STEADY INTERFACIAL WAVES OVER A NON-FLAT BOTTOM

YUNGJOO LEE, TIMOTHY MOK, HAOCHUAN WANG

Abstract. In this paper, we consider the existence of two-dimensional steady waves on the in-terface between two immiscible fluids where the lower fluid is taken to lie above an impermeable,non-flat boundary. We construct an asymptotic model to investigate the case where the bottomis not identically flat. Using Implicit Function and Bifurcation theoretic arguments, we prove theexistence of a family of stationary solutions to the approximate model that are of small amplitudeand small circulation. We also provide some numerical results.

Contents

1. Introduction 12. Mathematical background 52.1. Banach spaces 52.2. The Frechet derivative 52.3. Implicit Function Theorem 52.4. Bifurcation theory 63. Equivalent formulation of the problem 64. Flattening the domain 74.1. Fluid in the upper region 84.2. Fluid in the lower region 85. Existence of small amplitude flows in a region with a non-flat free surface and a flat

bottom 95.1. Numerical analysis of solutions 116. Approximate model 136.1. Approximate solutions 146.2. Approximate expression for the Dirichlet-Neumann operator 167. Small amplitude flows over a non-flat bottom 167.1. Non-existence of small-amplitude flows with non-flat bottom and flat free surface 167.2. Existence of small amplitude flows in a region with a non-flat free surface and a

non-flat bottom 21Acknowledgement 21References 22Appendix A. Numerical Results 23

1. Introduction

Stokes [4] study of periodic water waves in a region of infinite depth heralded much interest inthe field of steady waves. In the early twentieth century, Nekrasov [3] and Levi-Civita [2] firstproved, rigorously, the existence of these waves. Struik [5] later extended the results of Nekrasov

Date: November 23, 2012.

1

and Levi-Civita to regions with an impermeable, flat bottom. Since then, the majority of researchin the field of steady waves has continued to be focused on the case where the bottom is flat.

In this paper, we extend these classical results by studying the case when the lower boundaryof the fluid domain need not be flat. Specifically, we investigate the existence of two-dimensionalsteady waves on the interface between two immiscible incompressible, and irrotational fluids wherethe lower fluid is taken to lie above an impermeable boundary. This scenario arises naturally inthe study of various physical phenomenon such as waves in lee of a mountain and current flow overnon-flat ocean bed.

Let us state the problem mathematically. We define the Cartesian coordinates (x, y) such that thex-axis lies in the direction of wave propagation and the y-axis points vertically upward. Additionally,we assume the free surface between the two fluids is given as the graph of a function η(x) and liesabout y = 0 while the impermeable bottom, y = ζ(x), lies about y = −h. Under this coordinatesystem, the less dense fluid with density ρ1 occupies the domain Ω1 where

Ω1 := (x, y) : y > η(x)and the denser fluid with density ρ2 occupies the domain Ω2 where

Ω2 := (x, y) : ζ(x) < y < η(x).Figure 1 provides a useful graphical representation of the above system.

y = η(x)

y = ζ(x)

x-

6y

Ω2

Ω1

Figure 1. Domain with non-flat bottom and non-flat free surface

The steady Euler’s equations for incompressible and irrotational flow govern the motion in ourtwo-fluid system. Within the region Ωi (for i = 1, 2), we have:

(1.1a) ∂xui + ∂yvi = 0

(1.1b) ∂yui − ∂xvi = 0

(1.1c) ui∂xui + vi∂yui = − 1

ρi∂xPi

(1.1d) vi∂xui + vi∂yvi = − 1

ρi∂yPi + g

2

where ui is the i-th fluid velocity in the horizontal x-direction, vi the velocity in the vertical y-direction, Pi the pressure, and g the gravitational constant of acceleration. In addition, we haveBernoulli’s law which states that:

(1.2) u2i + v2i + 2gρiy + 2Pi = Qi,

where Qi is an arbitrary constant, throughout the fluid. On the boundary, we have:

(1.3a) P1 = P2 on y = η

(1.3b) vi = uiηx on y = η

(1.3c) v2 = u2ζx on y = ζ

(1.3d) u1 → U as y →∞.(1.3a) is the dynamic boundary condition which requires that the pressure, in the absence ofsurface tension, be continuous across the free-surface. (1.3b) is the kinematic boundary conditionthat necessitates that a particle which starts on the free-surface stays there. (1.3c) and (1.3d)encapsulate the zero normal-flow conditions at y − ζ and as y →∞.

Altogether, Euler’s equations in (1.1)–(1.3) are a complicated system that involves multipleunknowns. With some manipulation — which we flesh out in Section 3 — Euler’s equations canbe rewritten in a more tractable form. First, the incompressibility condition, ux + vy = 0, permitsus to introduce the pseudo relative stream function, ψ, defined, up to a constant, as

ψx =√ρv and ψy = −√ρu.

In terms of the stream function, our original system in (1.1)–(1.3) is equivalent to:

(1.4a) ∆ψ1 = 0 in Ω1

(1.4b) ∆ψ2 = 0 in Ω2

(1.4c)1

2(|∇ψ1|2 − |∇ψ2|2) + g(ρ1 − ρ2)η = Q on y = η

(1.4d) ψ1 = ψ2 = C on y = η

(1.4e) ∂yψ1 =√ρ1U as y →∞

(1.4f) ψ2 = 0 on y = ζ

where U is the horizontal velocity of the fluid in the upper region as y → ∞ and Q and C arearbitrary constants.

Second, we observe that while the homogeneous Laplace equations in (1.4) is well posed, we donot know, a priori, the domains Ω1 and Ω2. To get around this difficulty, we would like to pushthe problem to the free-surface and hence restate the system in terms of η and C — the trace ofψ on y = η. Thus, to further simplify (1.4), we can introduce the Dirichlet-Neumann operators,N1 : X × R→ L (R, Y ) and N2 : X × R→ L (R, Y ), defined as:

N1(η, U)C :=∂Ψ1

∂n|y=η and N2(η, ζ)C :=

∂Ψ2

∂n|y=η.

Here, X and Y denote the Banach spaces:

X := C2+α(R), 2π-periodic, even functionsY := C1+α(R), 2π-periodic, even functions,(1.5)

3

Ψ1 solves the system:

∆Ψ1 = 0 in Ω1

Ψ1 = C on y = η(1.6)

∂yΨ1 =√ρ1U as y →∞,

and Ψ2 solves the system:

∆Ψ2 = 0 in Ω1

Ψ2 = C on y = η(1.7)

∂yΨ2 =√ρ1U as y →∞.

