Two-dimensional steady edge waves. Part I: Periodic waves

TWO-DIMENSIONAL STEADY EDGE WAVES.

PART I: PERIODIC WAVES

MATS EHRNSTRÖM, JOACHIM ESCHER, AND BOGDAN-VASILE MATIOC

Abstract. We prove existence and uniqueness for two-dimensional steadywater waves propagating along the beach. For small periodic shorelinedata, global solutions vanishing in the seaward direction are found. Inaddition, we prove a priori properties of solutions, well-adapted to thephysical background.

1. Introduction

The mathematical study of water waves is a rich and challenging topic,which in recent years has seen an intensified interest from mathematiciansworking on nonlinear partial differential equations. In particular, the exactgoverning equations have been investigated in new directions [7, 10, 11, 13,14, 18, 19], and a number of new and promising model equations has beenderived, discussed, and exploited [8, 12, 22, 30, 31].

In spite of this, a type of waves that has yet to be fully understood is theclass of edge waves. Those are regular (essentially) two-dimensional wavespropagating along the beach, and vanishing fast in the direction perpendic-ular to the shoreline [3, 32, 33]. In the first respect they resemble waves ina canal, but there is no finite channel width; edge waves reside on a semi-infinite interval in the direction away from beach.

Possibly due to their unusual nature, there is relatively little mathematicalwork on edge waves, and indeed no general exact existence theory (thoughfrom the more applied point of view, there are many findings). Interestingly,the edge wave problem admits a family of explicit solutions [3,38], recoveredfrom the Gerstner wave [4,25,27], and therefore of rotational nature. Using aformal expansion in the (small) amplitude parameter, Whitham later foundweakly nonlinear solutions [37]. In the recent literature, a rigorous result ona one-dimensional model for edge waves was presented in [29].

The impetus for the present work comes from [32]. There a quasilineartwo-dimensional model equation for edge waves propagating over a slowly

2000 Mathematics Subject Classification. 35B10; 35M10; 76B03; 76B15.Key words and phrases. Existence; Uniqueness; A priori estimates; Water waves; Edge

waves; Periodic solutions.The authors are thankful for constructive suggestions made by the referees. In par-

ticular, their thorough reading of the manuscript helped in improving its quality, andenhanced the hydrodynamical aspects.

1

2 EHRNSTRÖM, ESCHER, AND MATIOC

y = 0

y ց −∞

Figure 1. The edge-wave solution ψ(x, y) = sinx exp y.

varying depth was deduced. Starting with the governing equations for watermotion over a sloping bed profile z = b(y) with b′(0) = 1, the author con-siders a slowly varying slope b(εy). Here z denotes the vertical position, andthe y-variable denotes the position in the seaward direction perpendicular tothe beach. A small amplitude of order δ << 1 is then also considered, andsolutions are sought that can be expanded in terms of δ and ε. By retainingorders up to O(δ) and O(ε), but neglecting those O(δε), a model equationis found. It describes the propagation of steady edge waves in the small-amplitude limit, when the slope of the beach is small and slowly varying. Afinal reduction yields a model equation that can be written in normalizedform as

∆ψ + ∂x|∇ψ|2 = 0,

where ψ is the normalized stream function, i.e. ∇ψ is the velocity field,and the Cartesian coordinates are chosen so that x denotes the positionalong the shoreline, and (as above) y measures the position in the directionperpendicular to the beach. Here both x and y are rescalings of the physicalvariables (for a detailed derivation, cf. [33, Section 9]). An important featureof this model is that it does not presuppose irrotationality, whence waves withvorticity are included in the analysis (cf. also the discussion in [20]). Theauthor notes that, for any constant c ∈ R, the function ψc := c sinx exp yconstitutes a solution. Such a solution functionally agrees with the two-dimensional wave potential for the classical Stokes edge wave (see, e.g., [2,33]), and has a similar functional dependence to the explicit solutions foundin [3]. Notice also that this resembles the behaviour of linear deep-waterwaves [6] (for the corresponding nonlinear waves, cf. [5, 28]). The existenceof other solutions, and their relation to this model solution, is left as an openproblem in [32].

