ0 = λ+k (resp. λ0 = λ−

k ). If �+ ≥ 0 (resp. �− ≥ 0), then thesequence (λ+

l )l≥1 (resp. (λ−l )l≥1) is strictly monotone and zero is a simple eigen-

value of ∂ηF(λ0, 0). On the other hand, if�+ < 0 (resp.�− < 0), then, the sequence(λ+(l))l≥N (resp. (λ−(l))l≥N ) is monotone (and unbounded) provided that N is suffi-ciently large. Moreover, they may exist at most a further integer p such that 1 ≤ p �= kand λ0 = λ+

p (resp. λ0 = λ−p ). Thus, if �+ < 0 (resp. �− < 0), then zero may be a

double eigenvalue of ∂ηF(λ0, 0). More precisely, setting Tl := tanh(l)/ l for l ≥ 1,there exists a discrete set of values for the square of the vorticity constant

ωl,p :=((g + σ l2)Tl − (g + σ p2)Tp

)2

σTl Tp(Tl − Tp)(p2 − l2), l, p ≥ 1, (3.11)

with the following property: we have that either λ+l = λ+

p or λ−l = λ−

p if and only if

ω2 = ωl,p. Moreover, if λ+l = λ+

p for a particular vorticity constant ω = ω, then we

have λ−l = λ−

p when ω = −ω.The fact that zero can be an eigenvalue of multiplicity two is an impediment when

trying to apply Crandall and Rabinowitz’s theorem on bifurcation from simple eigen-values or abstract global bifurcation results to our particular problem. However, whenzero is a double eigenvalue, we may reduce the dimension of the correspondingeigenspace to one by using the fact that the minimal period of F(λ, η) is a divi-sor of the minimal period of η for all (λ, η) ∈ U . More exactly, given p ∈ N withp ≥ 1 and m ∈ N, we define Cm+α

0,e,p(S) as being the subspace of Cm+α0,e (S) which

consists only of 2π/p−periodic functions. Setting Up := U ∩ (R × C2+α0,e,p(S)), it is

not difficult to see that f , and therefore also F, belong both to Cω(Up,C2+α0,e,p(S)).

We summarize this section with the following local bifurcation result.

Theorem 3.2 (Local bifurcation) Let k be a positive integer and λ0 ∈ {λ−k , λ

+k }. Then,

(λ0, 0) is a bifurcation point for the Eq. (3.6). More precisely, there exits ε > 0 anda real-analytic curve (�, η) : (−ε, ε) → U which consists only of solutions of (3.6).Given s ∈ (−ε, ε), the function η(s) is real-analytic and the bifurcating flows arenon-laminar, that is η(s) �≡ 0, for all s �= 0.

If λ �∈ {λ±k : 1 ≤ k ∈ N}, then (λ, 0) is not a bifurcation point.

Remark 3.3 Asymptotically, when s → 0, we have that

�(s) = λ0 + O(s) and η(s) = s cos(nx)+ O(s2) in C2+α0,e,n(S)

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B.-V. Matioc

whereby the positive integer n ≥ k is defined in the proof of Theorem 3.2. Particularly,if zero is a simple eigenvalue of ∂ηF(λ0, 0), then n = k.

Proof We first assume that λ0 �∈ {λ±k : 1 ≤ k ∈ N}. Then, μl(λ0) �= 0 for all

l ≥ 1, and therefore zero is in the resolvent set of ∂ηF(λ0, 0). As a consequence ofthe implicit function theorem we obtain that (λ0, 0) is not a bifurcation point.