(1.4) is, in turn, equivalent to:

(1.8)1

2(|N1(η, U)C|2 − |N2(η, ζ)C|2) + g(ρ1 − ρ2)η = Q.

Therefore, in order to prove the existence of steady waves, we seek solutions:

(η, ζ, U, C,Q) ∈ X ×X × R× R× Rsuch that (1.8) holds.

In line with the classical problem, our first theorem proves the existence of steady waves inregions with a non-flat free-surface and flat bottom:

Theorem 1.1 (Existence of small amplitude flows in a region with a non-flat free surface and aflat bottom). Consider the case where the impermeable bottom is identically flat: ζ ≡ −h. Thereexists δ > 0 and a C1 curve of solutions, parametrized by C, given by:

C = (η(C), C,Q(C)) : C ∈ (C∗ − δ, C∗ + δ) ⊂ X × R× R.bifurcating from the trivial solution at (0, C∗, Q(C∗)) such that (1.8) holds.

We construct, in Section 6, an approximate model to investigate the case where the bottom isnot identically flat. Following the method of Duchene [1], we let ζ = −h + εζ(1) and perform anasymptomatic expansion on N2. We obtain an approximate expression:

(1.9)1

2(|N1(η, U)C|2 − |N app

2 (η, ζ)C|2) + g(ρ1 − ρ2)η = Q.

where N app2 is a linear in ζ approximation of N2 near ζ = −h. See Section 6 for details.

Theorem 1.2 (Non-existence of small-amplitude flows with non-flat bottom and flat free surface).Consider the approximate problem in a two-fluid region that consists of a flat free interface and anon-flat bottom, and extends infinitely above. There does not exist any steady, incompressible andirrotational flow in such region.

Theorem 1.3 (Existence of small amplitude flows in a region with a non-flat free surface and anon-flat bottom). There exists small amplitude steady waves in a region with a non-flat free surfaceand non-flat bottom. More precisely, there exist a δ > 0 and a curve of solutions (C, η(C)) suchthat (1.9) holds for all C ∈ (−δ, δ).

Looking ahead, the structure of this paper is as follows. In Section 2, we first acquaint thereader with the mathematical background necessary to understanding the rest of this paper. Next,in Section 3, we prove the equivalence among (1.1)-(1.3), (1.4), and (1.9) and, in Section 4, flattenthe domains Ω1 and Ω2 to a strip. At this point, we have all the required tools to furnish the proofof Theorem 1.1; we do so in Section 5. We also presents some numerical results in this section.Moving on, we derive our approximate model in Section 6. Finally, we provide, in Section 7, theproofs of our Theorems 1.2 and 1.3.

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2. Mathematical background

This section provides a brief overview of the key concepts covered in this paper.

2.1. Banach spaces.

Definition 2.1. (Convergent and Cauchy sequences). Let V be a normed vector space, x ∈ V anda xn ⊂ V be a sequence in V .

(1) We say that xn converges to x in norm provided that, for all ε > 0, there exists N > 0such that ‖xn−x‖ < ε for all n > N. A sequence xn ⊂ V is said to be convergent if thereexists x ∈ V that xn approaches in norm.

(2) A sequence xn ⊂ V is said to be Cauchy provided that, for all ε > 0, there exists N > 0such that |xn − xm‖ < ε for all m,n > N.

Definition 2.2. (Banach Spaces) A normed vector space in which every Cauchy sequence is con-vergent is called a Banach space.

Definition 2.3. (Maps between Banach spaces) If X and Y are Banach spaces with norms ‖ · ‖Xand ‖ · ‖Y , respectively, then a function F : X → Y is said to be continuous if for every x1, x2 ∈ X,and every ε > 0, there exists δ > 0 such that

‖x1 − x2‖X < δ implies ‖F(x1)−F(x2)‖Y < ε.

This is denoted C0(X;Y ) for a set of continuous mappings from X to Y .

2.2. The Frechet derivative.

Definition 2.4. (The Frechet derivative) Let X and Y be Banach spaces and let F ∈ C(X;Y ) begiven. We say that F is Freechet differentiable at x ∈ X provided that there exists a bounded linearoperator Lx : X → Y such that

limh→0

‖F(x+ h)−F(x)− Lx(h)‖Y‖h‖X

= 0,

where ‖ · ‖X and ‖ · ‖Y are the norms on X and Y , respectively, and we denote (DF)(x) := Lx orsometimes DxF(x). If F is differentiable at every x ∈ X, we say that it is Frechet differentiable.If, for each x ∈ X, the derivative DxF(x) is continuous, we say that F ∈ C1(X;Y ).

2.3. Implicit Function Theorem.

Theorem 2.1 (Implicit function theorem on Banach spaces). Let W,X and Y be Banach spacesand F ∈ C1(W ×X;Y ). Suppose that for some (w0, x0) ∈W ×X we have

F(w0, x0) = 0, and DxF (w0, x0) is bijective.

Then there exists an open set O ⊂W with w0 ∈ O, and a C1 function ϕ : O → X such that

ϕ(w0) = x0,

and

F(w,ϕ(w)) = 0, for all w ∈ O.

Moreover, if F(w, x) = 0 and w ∈ O, then x = ϕ(w).

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2.4. Bifurcation theory.

Definition 2.5. Let X and Y be Banach spaces and suppose that L : X → Y is a bounded linearoperator. Suppose that

dim N (L) <∞, dim R(L)c <∞,where N (L) is the null space and R(L)c is the complement of the range. If we also know that therange is a closed set in Y , then we say that L is a Fredholm operator.

Definition 2.6. Let L be a bounded linear operator between Banach spaces. If L is Fredholm, wedefine the index of L, denoted indL, to be the integer

indL := dim N (L)− cod R(L).

Theorem 2.2 (Crandall–Rabinowitz). Let X and Y be Banach spaces and let F : R×X → Y bea Fredholm operator of index 0. Suppose that for all λ,

F(λ, 0) = 0,

and that there exists λ0 such that

N (DxF(λ0, 0)) is one-dimensional.