The aim of this investigation is to address this question in a rigorousmanner. In addition to establishing existence of small periodic solutions,we present a number of a priori properties of solutions. In particular, it isshown that solutions with a slope bounded by 45 always vanish fast in theseaward direction; that an odd run-up pattern yields odd solutions boundedin terms of ψc; and that (up to translation) ψc constitute all bounded evensolutions. It is well-known that water waves are often associated with ahighest slope: this could be a limiting wave with a finite slope [1, 15, 35],or a wave developing a vertical slope, beyond which wave-breaking occurs(see e.g. [9, 21, 36] for related questions). It is therefore not surprising that

PERIODIC EDGE WAVES 3

our results, in particular the existence result, are valid for small wave slopes.Indeed, there is a transformation mapping small edge wave solutions to largeones of a different period, while preserving the slope. Though we did notdetermine the greatest possible slope for waves of a certain absolute size, ourresults are consistent with such a bound.

Mathematically, we deploy a range of different methods, the theoretic bulkbeing elliptic estimates. Since the edge wave equation (2.1a) is in generalnot elliptic, it is crucial to reduce the domain of solutions. Notice, though,that we deduce many properties of the wave that apply in the case wherethe equation is not elliptic. A related problem is the unbounded nature ofthe fluid domain; because of this complication, we deduce uniform estimatesin finite domains. Some of the techniques used in this paper are adapted, in-spired, or just resemblant of techniques previously developed by the authors.For example, to prove the structure of odd solutions, we use an approachsimilar to that of [10] (see also [18]); for uniqueness a trick like in [17]; andas in [19]—where the precise form of the stream function is presented—weobtain the precise form of the even solutions. The proof that the solutionvanishes at infinity is based on a concept from [16] and [23] (see also [24,39]).

Notice that the paper at hand is the first part of two concerning the edge-wave equation (2.1a). In a follow-up article [20], we extend existence tosolitary waves. The qualitative validity of the two-dimensional edge-waveequation as a model for water waves is also discussed.

The disposition of part I is as follows. After having established the math-ematical setting in Section 2, the properties of solutions are proved in Sec-tion 3. Section 4 contains the main existence and uniqueness result forperiodic solutions, and in Section 5 we briefly comment on the modellingaspects of equation (2.1a).

2. Preliminaries

We consider solutions ψ ∈ BUC 2+α(Ω) of the edge wave problem

∆ψ + ∂x|∇ψ|2 = 0, (x, y) ∈ Ω, (2.1a)

in the fluid domain

Ω := R × (−∞, 0).

Here, BUC 2+α(Ω) denotes the space of all maps from Ω to R which havebounded and uniformly continuous derivatives up to order 2, and in additionuniformly α−Hölder continuous derivatives of order 2, for some given α ∈(0, 1). When the boundary data is 2L-periodic,

ψ(x+ 2L, 0) = ψ(x, 0) for all x ∈ R, (2.1b)

we shall require that

supx∈R

ψ(x, y) → 0 as y → −∞. (2.1c)

The following result is immediate.


Proposition 2.1. If ψ solves (2.1a), then the transformation

φλ(x, y) := λψ

(

x

λ,y

|λ|

)

, λ 6= 0, (2.2)

also constitutes a solution of (2.1a).

For periodic solutions there is consequently no loss of generality in re-stricting attention to the period 2π. We shall identify the periodic solutionsψ of (2.1) with functions u defined on the semi-infinite cylinder

Σ := S × (−∞, 0).

Here S denotes the unit circle R/2πZ. Thus, from now on, u will alwaysdenote a periodic solution of (2.1a), while ψ shall be used to denote generalsolutions on Ω. Equation (2.1) is then naturally supplemented by an initialcondition

u|y=0 = f on S,

where f ∈ C2+α(S). Since the function f describes a steady wave propagat-ing along the beach, we shall in Section 4 make the normalization

∫

S

f(x)dx = 0. (2.3)

Indeed, for periodic solutions (2.3) constitutes a necessary condition for(2.1c) to hold true, since the conservation law

∫

S

u(x, y) dx =

∫

S

f(x) dx (2.4)

holds for all y ∈ (−∞, 0] (cf. the proof of Theorem 3.2).It should also be noted that though the equation (2.1a) in general changes

type, it is uniformly elliptic as long as

1 + 2ux − u2y ≥ δ > 0. (2.5)

This is the cornerstone of the existence and uniqueness results in Section 4,but many a priori properties will be proved irregardless of (2.5) in Sec-tion 3. Indeed, it can be seen from the proof of Theorem 3.1 ii) that theeven solutions of (2.1a) solve

∆ψ = 0,

∂x|∇ψ|2 = 0.