Assume now that λ0 satisfiesμk(λ0) = 0,meaning that zero is an eigenvalue of thelinear operator ∂ηF(λ0, 0) ∈ L(C2+α

0,e (S)). If the vorticity constant is chosen such thatzero is a double eigenvalue, that is ω is one of the solutions of the equation ω2 = ωk,p

for some p �= k, then we define n := max{k, p}. In this case we have that eitherλ0 = λ+

k = λ+p or λ0 = λ−

k = λ−p . If zero is a simple eigenvalue, set n := k. We

shall consider now the bifurcation problem for the equation (3.6) and the restrictionF : Un → C2+α

0,e,n(S). The advantage of restricting the domain and the range of F isthat in this way we eliminate the mode cos(lx), with l := min{k, p}, from the kernelof ∂ηF(λ0, 0) in the case when λ0 is a double eigenvalue. As noticed before, we haveF(λ, 0) = 0 for all λ ∈ R, while the Fréchet derivative ∂ηF(λ0, 0) ∈ L(C2+α

0,e,n(S)) isgiven by

∂ηF(λ0, 0)[η] =∞∑

l=1

μln(λ0)al cos(lnx) if η =∞∑

l=1

al cos(lnx), (3.12)

with μln defined by (3.9). Due to our choice of n, we have that μn(λ0) = 0 andμln(λ0) �= 0 for all l ≥ 2. Particularly, we conclude that ∂ηF(λ0, 0) ∈ L(C2+α

0,e,n(S))

is a Fredholm operator of index zero, and

Ker∂ηF(λ0, 0) = span{cos(nx)}, Im∂ηF(λ0, 0)⊕ span{cos(nx)} = C2+α0,e,n(S).

Finally, the transversality condition needed in order to apply the theorem on bifurcationfrom simple eigenvalues due to Crandall and Rabinowitz [8] to our bifurcation problemis also satisfied because, due to

|∂λμn(λ0)| = 1

σn2

√

ω2 + 4n(g + σn2)

tanh(n)�= 0,

we have ∂ληF[cos(nx)] = ∂λμn(λ0) cos(nx) �∈ Im∂ηF(λ0, 0). We conclude that thereexists a real-analytic curve (�, η) of solutions of (3.6) with the properties stated inRemark 3.3. The additional regularity of η(s), for |s| < ε, is obtained from Proposition2.1. This finishes the proof. ��Flows with stagnation points Let λ0 ∈ {λ−

k , λ+k }, k ≥ 1, be one of the bifurcation

points for the equation (3.6). For simplicity we assume that μk(λ0) = 0 is a simpleeigenvalue of ∂ηF(λ0, 0). The velocity field of the steady laminar flow correspondingto this parameter is given by the relation

(u − c, v)(x, y) = (ωy + λ0, 0), (x, y) ∈ �, (3.13)

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Global bifurcation for waves with capillary effects

Fig. 1 The streamline pattern for the solutions (�(s), η(s)) when they bifurcate from a laminar flow withstagnation points inside in the case when λ0 > 0. There are two stagnation points within a period: onelocated inside the critical layer and one on its boundary

and it follows readily from this relation that the flow possesses stagnation points ifand only if

λ0ω > 0 and ω2 ≥ (g + σk2)tanh(k)

k − tanh(k). (3.14)

In fact, if (3.14) is satisfied, this laminar flow contains a streamline y = −λ0/ωwhichconsists entirely of stagnation points. This feature is displayed also by the bifurcatingsolutions (�(s), η(s)). Indeed, assuming that (3.14) is satisfied, arguments similar tothose presented in [11,40] show that the flow (�(s), η(s)) contains stagnation pointsfor all |s| < ε, provided that ε is small. A qualitative picture of the streamline patternfor the solution bifurcating from a laminar flow with stagnation points inside, thatis when y = −λ0/ω ∈ (−1, 0), is presented in the Fig. 1. The flow in this picturecontains two stagnation points per period: one located in the middle of Kelvin’s cat’seye vortex [35] and the other one on the separatrices enclosing the critical layer ofclosed streamlines.

4 Global bifurcation

The main goal of this section is to show that every local curve (�, η) belongs to aglobal continuum C, consisting of solutions of (2.1), and to describe the behavior ofthe solutions along this continuum. To this end, we shall prove first the followingglobal bifurcation result.