If

DλDxF(λ0, 0)u 6∈ R(DxF(λ0, 0)),

for any u ∈ N (DxF(λ0, 0)), then, there exists ε > 0 and a C1 curve

C = (λ, x(λ)) : λ ∈ (λ0 − ε, λ0 + ε) ⊂ R×X,

such that x(0) = 0 and F(λ, x) = 0 for each (λ, x) ∈ C . Moreover, x(λ) 6= 0 for λ 6= λ0.

3. Equivalent formulation of the problem

As previously discussed, Euler’s equations in (1.1)–(1.3) are an unwieldy system to work with.Instead, it is more convenient to conduct our analysis on (1.8). Thus, in this section, we prove theequivalence among Eulers equations in (1.1)–(1.3), the stream function system in (1.4), and theformulation in terms of the Dirichlet-Neumann operators in (1.8). The blueprint for this sectionis as follows. We will show, separately, the equivalence between (1.1)–(1.3) and (1.4), and thatbetween (1.4) and (1.8). Combined, these two results prove the equivalence among the threedifferent formulations of the problem.

Proposition 3.1. Euler’s equations in (1.1)–(1.3) are equivalent to (1.4)

Proof. We first show that (1.1)–(1.3) imply (1.4). (1.4a) and (1.4b) arise because of the irrotation-ality condition which necessitates that

∂yui + ∂xvi =1√ρi

∆ψi = 0.

Rewriting Bernoulli’s law in (1.2) in terms of the pseudo sream function, we have:

1

2|∇ψ1|2 + gρ1η −Q1 = −P1 on y = η

1

2|∇ψ2|2 + gρ2η −Q2 = −P2 on y = η.

6

Coupled with the dynamic boudary condition, P1 = P2, we can readily check that (1.4c) holds,with Q = Q1 −Q2. Moreover, once we state (1.3b)–(1.3d) in terms of the stream function:

∂xψi = −∂yψiηx on y = η

∂xψ2 = −∂yψ2ζx on y = ζ

∂yψ1 →√ρ1U as y →∞,

it becomes evident that ψi must be constant on the free-surface and at bottom; we can pickconstants such that (1.4d) – (1.4f) hold.

We next show that (1.4) implies (1.1)–(1.3). Suppose we begin with a solution ψi to the systemin (1.4) and define the velocity field by (u, v) = 1√

ρ∇⊥ψ. Moreover, we can define the gradient of

its pressure to be:

ρi(ui∂xui + vi∂yui) = −∂xPiρi(vi∂xui + vi∂yvi − g) = −∂yPi.

Then, we can confirm that the irrotationality condition and Euler’s momentum equations followdirectly from the homogeneous Laplace equation. In addition, we can always pick constants Q in(1.4c) such that the dynamic boundary condition, P1 = P2, and Bernoulli’s law hold. Lastly, todervive the remaining boundary conditions in (1.3b)–(1.3d), we take tengential derivatives of thetraces of ψi on the respective boundaroes.

The above results prove the equivalence betwwen Euler’s equations and the stream functionsystem. We refer the reader to Yih [6] for a more detailed discussion on the pseudo relative streamfunction.

Proposition 3.2. The system in (1.4) is equivalent to (1.8).

Proof. We can decompose the trace ∇Ψi on the interface η into its normal and tangential compo-nents:

∇Ψi = (∇Ψi · n)n + (∇Ψi · t)t

where n is the outward unit normal vector and t is the unit tangent vector. Now consider (1.4c).Because Ψi = C on η, the tangential component of its gradient vanishes, and thus

|∇Ψi|2 = |∇Ψi · n|2 on y = η

which in turn allows us to rewrite (1.4c) as:

1

2(|∇Ψ1 · n|2 − |∇Ψ2 · n|2) + g(ρ1 − ρ2)η = Q.

The equivalence of this to (1.8) follows from our definitions of the Dirichlet-Neumann operators.

Remark 1. On occasion, it is more convenient to work with the non-normalized Dirichlet-Neumannoperator

G(η)C =√

1 + η2xN (η)C.

4. Flattening the domain

It is difficult to obtain expressions for N1(η, U) and N2(η, ζ) as the systems in (1.6) and (1.7) areunwieldy to solve for general η and ζ. It is therefore convenient to introduce a change in coordinatesto flatten the domain to a strip. Before we proceed with this flattening, it is helpful to observe thatthe systems in (1.6) and (1.7) are independent and we can, thus, flatten the respective domainsseparately.

7

4.1. Fluid in the upper region. To simplify the problem in (1.6), we let ξ = Ψ1 +√ρ1Uy and

observe that the original problem is equivalent to

∆ξ = 0 in Ω

ξ = C +√ρ1Uη on y = η(4.1)

ξy = 0 as y →∞.

Let w = x and z = y − η. With this change of variables, we note that the set y = η(x)corresponds to the set z = 0. We aim to rewrite the system in (1.6) in terms of the newvariables. Calculating the partial derivatives, we have, by the chain rule:

∂x = ∂w − ηw∂z and ∂y = ∂z.

Thus,

(4.2) ∆(x,y) = ∂2w + ∂2z − 2ηw∂w∂z − ηww∂z + η2w∂2z .

We denote the operator in (4.2) as L1.Therefore, (4.1) written in the (w, z)-variables is:

L(w,z)ξ = 0 in Ω

ξ = C +√ρ1Uη on z = 0(4.3)

ξz = 0 as z →∞

where ξ is a function in the new w and z variables.Also, the Dirichlet-Neumann operator. G1(η, U), written in our new variables is:

G1(η, U)C = [∂n(x,y)Ψ1]|y=η(x)

= [(−ηw(∂w + ηw∂z) + ∂z)Ψ1]|z=0

= [(−ηw(∂w + ηw∂z) + ∂z)(ξ −√ρ1U(z + η))]|z=0.

4.2. Fluid in the lower region. Let w = x and z = −h(y−η(x))ζ(x)−η(x) . With this change of variables,

we note that the sets y = η(x) and y = ζ(x) correspond to the sets z = 0 and z = −hrespectively. We rewrite the system in (1.7) in terms of the new variables. Calculating the partialderivatives, we have, by the chain rule:

∂x = ∂w +hηw − z(ζw − ηw)

ζ − η∂z and ∂y = − h

ζ − η∂z.

Thus,

(4.4) ∆(x,y) = (∂w +hηw − z(ζw − ηw)

ζ − η∂z)

2

+ (− h

ζ − η∂z)

2

.