Up to the addition of any real constant, the entire set of classical solutions ofthis system are the affine functions, together with the family of phase-shiftedsolutions ψ(x, y) = λ exp(cy) sin(cx+β), for any λ, β, c ∈ R. In general, thesesolutions do not satisfy (2.5). Still, as can be seen from Theorem 3.1, theyare in some sense the archetypal solutions of (2.1).


3. A priori properties of solutions

We prove here restrictive properties of solutions, some of which will helpus establish existence of asymptotically vanishing solutions in Section 4.

Theorem 3.1.

i) Let u be a 2π-periodic solution of (2.1) satisfying (2.5). If the bound-ary data f(x) is odd, then u is everywhere odd, and satisfies

|u(x, y)| ≤ ‖f‖C1(S)| sin (x)| exp (y) in Σ. (3.1)

ii) Up to the scaling (2.2) and an additive constant, the even solutionsof (2.1a) are exactly u(x, y) = λ exp(y) cos(x), for any λ ∈ R.

We remark that the inequality (3.1) is the best possible, since the solutionu(x, y) = sinx exp y fulfills it. When the solutions are not odd, one can stillobtain a general result for their asymptotic behaviour:

Theorem 3.2. If u is a 2π-periodic solution of (2.1a) and (2.1b) satisfying1 + ux ≥ δ > 0, then

u(x, y) → c0 :=1

2π

∫

S

f(x) dx as y → −∞,

uniformly for x ∈ S, and

‖u− c0‖2H1(Σ) :=

∫∫

Σ

(

|u− c0|2 + |∇u|2

)

dx dy <∞. (3.2)

If furthermore u satisfies |ux| ≤ 1/4, with boundary data f(x), we have theuniform estimate

‖u− c0‖H1(Σ) ≤ 2‖f‖H1(S).

For the proof of Theorem 3.1 we shall need the following lemma.

Lemma 3.3. Let u and v be two periodic solutions of (2.1). If v satisfies (2.5),then there exists a uniformly elliptic operator with L(u− v) = 0, so that thestrong maximum principle holds for u− v. In particular, the maximum andminimum of u− v are attained at y = 0.

Corollary 3.4. If u is a periodic solution of (2.1a) and (2.1b) with 1 + ux ≥δ > 0, then u obtains its maximum (and minimum) along y = 0. If (2.5) isfulfilled, then also ux and uy obtain their maximum and minimum on y = 0.

Proof of Lemma 3.3. Since both u and v solve (2.1), by adding and subtract-ing terms, we find an operator L, with

L((Dv,D2), (D2u,D))(u − v)

:= [(1 + 2vx)∂xx + 2vy∂xy + ∂yy + 2uxx∂x + 2uxy∂y] (u− v)

= ∆(u− v) + ∂x

(

|∇u|2 − |∇v|2)

= 0.

In view of (2.5), L is uniformly elliptic in Σ. The strong maximum prin-ciple for uniformly elliptic operators implies that neither the maximum or


minimum of u − v is attained at an interior point, unless u − v ≡ 0 van-ishes identically. In view of the boundary conditions (2.1c) and (2.1b), theproposition follows.

Proof of Corollary 3.4. Note that u is both a sub- and a supersolution, since

[∆ + 2uxx∂x + 2uxy∂y] u = 0.

By differentiating (2.1a) once with respect to x or y, we obtain that ux oruy, respectively, lies in the kernel of the operator

(1 + 2ux)∂xx + 2uy∂xy + ∂yy + 2uxx∂x + 2uxy∂y,

which is uniformly elliptic if (2.5) is satisfied. Note that u fulfills Theo-rem 3.2, which also implies that |∇u| → 0 as y → −∞, uniformly for x ∈ S

(recall that second derivatives of u are uniformly bounded in Σ). In view ofperiodicity, the strong maximum principle then yields the assertion.

Proof of Theorem 3.1.i) Let v(x, y) := −u(−x, y) be the odd reflection, which according to

Proposition 2.1 is a solution as well. It then follows from Lemma 3.3 that

‖u(x, y) + u(−x, y)‖∞ ≤ ‖u(x, 0) + u(−x, 0)‖∞ = 0,

meaning that u is everywhere odd.As a consequence of Lemma 3.3, we see that if u(x, 0) is 2π-periodic, so

is u(x, y). Thus the function

w(x, y) := u(x, y) − λ sinx exp y

is odd and well-defined on S. At the vertical boundaries, we thus have that

w(0, y) = w(π, y) = 0,

in addition to the asymptotic behaviour

limy→−∞

supx∈R

w(x, y) = 0.