Theorem 4.1 Let the assumptions of Theorem 3.2 be satisfied and let C be the maximalconnected component of the set

{(λ, η) : (λ, η) ∈ U , η �= 0, and F(λ, η) = 0}

containing (λ0, 0). Then, we have:

(i) C is unbounded in R × C2+α0,e (S), or

(ii) C contains a second bifurcation point (λ1, 0) with λ1 �= λ0, or(iii) inf(λ,η)∈C min[0,2π ] η = −1.

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This global bifurcation result is the main step in the proof of our main theorem. Inorder to prove Theorem 4.1, we shall use the following version of the global Rabinowitzbifurcation theorem.

Theorem 4.2 Let X be a Banach space and O ⊂ R × X a bounded and open set.Assume that the function F(λ, x) := x + f (λ, x) belongs to C1(O, X), that addi-tionally f : O → X is completely continuous, and let S denote the set of nontrivialsolutions of the equation F(λ, x) = 0. If ∂x F(λ, 0) has an odd crossing number atλ = λ0, with (λ0, 0) ∈ O, then (λ0, 0) ∈ S and the connected component C to which(λ0, 0) belongs

(i) intersects the boundary of ∂O, or(ii) contains some (0, λ1) ∈ O with λ0 �= λ1.

Proof The proof is similar to that of Theorem II.3.3 in [23]. ��We clarify now the assumptions of Theorem 4.2. It is important for bifurcation at

(λ0, 0) how the eigenvalue 0 perturbs for ∂x F(λ, 0) when λ varies in a neighborhoodof λ0. Defining the 0-group of ∂x F(λ, 0) as being the set consisting of the perturbedeigenvalues of ∂x F(λ, 0) near 0 (they depend continuously on λ), we let σ<(λ) = 1if there are no negative real eigenvalues in the 0−group of ∂x F(λ, 0) and we setσ<(λ) = (−1)m1+...+mk if μ1, . . . , μk are all negative real eigenvalues in the 0-grouphaving algebraic multiplicities m1, . . . ,mk, respectively. If for some small δ > 0

∂x F(λ, 0) is bijective for λ ∈ (λ0 − δ, λ0) ∪ (λ0, λ0 + δ)

and σ<(λ) changes at λ = λ0,(4.1)

then ∂x F(λ, 0) has an odd crossing number at λ = λ0, cf. Definition II.3.1 in [23].As a consequence of ∂x F(λ, 0) having an odd crossing number at λ = λ0, the indexi(∂x F(λ, 0), 0) jumps at λ = λ0 from 1 to −1 or vice versa.

Proof of Theorem 4.1 As in the proof of Theorem 3.2, we shall consider the bifurcationproblem for the equation (3.6) and the following restriction F : Un → C2+α

0,e,n(S).

Defining Cn := C ∩ C2+α0,e,n(S) ⊂ Un, we assume by contradiction that none of the

alternatives (i)− (i i i) in Theorem 4.1 is true for Cn . Then, we claim that Cn is a closedsubset of Un . Indeed, let (λl , ηl) ∈ Cn satisfy (λl , ηl) → (λ, η) in R × C2+α

0,e,n(S).

Because, ηl ≥ −1 + ε, for some small ε > 0 and uniformly in l, we conclude thatη ∈ V. On the other hand, we have that

x∫

0

�(λl , ηl) dt = η′l(x)

(1 + η′2l (x))

1/2for x ∈ R

and all l ∈ N, the norm ‖η′l‖0 being bounded uniformly in l. This implies that the limit

point (λ, η) belongs to Un . Passing to the limit l → ∞ in the equation F(λl , ηl) = 0,we finally conclude that (λ, η) ∈ Cn .