We denote the operator in (4.5) as L2.Therefore, our system in (1.7) reduces to:

L2Ψ = 0 in Ω

Ψ = C on z = 0(4.5)

Ψ = 0 on z = −h

where Ψ is Ψ expressed in the new (w, z)-variables.8

Also, the Dirichlet-Neumann operator. G2(η, ζ), written in our new varibles is:

G2(η, ζ)C = [∂n(x,y)Ψ2]|y=η(x)

= [(−ηw∂w −hη2w − zηw(ζw − ηw)− h

ζ − η∂z)Ψ2]|z=0.

5. Existence of small amplitude flows in a region with a non-flat free surfaceand a flat bottom

In this section, we investigate the classical case where the region Ω2 lies above an impermeableflat bottom, ζ ≡ −h (See Figure 1). To simplify our analysis, we consider the single fluid case (i.e.ρ1 = 0). Under this scenario, (1.8) reduces to:

1

2|N2(η,−h)C|2 + gρ2η = Q.(5.1)

With this in mind, it is useful to define a function, F , as

F(η, C,Q) :=1

2|N2(η,−h)C|2 + gρ2η −Q(5.2)

whose zero set corresponds to the solution set, (η, C,Q) ∈ X × R× R, of (5.1).

When η ≡ 0, we note that there exist trivial laminar flows. By (4.5), Ψ2 = C(h+z)h and, hence,

N2(0,−h)C = Ch . Therefore, the set of trivial solutions is given by:

(0, C,Q(C)) ∈ X × R× R,

where Q(C) = C2

2h2. We would like to show that there exist small amplitude flows such that η 6≡ 0.

Proof of Theorem 1.1. We look for solutions to (5.1) near η = 0, C = C0, and Q =C2

02h2

. Takingthe Frechet derivative of F with respect to η, and evaluating it at (0, C0, Q(c0)), we have:

DηF(0, C0, Q(C0))v = N (0,−h)C0〈DηN2(0,−h)C0, v〉+ gρ2v.

Here:

〈DηN2(0,−h)C0, v〉 = [∂z˙Ψ2 −

v

h∂zΨ2]z=0

where:

Ψ2 =C0(h+ z)

h

and, from (4.5), ˙Ψ2 solves the system:

∆ ˙Ψ2 =

[2

h(h+ z)vw∂w∂z +

1

h(h+ z)vww∂z − v∂2z

]Ψ2

˙Ψ2 = 0 on z = 0

˙Ψ2 = 0 on z = −h.Thus:

˙Ψ2 = −∑k≥0

C0

h2vk [h cosh(kz) + h coth(kh) sinh(kz)− (h+ z)] cos(kw).

In summary:

DηF(0, C0, Q(C0))v = −C20

h2

∑k≥0

[k coth(kh)− gh2

C20

]vk cos(kw).(5.3)

9

For C0 such that k coth(kh) − gh2

C20

6= 0, DηF(0, C0, Q(C0))v is an isomorphism. We can thus

conclude by Implcit Function Theorem that there exists δ > 0 and a C1 curve

C = (η(C), C,Q(C)) : C ∈ (C0 − δ, C0 + δ) ⊂ X × R× R

such that (5.1) holds.

Conversely, for C0 such that k coth(kh) − gh2

C20

= 0, DηF(0, C0, Q(C0))v fails to be an isomor-

phism and the Implicit Function Theorem, consequently, fails to prove the existence of non-trivialsolutions. In this case, we would need to use the Crandall–Rabinowitz theorem to establish theexistence of a curve bifurcating from the trivial solutions. Without loss of generality, assume thatthere exists a

C∗ =

(ρgh2

coth(h)

) 12

.(5.4)

such that (5.3) vanishes for k = 1. Over the next three lemmas, we will verify the hypothesis ofthe Crandall–Rabinowitz theorem and thus prove the existence of small amplitude flows over a flatbottom.

Lemma 5.1. N (L) = span v∗, where v∗(x) = cosx.

Proof. From our definition of C∗ in (5.4), and the computation of DηF in (5.3), we see that the null

space of L is nontrivial becauseρgh2

coth(h)vanishes. More specifically, because it vanishes at k = 1,

cosx will be the null space. So, N (Fη(0, Q0(C∗))) = Spanv∗, where v∗(x) = cosx. Therefore,dim N (L) = 1 since v∗ vanishes only at k = 1.

Next, we show that L is a Fredholm operator of index 0 (cf. Definition 2.6).

Lemma 5.2. The range of L is span v∗⊥.

Proof. To identify the range of L, we want to find for which u ∈ Y one can solve:

DηF(0, C,Q0(C∗))v = u, for u ∈ Y.

Taking the Fourier transform yields:

(5.5) − C2∗h2

∑k≥0

(k coth(kh)− gh2

C2∗

)vk cos(kw) =

∑k

uk cos(kw).

If k coth(kh)− gh2

C2∗

is non-zero for some k ≥ 0, then we can solve for vk. Since this quantity is 0 for

k = 1, we have u1 = 0. If u1 6= 0, then there is no v satisfying (5.5) can exist. Therefore, we can

solve for v if and only if u is orthogonal to cos(w). Therefore, the range of L will equal span v∗⊥and we have that dim span v∗ = 1.

Lemma 5.2 implies that the Fredholm index is 0 as dim span v∗ = cod R = 1. Lastly, we needto verify the transcritical condition of the Crandall–Rabinowiz Theorem 2.2, which is done in thefollowing lemma.

Lemma 5.3. DCDηF(0, Q0(C∗))v∗ /∈ R(DηF(0, Q0(C∗))); it satisfies the transcritical condition.

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Proof. Taking the derivative of DηF with respect to C, and then evaluating at (η = 0, Q0(C∗)) wefind:

DCDηF(0, Q0(C∗))v∗ = DC

[−C

2∗h2

∑k

(k coth(kh)− ρgh2

C2∗

)v∗k cos(kw)

]

= −2C∗h2

∑k

(k coth(kh)− ρgh2

C2∗

)v∗k cos(kw) +

C2∗h2

∑k

(ρgh2

C3∗· 2C∗

)v∗k cos(kw)(5.6)

= −2C∗h2

∑k

[(k coth(kh)− ρgh2

C2∗

)+ ρg

]v∗k cos(kw).

So by taking inner product with cos(w),

(DCDηF(0, Q0(C∗))v∗, cos(w))L2 = cothhπ 6= 0

Hence, DCDηF /∈ R(DηF), by Lemma 5.2.