Moreover, according to the mean value theorem, we have that

w(x, 0) = x(ux(ξ, 0) − λ cos ξ) ≤ x(‖ux‖∞ − λ cos x) < 0,

whenever λ > ‖ux‖∞, and 0 < ξ < x < arccos (‖ux‖∞/λ). Further, whenarccos (‖ux‖∞/λ) ≤ x ≤ π/2, we have that

w(x, 0) ≤ ‖u‖∞ − λ√

1 − (‖ux‖∞/λ)2 < 0,

if we simply require that λ2 > ‖u‖2∞

+‖ux‖2∞

. Since the same analysis holdsfor x ∈ (π/2, π), we conclude that

w(x, 0) ≤ 0, x ∈ S,

holds for such λ. According to Lemma 3.3, we have that w(x, y) ≤ 0 every-where in Σ, meaning that

u(x, y) ≤ λ sinx exp y, in Σ.


Note that, according to Corollary 3.4, λ can be bounded by the C1-norm ofthe boundary data f on S. Since the same argument now can be applied tow(x, y) := u(x, y)+λ sin x exp y, we see that oddness and periodicity implies(3.1).

ii) Let x be the variable of concern. If ψ is even, the term ∆ψ is even,while −∂x|∇ψ|

2 is odd. Since they are equal, they must both vanish. Thusψ solves

ψxx = −ψyy,

ψxψxx = −ψyψxy.

Consider the set Ω := (x, y) ∈ Ω: ψx(x, y), ψy(x, y) 6= 0. By applying thefirst equality to the second, we obtain that

ψyy

ψy

=ψxy

ψx

,

and consequently

∂y log

∣

∣

∣

∣

ψy

ψx

∣

∣

∣

∣

= 0.

This implies that

ψy(x, y) = h(x)ψx(x, y) in Ω, (3.3)

for some function h ∈ C1+α. By differentiating this equality separately withrespect to x and y, we obtain that

ψxy = hxψx + hψxx,

ψyy = hψxy.

Using the fact that ψ is harmonic, this leads to

(1 + h2)ψxx + hhxψx = 0,

or, equivalently,

∂x log∣

∣

∣ψx

√

1 + h2∣

∣

∣= 0.

Hence, ψx = g(y)/√

1 + h(x)2 for some function g ∈ C1+α, and in view of(3.3),

ψ(x, y) = g(y)

∫ x

x0

dξ√

1 + h(ξ)2+ β(y)

=h(x)

√

1 + h(x)2

∫ y

y0

g(ζ) dζ + β(x).

Insertion of x = x0 and y = y0 yields that

ψ(x, y) = g(y)

∫ x

x0

dξ√

1 + h(ξ)2+

h(x0)√

1 + h(x0)2

∫ y

y0

g(ζ) dζ + γ (3.4)

=h(x)

√

1 + h(x)2

∫ y

y0

g(ζ) dζ + g(y0)

∫ x

x0

dξ√

1 + h(ξ)2+ γ, (3.5)


for some γ ∈ R. Since (2.1a) is invariant under addition of any constant,we may as well consider γ = 0. By differentiating the equality (3.4) withrespect to y, we obtain that

g′(y)

∫ x

x0

dξ√

1 + h(ξ)2+

(

h(x0)√

1 + h(x0)2−

h(x)√

1 + h(x)2

)

g(y) = 0.

This is a separable equation, g′(y)/g(y) = F (x), for a function F dependingonly on x. Since the left-hand side does not depend on x, neither does theright-hand side. In other words,

g(y) = exp(cy), c ∈ R,

up to multiplication by some real constant. In the case when c = 0 we findthat ψ is an affine function depending only on x. If c 6= 0, this implies thatψ(x, y) = h(x)g(y) is separable, and moreover, (2.1a) reads as

exp(cy)(

h′′(x) + c2h(x))

+ exp(2cy)h′(x)(

h′′(x) + c2h(x))

= 0.

Since h(x) is even, and since for c 6= 0 the functions exp(cy) and exp(2cy)

are linearly independent, we obtain that in each connected component of Ω,the solution is given by λ sin(cx) exp(cy).