Next, we prove that Cn is a compact subset of Un . To this end, let ((λl , ηl))l≥0 be asequence in Cn . Since (λl)l is bounded in R,we may assume (after possibly extracting

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Global bifurcation for waves with capillary effects

a subsequence) that λl → λ for some λ ∈ R. Moreover, there exists ε > 0 such thatmin[0,2π ] ηl ≥ −1 + ε and ‖ηl‖2+α ≤ 1/ε for all l ∈ N. By Schauder’s estimate, cf.[13, Theorem 6.6], we find a positive constant C such that ‖T (λl , ηl)‖2+α ≤ C for alll ∈ N. Invoking (3.1) and (3.3), we conclude that supl ‖κ(ηl)‖1+α < ∞, so that (ηl)

is a bounded sequence in C3+α(S). This shows that a subsequence of (ηl)l convergesin C2+α(S), and yields the desired claim.

Consequently, we may cover Cn by a finite number of balls Bi , i = 1...N , having theproperty that Bi ⊂ Un and inf(λ,η)∈O min[0,2π ] η = −1 + ε for some ε > 0, wherebywe set O := ∪i=1...N Bi . We verify next that the restriction of F to O satisfies allthe assumptions of Theorem 4.2. First, let us note that the arguments presented aboveshow that f is a completely continuous map. Next, we study how the eigenvaluesμln(λ) perturb when λ crosses λ = λ0. Since μln(λ) →l→∞ 1 uniformly on boundedλ intervals, and due to our choice of n, we may find δ > 0 with the property thatμln(λ) �= 0 for all l ≥ 1 and all λ ∈ (λ0 − δ, λ0)∪ (λ0, λ0 + δ). Thus, ∂ηF(λ, 0) is anisomorphism for all λ ∈ (λ0 − δ, λ0)∪ (λ0, λ0 + δ). On the other hand, we know that

|∂λμn(λ0)| �= 0,

and therefore μn(λ) changes sign when λ crosses the value λ0. Since the 0-group of∂ηF(λ, 0) consists only of the eigenvalue μn(λ), we conclude that ∂ηF(λ, 0) has anodd crossing number at λ = λ0. We are thus in the situation of applying Theorem4.2. Since Cn contains no other bifurcation point of (3.6) in Un , we obtain then thatCn must intersect the boundary of O, which is a contradiction. This leads us to thedesired conclusion. ��

Finally, we come to the proof of our main result Theorem 2.2.

Proof of Theorem 2.2 If λ0 is chosen such that λ0 �∈ {λ±k : 1 ≤ k ∈ N}, then, as

shown in the proof of Theorem 3.2, we are in the situation (a).Assume now thatλ0 ∈ {λ−

k , λ+k } for some k ≥ 1, and let n ≥ k be the corresponding

integer defined in the proof of Theorem 3.2. Thus, we have λ0 ∈ {λ−n , λ

+n }. Moreover,

we define Cn := C ∩ Un .Let us exclude the alternatives (i) and (i i i) in Theorem 4.1. Then, considering the

equationF(λ, η) = 0 in C2+α

0,e,n(S) (4.2)

for the restriction F : Un → C2+α0,e,n(S), we infer from the proof of Theorem 4.1 that

Cn intersects the set of trivial solutions a second time at (λ1, 0) for some λ1 �= λ0.

There are two alternatives.It may happen that {λ0, λ1} = {λ−

n , λ+n }. In this case, (λ1, 0) ∈ Cn ⊂ C, and

it follows readily from (3.13) and the inequality λ0λ1 < 0 that if the laminar flowdetermined by (λ0, 0) contains stagnation points, then the laminar flow correspondingto (λ1, 0) has no stagnation points.