From our first two lemmas, we have shown that the dimension of the null space and the codi-mension of the range both equal 1. From Definition 2.6, we have

indL = 1− 1 = 0,

which proves that DηF(0, Q0(C∗)) is a Fredholm operator of index 0. Moreover, the transcriticalcondition of the Crandall–Rabinowitz Theorem is satisfied by Lemma 5.3. Therefore, by Crandall–Rabinowitz, there exists a curve of solutions bifurcating from T at (0, Q0∗(C∗)). There exists δ > 0and a C1 curve

C = (η(C), C,Q(C)) : C ∈ (C∗ − δ, C∗ + δ) ⊂ X × R× R.

5.1. Numerical analysis of solutions. After we have proven existence of solution for the ap-proximate problem presented in Section 5, we show in this section what these solutions look likeby using AUTO. AUTO-07p, the newest version of AUTO, is a package that runs on Linux/Unix,Mac OS X and Windows. After proper installation, AUTO-07p can give a bifurcation analysis foralgebraic systems. For a detailed overview of AUTO’s capabilities, please visit their website athttp://indy.cs.concordia.ca/auto/.

Before using AUTO to find bifurcation branches, we need to simplify our equations into somethingthat AUTO can understand. Recall that the equation we need to solve is the following

1

2|N (η)C|2 + ρgη + σκ−Q = 0,

where κ is the mean curvature and σ > 0 is the coefficient of surface tension. We include thesurface tension to simplify the numerics; and take σ = 1. Multiplying both sides by 2 and writingthe Dirichlet-to-Neumann operator as its Fourier cosine series, we have[

C

h+C

h

∞∑k=0

k coth(kh)ηk cos(kx)

]2+ 2ρgη − 2ση′′ − 2Q = 0,

where −η′′ is the approximation of κ It follows that

C2

h2

1 + 2

∞∑k

coth(kh)ηk cos(kh) +

∞∑k=0

∞∑j=0

k coth(kh)ηkj coth(kh)ηj cos(kx) cos(jx)

+2ρgη − 2ση′′ − 2Q = 0.

11

Moreover, taking inner product in L2x of the left hand side with cos(`x), and using the fact that

cos(kx) cos(jx) =1

2[cos(k + j)x+ cos(k − j)x] ,

we get(5.7)

C2

h2

δ0,` + 2` coth(`h)η` +1

2

∑`=k+j

or `=|k−j|

k coth(kh)j coth(jh)ηkηj

+ 2ρgη` + 2σ`2ηl − 2Qδ0,` = 0.

For example, for ` = 0, we have k = 1 and j = 1. Thus, the above expression is the same as

C2

h2

[1 +

1

2coth2(h)η21

]− 2Q = 0,

which is consistent with the results shown in section 5. That is, for η1 = 0, the trivial solutionsatisfies

Q =1

2

C2

h2.

Moreover, for ` = 1, k = 0 and j = 1 (which gives the same outcome as k = 1 and j = 0). Then5.7 becomes [

C2

h2coth(h) + g + σ

]η1 = 0.

It follows that η1 = 0 and we get the trivial solution again. If we want to approximate η up to η2,then we have to solve the following system of equations

C2

h2[4 coth(2h)η2 + coth2(h)η21

]+ 2gη2 + 8ση2 = 0 for ` = 1 + 1 = 2

C2

h2

2 coth(h)η1 +

1

2[2 coth(2h) coth(h)η2η1]

+ 2gη1 + 2ση1 = 0 for ` = 2− 1 = 1

C2

h2

1 +

1

2

[coth2(h)η21 + 4 coth2(2h)η22

]− 2Q = 0 for ` = 2− 2 = 1− 1 = 0.

At this point, we can use AUTO-07p to analyse the above algebraic system. First, we need totransform the system into something that AUTO-07p can understand. The input to AUTO-07p iswritten in Fortran 90 and the corresponding name for the unknowns are given below.

C η1 η2 QPAR(1) U(1) U(2) U(3)

Table 1.

The source code is shown below as reference.

F(1)= (PAR(1)**2)*(1.00 + (U(1)**2)/(DTANH(1D0)**2) + 4.00*(U(2)**2)/(DTANH(2D0)**2))

- 2*U(3)

F(2)= (PAR(1)**2)*(-2.00*U(1)/DTANH(1D0) + U(1)*U(2)/(DTANH(2D0)*DTANH(1D0)))

+ 20.00*U(1) + 2.00*U(1)

F(3)= (PAR(1)**2)*(-4.00*U(2)/DTANH(2D0) + (U(1)**2)/(DTANH(1D0)**2))

+ 20.00*U(2) + 8.00*U(2)12

AUTO is then told to vary C, solve for η1, η2 and Q, and look for bifurcation branches, startingat C = 2.5, η1 = 0, η2 = 0, and Q = 3.1. The results are attached in the appendix.

Figure 2. Bifurcation diagram

We can also produce graphs of the corresponding solutions using Matlab. AUTO outputs valuesof η1 and η2 along the bifurcation curve, and Figure 2 is the graph of η1 cos(x) + η2 cos(2x) forthese values. Similarly, Figure 3 shows results for the second bifurcation curve. In the appendix,we include the computation of solutions for the three remaining curves.

Figure 3. Solution Branch #1

6. Approximate model

In order to investigate the case where the bottom, ζ, is not identically flat, we seek asymptoticexpansions, near η = 0 and ζ = −h, for N2(η, ζ).

13

6.1. Approximate solutions. We look for approximate solutions of the form

Ψapp = Ψ(0) + εΨ(1).

(It is worth noting that Ψ(0) is the solution when ζ = −h.) To this end, we first expand the operators(4.4) in terms of ε. Next, we plug in our approximate solution and solve for the respective orders

of ε to obtain expression for each Ψ(i).We perform a Taylor series expansion of the operators in (4.4). Let ζ = −h + εζ(1). Hence, we

know that:

∂2x = (∂w +hηw − z(ζw − ηw)

ζ − η∂z)

2

= (∂w +(h+ z)ηw − zεζ(1)w

−h+ εζ(1) − η∂z)

2

= (∂w −1

(h+ η)2[(h+ z)ηw − zεζ(1)w ][h+ η + εζ(1)]∂z)

2

+O(ε2)

= (∂w −1

(h+ η)2[(h+ z)(h+ η)ηw + ε[(h+ z)ηwζ

(1) − (h+ η)zζ(1)w ]]∂z)2

+O(ε2).