Now, Ω\Ω is the union of the two closed sets X := (x, y) ∈ Ω: ψx(x, y) =0 and Y := (x, y) ∈ Ω: ψy(x, y) = 0. If any of them has no interior, the

fact that u ∈ BUC 2+α(Ω) implies that it can be neglected. So assumethat either X or Y contains an open set. In that set, we have either thatψ(x, y) ≡ ψ(x), or that ψ(x, y) ≡ ψ(y). In both cases, insertion into (2.1a)yields that ψ is an affine function of one variable. In view of continuity offirst derivatives, the only places Ω can meet Ω are the vertical lines cx = kπ,k ∈ Z. But there the affine functions of one variable cannot match anyfunction of the type λ sin(cx) exp(cy) in BUC 2+α(Σ) (unless they are bothvanishing.) Hence ψ(x, y) = λ sin cx exp cy everywhere in Ω. (The case when

c = 0 and ψ(x, y) ≡ ψ(x) is an affine function in Ω leads to a contradictionunless ψ is constant). That c = 1 finally follows from the fact that ψ can bescaled to a 2π-periodic function u.

Proof of Theorem 3.2. We proceed as in [23] and [16] and consider for u theexpansion

u(x, y) =∞∑

k=−∞

ck(y) exp(ikx), so that

∫

S

u(x, y) dx = 2πc0(y).


Periodicity then yields that

0 =

∫

S

(

∆u+ ∂x|∇u|2)

dx

=

∫

S

∞∑

k=−∞

(

c′′k(y) − k2ck(y))

exp(ikx)

= 2πc′′0(y),

whence c0(y) is an affine function. In view of the fact that ψ is bounded, wemust however have that

c0(y) =1

2π

∫

S

u(x, 0) dx for all y ∈ (−∞, 0].

This implies that for any y ≤ 0, and in any period, there exists at least onepoint for which u(x0, y) = c0. Using the fact that u solves (2.1a) and isperiodic, we see that

∂2y

1

2

∫

S

|u− c0|2 dx =

∫

S

(

u2y + (u− c0)uyy

)

dx (3.6)

=

∫

S

(

u2y + (c0 − u)

(

uxx + ∂x|∇u|2)

)

dx (3.7)

=

∫

S

(1 + ux)|∇u|2 dx (3.8)

≥ δ

∫

S

u2x dx ≥ δ

∫ ξ

x0

u2x dx, x0 ≤ ξ ≤ x0 + 2π.

(3.9)

According to the fundamental theorem of calculus and Hölder’s inequality,we have that

|u(ξ, y) − u(x0, y)|2 =

∣

∣

∣

∣

∫ ξ

x0

ux dx

∣

∣

∣

∣

2

≤

(∫ ξ

x0

|ux| dx

)2

≤ (ξ − x0)

∫ ξ

x0

u2x dx ≤ 2π

∫ ξ

x0

u2x dx,

and consequently that

∂2y

∫

S

|u− c0|2 dx ≥

δ

π|u(ξ, y) − c0|

2,

which by periodicity is valid for arbitrary ξ ∈ S. Hence we may choose ξ sothat |u(ξ, y) − c0| equals its mean over a period. This leads to

∂2y

∫

S

|u− c0|2 dx ≥ 2δ

∫

S

|∇u|2 dx ≥δ

2π2

∫

S

|u− c0|2 dx ≥ 0. (3.10)


Recall from (3.6) that ∂2y

∫

S|u− c0|

2 dx ≤ C∫

S|∇u|2 dx, for some C ∈ R.

Since u is bounded, there is no other alternative than that

0 ≤ ∂ky

∫

S

|u− c0|2 dx→ 0, as y → −∞, for k = 0, 1, 2.

By integrating (3.10) once, we obtain (3.2).To prove the bound in the case when ‖ux‖∞ ≤ 1/4, define

h(x, y) := f(x) exp(y),

so that u(·, y)−h(·, y) is a periodic function, vanishing at y = 0 and inH1(Σ),according to what we just proved. By partial integration with respect to thetest function u− h, we thus see that

0 = −

∫∫

Σ

(

∆u+ ∂x|∇u|2)

(u− h) dx dy

=

∫∫

Σ

(

|∇u|2 −∇u · ∇h+ |∇u|2(ux − hx))

dx dy.

Using Young’s inequality this implies that∫∫

Σ(1 + ux − hx) |∇u|2 dx dy ≤

1

2

∫∫

Σ

(

ε|∇u|2 +1

ε|∇h|2

)

dx dy,

meaning, in view of Corollary 3.4, that∫∫

Σ(1 − 2‖ux‖∞ − ε/2) |∇u|2 dx dy

≤

∫∫

Σ(1 + ux − fx(x) exp(y) − ε/2) |∇u|2 dx dy

≤1

2ε

∫∫

Σexp(2y)

(

ux(x, 0)2 + u(x, 0)2)

dx dy

=1

4ε

∫

S

(

fx(x)2 + f(x)2)

dx.