The second alternative is that λ1 /∈ {λ−n , λ

+n }. Since (λ1, 0) is a bifurcation point

for (4.2), the derivative ∂ηF : C2+α0,e,n(S) → C2+α

0,e,n(S) is not invertible. Consequently,there exists an integer l1 ≥ 1 such that μl1n(λ1) = 0, cf. (3.12). In fact, the relationλ1 /∈ {λ−

n , λ+n } implies that l1 ≥ 2. We now set k1 := l1n, and define n1 := k1 if zero is

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B.-V. Matioc

a simple eigenvalue of ∂ηF : C2+α0,e,n(S) → C2+α

0,e,n(S), respectively n1 := max{k1, p1n}if, for some p1 ≥ 3, we also have λ1 ∈ {λ−

p1n, λ+p1n}. Studying the bifurcation problem

F(λ, η) = 0 for the restriction F : Un1 → C2+α0,e,n1

(S), the arguments presented in theproof of Theorem 4.1 ensure that Cn1 := Cn ∩ Un1 contains a further bifurcation point(λ2, 0). Again, we have two possible alternatives: {λ1, λ2} = {λ−

n1, λ+

n1}, or, when

λ2 /∈ {λ−n1, λ+

n1}, μl2n1(λ2) = 0 for some l2 ≥ 2. Moreover, in the latter case we have

λ2 �= λi for i = 0, 1.Because we excluded (i), this procedure has to stop after a finite number of steps,

otherwise we find a sequence (pl)l≥1 ⊂ N\{1} and a further sequence (λnl )l≥1 ⊂ {λ±k :

1 ≤ k ∈ N} with (λnl , 0) ∈ C, λnl ∈ {λ−pl ...p1n, λ

+pl ...p1n}, and λnl �= λnm for all l �= m.

Since |λ±pl ...p1n| → ∞ as l → ∞, cf. (3.10), this contradicts the boundedness of C.

Consequently, if (i) and (i i i) are excluded, then C contains two bifurcation points(λ1, 0) and (λ2, 0) with the property that, for some p ≥ 1, {λ1, λ2} = {λ−

p , λ+p }.

Thus, if the flow determined by (λ0, 0) contains stagnation points, then on the branchconnecting (λ1, 0) and (λ2, 0), the waves will lose this property.

In order to prove the property (2.4), let us now exclude the alternatives (i i) and(i i i) of Theorem 4.1. Moreover, we presuppose that

sup(λ,η)∈C

|λ| < ∞.

We have to show that sup(λ,η)∈C ‖η′‖β = ∞ for all β ∈ (0, 1). To this end, weproceed by contradiction and assume that there exists some fixed β ∈ (0, 1) withsup(λ,η)∈C ‖η′‖β < ∞. Since the mean depth of the fluid is kept fixed, we have

‖η‖0 ≤ 2π‖η′‖0 for all η ∈ C2+α0,e (S). Therefore sup(λ,η)∈C

(|λ| + ‖η‖1+β) =: K0 is

finite. Introducing the functions

a1 := − (1 + y)η′

1 + ηand a2 := 1

1 + η(4.3)

we see that, given (λ, η) ∈ C, the map T (λ, η) is the classical solution of the problem

∂i (ai j∂ jv)+ c1∂1v + c2∂2v = ω in �, v = (λ− ω/2)y on ∂�, (4.4)

whereby

a11 = 1, a12 = a21 = a1, a22 = a21 + a2

2 , c1 = −∂2a1, c2 = −a1∂2a1.

(4.5)Using Schauder estimates for the weak solutions of (4.4), cf. [13, Theorem 8.33], weobtain that

‖T (λ, η)‖1+β ≤ C(μ, K )(‖T (λ, η)‖0 + ‖(λ− ω/2)y‖1+β + |ω|) . (4.6)

The constant K in (4.6) is chosen to satisfy max{‖ai j‖β, ‖ci‖0} ≤ K and, due to ourassumption, it depends only on K0. Moreover, μ is the positive constant of ellipticityof the operator A(η) and, since the alternative (i i i) of Theorem 4.1 does not occur,