By the above expression, we mean that ∂2x equals the operator in parenthesis plus ε2 multiplying abounded operator. So, dropping O(ε2) terms, we have:

= ∂2w −2

(h+ η)2[(h+ z)(h+ η)ηw + ε[(h+ z)ηwζ

(1) − (h+ η)zζ(1)w ]]∂w∂z

− ∂w[1

(h+ η)2[(h+ z)(h+ η)ηw + ε[(h+ z)ηwζ

(1) − (h+ η)zζ(1)w ]]]∂z

+1

(h+ η)4[(h+ z)(h+ η)ηw + ε[(h+ z)ηwζ

(1) − (h+ η)zζ(1)w ]]2∂2z

+2

(h+ η)4[(h+ η)ηw + ε[ηwζ

(1) − (h+ η)ζ(1)w ]][(h+ z)(h+ η)ηw + ε[(h+ z)ηwζ(1) − (h+ η)zζ(1)w ]]∂z.

Expanding the expression further, we get,

= ∂2w −2

(h+ η)2[(h+ z)(h+ η)ηw]∂w∂z − ε

2

(h+ η)2[(h+ z)ηwζ

(1) − (h+ η)zζ(1)w ]∂w∂z

− 1

(h+ η)2[(h+ z)[(h+ η)ηww + η2w]]∂z +

2

(h+ η)3(h+ z)(h+ η)η2w∂z

− ε 1

(h+ η)2[(h+ z)(ηwwζ

(1) + ηwζ(1)w )− ηwzζ(1)w − (h+ η)zζ(1)ww]∂z

+ ε2

(h+ η)3[(h+ z)η2wζ

(1) − (h+ η)zηwζ(1)w ]∂z

+1

(h+ η)4[(h+ z)(h+ η)ηw]2∂2z + ε

2

(h+ η)4[(h+ z)(h+ η)ηw][(h+ z)ηwζ

(1) − (h+ η)zζ(1)w ]∂2z

+2

(h+ η)4(h+ z)(h+ η)2η2w∂z + ε

2

(h+ η)4(h+ η)ηw[(h+ z)ηwζ

(1) − (h+ η)zζ(1)w ]∂z

+ ε2

(h+ η)4[ηwζ

(1) − (h+ η)ζ(1)w ](h+ z)(h+ η)ηw∂z

14

Furthermore,

∂2y = [− h

ζ − η∂z]

2

= [− h

−h+ εζ(1) − η∂z]

2

= [h

(h+ η)2(h+ η + εζ(1))∂z]

2

+O(ε2)

=h2

(h+ η)2∂2z + ε

2h2

(h+ η)3ζ(1)∂2z +O(ε2).

Combining our above results, (4.4) becomes

L(w,z) = L(0) + εL(1) +O(ε2)

where

L(0) = ∂2w −2

(h+ η)2[(h+ z)(h+ η)ηw]∂w∂z −

1

(h+ η)2[(h+ z)[(h+ η)ηww + η2w]]∂z

+2

(h+ η)3(h+ z)(h+ η)η2w∂z +

1

(h+ η)4[(h+ z)(h+ η)ηw]2∂2z

+2

(h+ η)4(h+ z)(h+ η)2η2w∂z +

h2

(h+ η)2∂2z ,

and

L(1) = − 2

(h+ η)2[(h+ z)ηwζ

(1) − (h+ η)zζ(1)w ]∂w∂z +2

(h+ η)3[(h+ z)η2wζ

(1) − (h+ η)zηwζ(1)w ]∂z

− 1

(h+ η)2[(h+ z)(ηwwζ

(1) + ηwζ(1)w )− ηwzζ(1)w − (h+ η)zζ(1)ww]∂z

+2

(h+ η)4[(h+ z)(h+ η)ηw][(h+ z)ηwζ

(1) − (h+ η)zζ(1)w ]∂2z

+2

(h+ η)4(h+ η)ηw[(h+ z)ηwζ

(1) − (h+ η)zζ(1)w ]∂z

+2

(h+ η)4[ηwζ

(1) − (h+ η)ζ(1)w ](h+ z)(h+ η)ηw∂z +2h2

(h+ η)3ζ(1)∂2z .

To solve for Ψ(0)2 and Ψ

(1)2 , we plug our approximate solution into (4.5) and consider the respective

O(εi) terms. For O(1), we have:

L(0)Ψ(0)2 = 0

Ψ(0)2 = C on z = 0(6.1)

Ψ(0)2 = 0 on z = −h.

For O(ε), we have:15

L(0)Ψ(1)2 = −L(1)Ψ(0)

Ψ(1)2 = 0 on z = 0(6.2)

Ψ(1)2 = 0 on z = −h.

6.2. Approximate expression for the Dirichlet-Neumann operator. Using the results fromthe previous section, we now have an approximation for the Dirichlet-to-Neumann operator, G2:

G2(η,−h+ εζ(1))C = [∂n(x,y)Ψ2]|y=η(x)

= [(−ηx∂x + ∂y)Ψ2]|y=η(x)

= [(−ηw[∂w −1

(h+ η)2[(h+ z)(h+ η)ηw + ε[(h+ z)ηwζ

(1) − (h+ η)zζ(1)w ]]∂z]

+h

(h+ η)2[h+ η + εζ(1)]∂z)(Ψ

(0)2 + εΨ

(1)2 )]|z=0 +O(ε2)

= [−ηw∂wΨ(0)2 +

1

h+ η[(h+ z)η2w + h]∂zΨ

(0)2 − ε(ηw∂wΨ

(1)2 −

1

h+ η[(h+ z)η2w + h]∂zΨ

(1)2 )

+ ε1

(h+ η)2[(h+ z)η2wζ

(1) − (h+ η)ηwzζ(1)w + hζ(1)]∂zΨ

(0)2 ]|z=0 +O(ε2)

Define Gapp2 to be the operator found by taking the expression for G2 above and dropping the O(ε2)

terms. We define N app2 analogously. The approximate model (1.9) is found by replacing N2 and G2

by the approximate operators:

7. Small amplitude flows over a non-flat bottom

Having derived our approximate model in Section 6, we are now in a position to investigate smallamplitude flows over an impermeable non-flat bottom, −h+ εζ(1). In order to ultimately prove theexistence of small amplitude, small circulation flows over a region bounded by a non-flat free surfaceand a non flat-bottom, we first prove an auxiliary, albeit important, result: the non-existence ofsmall amplitude flows over a region bounded by a flat free surface and a non flat-bottom.