By choosing ε = 1/2, we obtain the estimate

‖∇u‖2L2(Σ) ≤ 2‖f‖2

H1(S). (3.11)

Then the one-dimensional Poincaré inequality (Wirtinger’s inequality) yieldsthe desired bound.

4. Existence and Uniqueness

We present here our main result, which shows that for a small enoughperiodic surface profile along the beach, there always exists an edge wavesolution which vanishes fast towards the ocean.

Theorem 4.1 (Existence of solutions for small data). There exists a constantK > 0, such that for given boundary data u(x, 0) = f(x) with ‖f‖C2+α(S) ≤K, the problem (2.1) has a solution u satisfying

‖u‖C

2+α(Σ) ≤ 1/4, and ‖u‖H1(Σ) ≤ 2‖f‖H1(S).


The solution u is unique within BUC 2+α(Σ), and if f is odd, then u vanishesexponentially fast as y → −∞, with u(·, y) being odd for all y ≤ 0.

For the proof of Theorem 4.1 we will need a (uniform) local existenceresult. To ease notation, let

Σn := S × (−n, 0), n ∈ N+.

Lemma 4.2. There exists a constant K > 0 such that whenever n ∈ N+ and

‖f‖C2+α(S) ≤ K, there exists a solution un ∈ BUC 2+α(Σn) of the problem

(1 + 2ux)uxx + 2uyuxy + uyy = 0 in Σn,

u = f on ∂Σn,

satisfying ‖un‖C 2+α(Σn) < 1/4.

Proof. Let n ∈ N+ be given and set

U :=

u ∈ BUC 2+α(Σn) : ‖u‖C

2+α(Σn) < 1/4

.

Notice that U is an open neighbourhood of 0 in BUC 2+α(Σn). Since therelation

∣

∣

∣

∣

1 + 2ux uy

uy 1

∣

∣

∣

∣

= 1 + 2ux − u2y >

1

4

holds true for all u ∈ U it follows that the operator Fu := ∆u + ∂x(|∇u|2)is strictly elliptic for all u ∈ U . Given f ∈ C2+α(S) we denote again by fits extension to Σn satisfying f(x, y) = f(x) for all (x, y) ∈ Σn. Notice that‖f‖C2+α(S) = ‖f‖

C2+α(Σn).

LetG : U → BUC 2+α(Σn) be the operator given byGu := u−f.Accordingto the method of continuity (see Theorem 17.28 in [26]), the proof is completeif we show that if we show that the set

E := u ∈ U : Pu = σP0 for some σ ∈ [0, 1]

is contained together with its closure in U . Here 0 also denotes the zeroelement of U , and Pu := (Fu,Gu). Let now u belong to the closure of E inBUC 2+α(Σn). Since [0, 1] is a compact interval we obtain that u must solvethe problem

(1 + 2ux)uxx + 2uyuxy + uyy = 0 in Σn,

u = (1 − σ)f on ∂Σ,

for some σ ∈ [0, 1].Such a solution u can be estimated above and below by its boundary

values. Hence standard estimates for solutions of elliptic Dirichlet problemsimplies the existence of a positive constant C with

‖u‖C

2+α(Σn) ≤ C‖f‖C2+α(S).

In general, C depends on ellipticity and the domain Σn. However, for u ∈ U ,we have a uniform ellipticity constant, and the dependence on Σn is through


its curvature (see estimate 6.36 in [26] and the remark preceding it, as well asTheorem 10.3.1 in [34]). Hence, the constant C does not depend on n, and if‖f‖C2+α(S) is small enough, then u ∈ E ⊂ U and the proof is complete.

Proof of Theorem 4.1. According to Lemma 4.2, there exists K > 0, suchthat for ‖f‖

C2+α(S) ≤ K, we obtain a sequence unn of solutions in Σn.

Let β < α. From ‖un‖C2+α(Σ1) < 1/4 we obtain, due to the compactness

of the embedding BUC 2+α(Σ1) → BUC 2+β(Σ1), a subsequence ug1(k)k

which converges in BUC 2+β(Σ1). Inductively, we obtain for every n ∈ N+ asubsequence ug1···gn(k)k of unn with the property that ug1···gn(k)k

converges in BUC 2+β(Σn).Using Cantor’s diagonal method we set vk := ug1···gk(k) for k ≥ 1. This

sequence converges in BUC 2+β(Σn) for every n ≥ 1. Given (x, y) ∈ Ω, wedefine

u(x, y) := limk→∞

vk(x, y).