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Global bifurcation for waves with capillary effects

we may also chose it to be independent of the solution (λ, η) ∈ C. Finally, observingthat

�(T (λ, η) ◦�−1

η − ωy2/2)

= 0 in �η,

the strong elliptic maximum principle ensures that

‖T (λ, η)‖0 ≤ C, (4.7)

the constant C depending only of ω and K0. Combining (4.6) and (4.7), we concludefrom (3.1)–(3.3) that sup(λ,η)∈C ‖η‖2+β < ∞. Moreover, invoking the Schauder esti-mate [13, Theorem 6.6], the latter bound yields that the solution T (λ, η) of (3.4)satisfies

sup(λ,η)∈C

‖T (λ, η)‖2+β < ∞.

In view of the relations (3.1)–(3.3), we now find sup(λ,η)∈C ‖η‖3+β < ∞. Thiscontradicts the fact that C is unbounded, and proves the desired assertion, namelyinfβ∈(0,1) sup(λ,η)∈C ‖η′‖β = ∞. The last claim of Theorem 2.2 is proven below. ��

We now assume that the flow described by (λ0, 0) contains stagnation points. Tocomplete the proof of Theorem 2.2, we are left to show that if only the alternative (2)of Theorem 2.2 occurs and if (2.5) is satisfied, then along a subcontinuum of C thewater waves will lose the property of containing stagnation points.

Proposition 4.3 Assume that the relations (2.4) and (2.5) are fulfilled. Then, for λsufficiently large, the flow corresponding to the solution (λ, η) ∈ C does not containany stagnation points.

Proof It suffices to prove that if ((λp, ηp))p ⊂ C is a sequence with |λp| → ∞, thenthe flow corresponding to (λp, ηp) ∈ C contains no stagnation points if p is large.Invoking (2.5), there exists β ∈ (0, 1) such that ‖η′

p‖β ≤ K for all p ∈ N, wherebyK > 0 is a constant independent of p. Dividing the equations of (4.4) by λp, we findthat vp := T (λp, ηp)/λp is the classical solution of the problem

∂i (ai j∂ jvp)+ c1∂1vp + c2∂2vp = ω/λp in �, vp = (1 − ω/(2λp))y on ∂�,

whereby ai j and ci , i, j = 1, 2, are defined by (4.3) and (4.5) with (ηp instead of η).In virtue of [13, Theorem 8.33] and of the strong maximum principle, we deduce thatthe sequence (vp)p is bounded in C1+β(�). Arzelà-Ascoli’s theorem now ensuresthe existence of a pair (η, v) ∈ C1+β/2(S) × C1+β/2(�) and of a subsequence (notrelabeled) such that ηp → η in C1+β/2(S) and vp → v in C1+β/2(�). Clearly, v isthe weak solution of the problem

∂i (ai j∂ jv)+ c1∂1v + c2∂2v = 0 in �, v = y on ∂�,

and therefore the function ψ := v ◦�−1η ∈ C1+β2/4(�η) is the weak solution of

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B.-V. Matioc

�ψ = 0 in �η, ψ = (y − η)/(1 + η) on ∂�η.

Then, by standard regularity results, we know that ψ ∈ C∞(�η). With η belongingto the class C1+β/2(S), the boundary ∂�η satisfies an exterior sphere condition at anyof its points. Since ψ is not constant it must attain its extreme values on the boundaryof �η. Taking into account that ψ = 0 on y = η(x) and ψ = −1 on y = −1, weconclude from Hopf’s lemma that ∂yψ > 0 on ∂�η. But, ∂yψ is also harmonic in�η,thus ∂yψ > 0 in�η. Consequently, we have that |∂yT (λp, ηp)| > 0 in� if p is large,and therefore the stream functionψp := T (λp, ηp)◦�−1

η satisfies min�η |∂yψp| > 0for all such p. This excludes the presence of stagnation points in the flow determinedby (λp, ηp) when p is large. ��

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