7.1. Non-existence of small-amplitude flows with non-flat bottom and flat free surface.

Proposition 7.1. Consider the approximate problem in a fluid region that is bounded by a flat freesurface and a non-flat bottom, there does not exist any steady, incompressible and irrotational flowin such region.

We will use this proposition to prove Theorem 1.2, which states the non-existence of small-amplitude flows in a two-fluid region with non-flat bottom and flat free surface. The proof for thistheorem can be found in Section 7.1.1. The proof of the above proposition is as follows:

Proof. Suppose that free surface is flat, i.e. η ≡ 0. We need only substitute η = 0 into (6.1) and(6.2), and we get

∆(w,z)Ψ(0) = 0(7.1)

Ψ(0) = 0 on z = −h

Ψ(0) = C on z = 0 ,16

which has a unique solution

(7.2) Ψ(0) =C(z + h)

h.

Also, Ψ(1) solves the system

−∆(w,z)Ψ(1) =

[2z

hζ(1)w ∂zw +

z

hζ(1)ww∂z +

2

hζ(1)∂2z

]Ψ0)(7.3)

Ψ(1) = 0 on z = −h

Ψ(1) = 0 on z = 0

Substituting the solution for Ψ(0) into (7.3), we get

∆(w,z)Ψ(1) = −Cz

h2ζ(1)ww.

Note that the right hand side can be rewritten

∆(w,z)Ψ(1) = −Cz

h2ζ(1)ww −

Cz

h2ζ(1)zz

= −∆(w,z)Cz

h2ζ(1)

Let u := Ψ(1) + Czζ(1)/h2, then we have that

∆(w,z)u = 0.

Moreover, the boundary conditions for Ψ(1) becomes

u = −Chζ(1) on z = −h

u = 0 on z = 0 .

By assumption, ζ(1), and hence u, is 2π-periodic and even. Expanding both as cosine series gives

u =∞∑k=0

u(k, z) cos(kw) and ζ(1) =∞∑k=0

ζ(1)k cos(kw).

Since u is harmonic, its Fourier cosine series coefficients satisfy the following ODE

−k2u(k) + uzz(k) = 0.

It follows that u is given byu = Ak cosh(kz) +Bk sinh(kz),

for some constants Ak, Bk ∈ R. Imposing the boundary conditions, we find that

u =C

hcsch(kh) sinh(kz)ζ

(1)k .

Substituting the above expression into the definition of u, we can solve for Ψ(1)

Ψ(1) =∞∑k=0

C

h2ζ(1)k [h csch(kh) sinh(kz)− z] cos(kw).

Next, we evaluate the Dirichlet-to-Neumann operator with η = 0 and ζ = −h + εζ(1), and notethat the normalized Dirichlet-to-Neumann operator is the same as the non-normalized one, becauseat η = 0,

N (0, ζ)C =1√

1 + 02G(0, ζ)C = G(0, ζ)C.

17

Thus,

N (0,−h+ εζ(1))C =∂Ψ

∂n

∣∣∣∣y=0

=∂Ψ

∂y

∣∣∣∣y=0

=h

h− εζ(1)[Ψ(0)z + εΨ(1)

z

]z=0

+O(ε2)

= (1 +ε

hζ(1))

[Ψ(0)z + εΨ(1)

z

]z=0

+O(ε2)

= (1 +ε

hζ(1))

C

h+ ε

∞∑k=0

C

h2ζ(1)k (kh csch(kh)− 1) cos(kw)

+O(ε2).

Thus, define the approximate Dirichlet-to-Neumann operator

(7.4) N app(0,−h+ εζ(1))C :=C

h

1 + ε

∞∑k=0

k csch(kh)ζ(1)k cos(kw)

.

Substituting (7.4) into the approximate model gives

(7.5)C

h

∣∣∣∣∣1 + ε∞∑k=0

k csch(kh)ζ(1)k cos(kw)

∣∣∣∣∣ =√Q

for some constants C, h and Q. Since we are only interested in what happens when ε is takensufficiently small, i.e. when

supw

∣∣∣∣∣ε∞∑k=0

k csch(kh)ζ(1)k cos(kw)

∣∣∣∣∣ < 1.

It follows that

1 + ε∞∑k=0

k csch(kh)ζ(1)k cos(kw) > 0.

Therefore, the equality (7.5) is the same the statement as

C

h

(1 + ε

∞∑k=0

k csch(kh)ζ(1)k cos(kw)

)−√Q = 0.

Observe that the inner product in L2w of the left-hand side of the above equality with cos(jw) for

positive integers j also vanishes, i.e.

(7.6)

(C

h

(1 + ε

∞∑k=0

k csch(kh)ζ(1)k cos(kw)

)−√Q, cos(jw)

)L2w

= 0 for j ≥ 0

First, consider when j = 0. Then (7.6) becomes∫ 2π

0

(C

h

(1 + ε

∞∑k=0

k csch(kh)ζ(1)k cos(kw)

)−√Q

)dw = 0

18

Evaluating the integral gives

0 =2πC

h− 2π

√Q+ ε

∞∑k=0

k csch(kh)ζ(1)k

∫ 2π

0cos(kw)dw

=2πC

h− 2π

√Q+ 2πεζ

(1)0 .

It follows that

ζ(1)0 =

h√Q− Cε

,

which equals to a constant. Next, consider when integer j > 0, (7.6) becomes

0 =

(C

h−√Q

)∫ 2π

0cos(jw)dw + ε

∞∑k=0

k csch(kh)ζ(1)k

∫ 2π

0cos(kw) cos(jw)dw

The first integral term vanishes for all j > 0, and the second integral vanishes for all k 6= j, hence,

0 = εj csch(jh)ζ(1)j

∫ 2π

0cos2(jw)dw.

It follows that

ζ(1)j = 0 for all j > 0.

Hence, only the zeroth order coefficient does not vanish for the Fourier cosine series of ζ(1), whichimplies that ζ(1) must equal some constant.