This function is well-defined since (x, y) ∈ Σk for k large enough. Moreover,the mapping u belongs to BUC 2+β(Ω) and u = f on S. It is also obviousthat u is a periodic solution of equation (2.1a). From regularity resultsfor elliptic problems (see [26]) we finally conclude that u ∈ BUC 2+α(Σ),and thus ‖u‖

C2+α(Σ) ≤ 1/4. But then the assumptions of Theorem 3.2

are fulfilled, and thus we obtain the desired asymptotic behaviour (2.1c),together with the H1-estimate. To prove uniqueness we apply Lemma 3.3,and Theorem 3.1 i) then finalizes the proof.

5. Concluding remarks

In the follow-up article [20] on solitary edge waves, we discuss the two-dimensional edge-wave equation as a model for water waves. We give heresome brief comments on the same topic, and in particular how vorticity mightbe interpreted in the the model (2.1a).

Since (2.1a) is a normalized model equation, the correspondence to theexact water waves is rather on the qualitative level. If one, however, regardsψ as a perturbed stream function, then

∆ψ = ω(x, y)

describes the effect induced by the vorticity (local curl) of the flow. In otherwords: the functions that simultaneously solve

∆ψ = 0 and ∂x|∇ψ|2 = 0,

are exactly the irrotational solutions of (2.1a). In light of this, the fact thatthose—modulo scaling, translation and addition of a constant—are exactlyof the form cos(x) exp(y), is a result which very well fits the general under-standing of irrotational water waves. On the other hand, if one accepts thatψ is a qualitative counterpart of the stream function, then all other solutions


of (2.1a) are rotational. In particular, any shoreline data f(x) that is not afundamental trigonometric function gives rise to edge waves with vorticity.

Finally, it is surprising that equation (2.1a), which has such a simpleform, carries so much structure: it contains strong symmetry propertiesand appropriate exponential asymptotic behaviour. It has the interestingpossibility of being type-changing, and its type depends on the gradient, i.e.the slope, of the solutions. Be it from a physical point of view, or a purelymathematical — it is certainly an equation that deserves further study.

References

[1] C. J. Amick, L. E. Fraenkel, and J. F. Toland, On the Stokes conjecture forthe wave of extreme form, Acta Math., 148 (1982), pp. 193–214.

[2] R. Caflisch and H. Lamb, Hydrodynamics (Cambridge Mathematical Library),Cambridge University Press, 1993.

[3] A. Constantin, Edge waves along a sloping beach, J. Phys. A, 34 (2001), pp. 9723–9731.

[4] , On the deep water wave motion, J. Phys. A, 34 (2001), pp. 1405–1417.[5] , The trajectories of particles in Stokes waves, Invent. Math., 166 (2006),

pp. 523–535.[6] A. Constantin, M. Ehrnström, and G. Villari, Particle trajectories in linear

deep-water waves, Nonlinear Anal. Real World Appl., 9 (2008), pp. 1336–1344.[7] A. Constantin, M. Ehrnström, and E. Wahlén, Symmetry for steady gravity

water waves with vorticity, Duke Math. J., 140 (2007), pp. 591–603.[8] A. Constantin and J. Escher, Well-posedness, global existence, and blowup phe-

nomena for a periodic quasi-linear hyperbolic equation, Commun. Pure Appl. Math.,51 (1998), pp. 475–504.

[9] , On the blow-up rate and the blow-up set of breaking waves for a shallow waterequation, Math. Z., 233 (2000), pp. 75–91.

[10] , Symmetry of steady deep-water waves with vorticity, European J. Appl. Math.,15 (2004), pp. 755–768.

[11] , Symmetry of steady periodic surface water waves with vorticity, J. Fluid Mech.,498 (2004), pp. 171–181.

[12] A. Constantin and D. Lannes, The hydrodynamical relevance of the Camassa—Holm and Degasperis—Procesi equations, Arch. Ration. Mech. Anal., 192 (2009),pp. 165–186.

[13] A. Constantin and W. A. Strauss, Exact periodic traveling water waves withvorticity, C. R. Math. Acad. Sci. Paris, 335 (2002), pp. 797–800.

[14] , Exact steady periodic water waves with vorticity, Comm. Pure Appl. Math.,57 (2004), pp. 481–527.

[15] , Rotational steady water waves near stagnation, Phil. Trans. R. Soc. Lond. A,365 (2007), pp. 2227–2239.

[16] S. Cui and J. Escher, Asymptotic behaviour of solutions of a multidimensionalmoving boundary problem modeling tumor growth, Comm. Partial Differential Equa-tions, 33 (2008), pp. 636–655.

[17] M. Ehrnström, Uniqueness for steady periodic water waves with vorticity, Int. Math.Res. Not., (2005), pp. 3721–3726.

[18] , Deep-water waves with vorticity: symmetry and rotational behaviour, DiscreteContin. Dyn. Syst., 19 (2007), pp. 483–491.

[19] , A new formulation of the water wave problem for Stokes waves of constantvorticity, J. Math. Anal. Appl., 339 (2008), pp. 636–643.


[20] M. Ehrnström, J. Escher, and B.-V. Matioc, Two-dimensional steady edgewaves. Part II: solitary waves., Preprint, (2008).

[21] J. Escher, Wave breaking and shock waves for a periodic shallow water equation,Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 365 (2007), pp. 2281–2289.

[22] J. Escher, Y. Liu, and Z. Yin, Global weak solutions and blow-up structure forthe Degasperis–Procesi equation, J. Funct. Anal., 241 (2006), pp. 457–485.

[23] J. Escher and B.-V. Matioc, On periodic Stokesian Hele–Shaw flows with surfacetension, European J. Appl. Math., 19 (2008), pp. 717–734.

[24] J. Escher and B.-V. Matioc, Stability of the equilibria for periodic Stokesian Hele-Shaw flows, J. Evol. Equ., 8 (2008), pp. 513–522.

[25] F. Gerstner, Theorie der Wellen samt einer daraus abgeleiteten Theorie der De-ichprofile, Ann. Phys., 2 (1809), pp. 412–445.

[26] D. Gilbarg and N. S. Trudinger, Elliptic partial differential equations of secondorder, Classics in Mathematics, Springer-Verlag, Berlin, 2001. Reprint of the 1998edition.

[27] D. Henry, On Gerstner’s water wave, J. Nonlinear Math. Phys., 15 (2008), pp. 87–95.

[28] , On the deep-water Stokes wave flow, Int. Math. Res. Not., 2008 (2008), pp. 1–7.[29] D. Henry and O. G. Mustafa, Existence of solutions for a class of edge wave

equations, Discrete Contin. Dyn. Syst. Ser. B, 6 (2006), pp. 1113–1119 (electronic).[30] H. Holden and X. Raynaud, A convergent numerical scheme for the Camassa-

Holm equation based on multipeakons, Discrete Contin. Dyn. Syst., 14 (2006), pp. 505–523.

[31] R. S. Johnson, Camassa–Holm, Korteweg–de Vries and related models for waterwaves, J. Fluid Mech., 455 (2002), pp. 63–82.

[32] , Some contributions to the theory of edge waves, J. Fluid Mech., 524 (2005),pp. 81–97.

[33] , Edge waves: theories past and present, Philos. Trans. R. Soc. Lond. Ser. AMath. Phys. Eng. Sci., 365 (2007), pp. 2359–2376.

[34] J. Jost, Partial differential equations, vol. 214 of Graduate Texts in Mathematics,Springer-Verlag, New York, 2002. Translated and revised from the 1998 Germanoriginal by the author.

[35] E. Varvaruca, On the existence of extreme waves and the Stokes conjecture withvorticity. arXiv:0707.2224, 2007.

[36] G. B. Whitham, Linear and nonlinear waves, Pure and Applied Mathematics (NewYork), John Wiley & Sons Inc., New York, 1974.

[37] G. B. Whitham, Nonlinear effects in edge waves, J. Fluid Mech., 74 (1976), pp. 353–368.

[38] C. S. Yih, Note on edge waves in a stratified fluid, J. Fluid Mech., 24 (1966), pp. 765–767.

[39] F. Zhou, J. Escher, and S. Cui, Bifurcation for a free boundary problem withsurface tension modeling the growth of multi-layer tumors, J. Math. Anal. Appl., 337(2008), pp. 443–457.


Institut für Angewandte Mathematik, Leibniz Universität Hannover, Welfen-

garten 1, 30167 Hannover, Germany.

E-mail address: [email protected]







Two-dimensional steady edge waves. Part I: Periodic waves

Documents

Transcript of Two-dimensional steady edge waves. Part I: Periodic waves