7.1.1. Non-existence of small amplituder flows over region with flat interface and non-flat bottom.See Figure 4. for a picture of the domain.

y = η(x)

y = ζ(x)

x-

6y

Ω2

Ω1

Figure 4. Domain with non-flat bottom and free flat surface

Proof of Theorem 1.2. The proof is almost identical to the one-fluid case. The only differencebetween the two situations is that the Bernoulli condition is now stated as

1

2|∇Φ1|2 −

1

2|∇Φ2|2 + g(ρ1 − ρ2)η = Q.

19

Or equivalently,

|N1(η, U)C|2 − |N app2 (0(η, ζ)C|2 + 2g(ρ1 − ρ2)η −Q′ = 0,

for Q′ = 2Q. Since we are considering the case in which η = 0, the above simplifies further to

|N1(η, U)C|2 − |N app2 (η, ζ)C|2 −Q′ = 0,

where N app2 is the same as the approximate form of Dirichlet-to-Neumann operator shown previ-

ously:

N app2 (0, ζ) = N (0,−h+ εζ(1))C :=

C

h

1 + ε

∞∑k=0

k csch(kh)ζ(1)k cos(kw)

.

and N1 for the system (4.3) can be computed. For η = 0, (4.3) becomes

L(w,z)ξ = 0

ξ = C on z = 0

ξz = 0 as z →∞

Recall that ξ is defined in section 4.1. Solving explicitly, we have

ξ =∑k≥0

√ρ1U [cosh(kz)− sinh(kz)]vk cos(kw).

Thus we get

N1(0, U)C =1√

1 + 02G1(0, U)C = G1(0, U)C = −√ρ1U.

Plugging the two identities into the approximate model, we get

0 = ρ1U2 − C2

h2

(1 + ε

∞∑k=0

k csch(kh)ζ(1)k cos(kw)

)2

−Q′

= ρ1U2 − C2

h2−Q′ − ε

(2C2

h2

∞∑k=0

k csch(kh)ζ(1)k cos(kw)

).

Intuitively, all the Fourier modes vanish except for k = 0 for the above equality to hold. The proofis similar to the proof given for 7.1 and is as follows. Take the inner product of both sides withcos(lw) for l ∈ N, and we get

0 =

∫ 2π

0

(ρ1U

2 − C2

h2−Q′

)cos(lw)dw − 2C2ε

h2

∫ 2π

0

∞∑k=0

k csch(kh)ζ(1)k cos(kw) cos(lw)dw.

For l = 0, we have that cos(lw) = cos(0) = 1. Then, the above becomes

0 = 2π

(ρ1U

2 − C2

h2−Q′

)− 2C2ε

h2

∞∑k=0

k csch(kh)ζ(1)k

∫ 2π

0cos(kw)dw

= 2π

(ρ1U

2 − C2

h2−Q′

)− 2π

2C2εζ(1)0

h3,

which implies that

ζ(1)0 =

ρ1U2h3 − C2h−Q′h3

2C2ε20

is a constant. Next, for l > 0, we have that

0 =

(ρ1U

2 − C2

h2−Q′

)∫ 2π

0cos(lw)dw − 2C2ε

h2

∞∑k=0

k csch(kh)ζ(1)k

∫ 2π

0cos(kw) cos(lw)dw

= −2C2ε

h2l csch(lh)ζ

(1)l

∫ 2π

0cos2(lw)dw,

which implies that

ζ(1)l = 0 for all l > 0.

Hence, it is proven ζ(1) must equal some constant. Moreover, small amplitude flows do not existfor non-flat bottom.

7.2. Existence of small amplitude flows in a region with a non-flat free surface and

a non-flat bottom. We fix ε = ε0, u1 = U , and ζ(1) = ζ(1)0 — where ζ

(1)0 is a non-constant,

2π-periodic, even function. It is appropriate that we define a function, F(η, C,Q), as

F(η, C,Q) =1

2(1 + η2w)[|G1(η)C|2 − |Gapp2 (η)C|2] + g(ρ1 − ρ2)η −Q

whose zero set is exactly the solution to (1.9). At η = 0, C = 0 and Q = 0, we observe that

(7.7) F(0, 0, 0) = 0

and hence (0, 0, 0) is a solution to (1.9).

Proof of Theorem 1.3. At C = 0, we have that, from (6.1) and (6.2) respectively, Ψ(0)2 = 0 and

Ψ(1)2 = 0. Hence, we know that ˙Ψ

(0)2 = 0 and ˙Ψ

(1)2 = 0. Therefore,

(7.8) DηF(0, 0, 0)v =∑k≥0

[ρ1U2k + g(ρ1 − ρ2)]vk cos(kw).

Therefore, for

k0 6=g(ρ2 − ρ2)ρ1U2

where k0 is some arbitrary integer, DηF(0, 0, 0)v is a bijective mapping from X×R×R to Y ×R×R.Therefore, applying the Implicit Function Theorem to (7.7) and (7.8), we can conclude that thereis a curve of solutions (C, η(C), Q(C)) such that (1.9) holds for all C ∈ (−δ, δ)

Furthermore, from Theorem 1.2, we know that, for C 6= 0 and ζ(1)0 6= a where a is some arbitrary

constant, it is impossible to have a flat free surface, η. Thus, η(C) is not identically flat.

Acknowledgement

This research was done under the S.U.R.E program at the Courant Institute of MathematicalSciences. We would like to thank our advisor Dr. Samuel Walsh for his insight and supportthroughout the research. We also appreciate help from Oliver Buler, Matthew Leingang, and BethMarkowitz.

21

References

[1] V. Duchene. Asymptotic shallow water models for internal waves in a two-fluid system with a free surface. SIAMJ. Math. Anal, 42, 2010.

[2] T. Levi-Civita. Determinazione rigorosa delle onde irrotazionali periodiche in acqua profonda. Math. Ann, 93:264–314, 1925.

[3] A. I. Nekrasov. On steady waves. Izv. Ivanovo-Voznesenk. Politekhn, 3, 1921.[4] G. Stokes. On the theory of oscillatory waves. Cambridge Phil. Soc., 8, 1847.[5] D. J. Struik. Determination rigoureuse des ondes irrotationelles periodiques dans un canal a profondeur finie.

Math. Ann, 95:595–634, 1926.[6] C. S. Yih. Dynamics of nonhomogeneous fluids. Macmillan, New York-London, 1965.

22

Appendix A. Numerical Results

Figure 5. output from AUTO

Figure 6. Solution Branch #2

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Figure 7. Solution Branch #3

Figure 8. Solution Branch #4

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