ON THE EXISTENCE OF STEADY PERIODIC CAPILLARY-GRAVITY
STRATIFIED WATER WAVES
DAVID HENRY AND BOGDANVASILE MATIOC
Abstrat. We prove the existene of small steady periodi apillary-gravity water waves
for general stratied ows, where we allow for stagnation points in the ow. We establish the
existene of both laminar and non-laminar ow solutions for the governing equations. This
is ahieved by using bifuration theory and estimates based on the elliptiity of the system,
where we regard, in turn, the mass-ux and surfae tension as bifuration parameters.
1. Introdution
In the following paper we prove the existene of small-amplitude two-dimensional steady
periodi apillary-gravity stratied water waves, where we do not exlude stagnation points
from the ow. Stratied water waves are heterogeneous ows where the density varies as
a funtion of the streamlines. Physially, stratiation is a very interesting phenomenon,
and uid density may be aused to utuate by a plethora of fators for example, salinity,
temperature, pressure, topography, oxygenation. Mathematially, allowing for heterogeneity
adds severe ompliations to the governing equations, making their analysis even more intratable. For apillary-gravity waves, the eets of surfae tension are inorporated in the
governing equations by adding a term proportional to the urvature of the wave prole into
the surfae boundary ondition, whih also ompliates matters signiantly. Nevertheless,
as we have shown in our paper [20℄, the remarkably nie regularity properties whih have
been reently proven to hold for a wide-variety of homogeneous ows [5,18,19,27,28℄, in the
main apply also to these stratied ows.
The rst results onerning the existene of small-amplitude waves for stratied ows were
obtained by Dubreil-Jaotin [13℄, in 1937. Interestingly, prior to this Dubreil-Jaotin suessfully adapted the famous Gerstner's water wave [1,17℄, whih is one of the few examples of
expliit solutions for the full governing equations whih exist, and whih desribes a rotational
water wave in a homogeneous uid, to the setting of a stratied uid in [11℄. More reently,
rigorous results onerning the existene of small and large amplitude stratied ows were
obtained via bifuration methods in [35℄, building on tehniques rst applied to rotational
homogeneous ows in [6℄. The existene of small and large amplitude waves for stratied
ows with the additional ompliation of surfae tension was then addressed in [37℄, building
on from loal existene results for rotational apillary-gravity waves ontained in [33℄.
All of the above existene results are heavily dependent on the absene of stagnation
points for steady water waves. The ondition whih ensures the lak of stagnation points for
a steady wave moving with onstant speed c is that all partiles in the uid have a horizontal
1991 Mathematis Subjet Classiation. 35Q35; 76B70; 76B47.
Key words and phrases. Stratied water waves; Existene; Streamlines.
1
2
D. HENRY AND B.V. MATIOC
veloity less than
c.
This is a physially reasonable assumption for water waves, without
underlying urrents ontaining strong non-uniformities, and whih are not near breaking [24℄.
Nevertheless, stagnation points are a very interesting phenomenon and ideally we would not
wish to exlude them from our piture. Stagnation points have long been a soure of great
interest and fasination in hydrodynamial researh, dating bak to Kelvin's work onerning
at's eyes and Stokes onjeture on the wave of greatest height (see [2, 30℄ for a thorough
survey of Stokes' waves).
Note that the physial meaning of a stagnation point is that of
a xed partile in the frame of referene moving with the wave-speed. However, there are
stagnation points where
u = c, v = 0
but the wave does not atually stagnate there, for
example the Stokes' wave of greatest height [2, 30℄. Of ourse, allowing for the presene of
stagnation points adds yet another major ompliation to the mathematial piture (see the
survey in [8℄) and we merely mention here that there have been two reent works [9, 34℄ where
dierent approahes were used to show the existene of small amplitude waves with onstant
vortiity where stagnation points or ritial layers our.
Mathematially rigorous work onerning the existene of small-amplitude waves for stratied ows whih also admitted ritial points was reently begun by one of the authors and
his ollaborators in [14℄.
Here the authors proved the existene of small amplitude waves
for stratied ows whih have a linear density distribution, and were also able to provide an
expliit piture of the ritial layers whih were present in the resulting ows. In the following
work, we prove the existene of small amplitude waves for stratied ows whih have a more
general density distribution funtion than that of [14℄ and under onditions whih ould not
be treated in [37℄.
The generality of the ows we admit ompliates the analysis immensely, and the resulting
governing equations whih emerge take the form of an overdetermined semi-linear ellipti
Dirihlet problem.
We then use and adapt various tools from the theory of ellipti equa-
tions and Fourier multiplier theory to prove, using the Crandall-Rabinowitz loal bifuration
theorem, the existene of small amplitude water waves.
The outline of the paper is as follows. In Setion 2 we desribe the Long-Yih [25, 31, 38℄
mathematial model for stratied apillary-gravity water waves, where we formulate the
governing equations equations in terms of the pseudo-streamfuntion.
Then, in Setion 3,
we establish the existene of laminar ow solutions for the governing equations. Finally, in
Setion 4, we prove, for a xed volume of uid, the existene of non-laminar ow solutions
for the governing equations using bifuration theory and estimates based on the elliptiity of
the system, ulminating in our main results, Theorem 4.3 and Theorem 4.6.
2.
The mathematial model
We now present the Long-Yih [25,31,38℄ formulation of the governing equations for the motion of a two-dimensional invisid, inompressible uid with variable density. The equations
will be formulated in terms of the veloity eld
funtion,
ρ
the variable density funtion, and
g
(u, v) of the uid, P
the pressure distribution
the gravitational onstant of aeleration, in
the uid domain
The symbol
funtions on
Ωη := {(x, y) : x ∈ S
and
−1 < y < η(t, x)}.
S stands for the unit irle, and funtions on S are identied with 2π−periodi
R. The funtion η desribing the wave prole is assumed to satisfy η(t, x) > −1
ON STRATIFIED CAPILLARY-GRAVITY WATER WAVES
for all
(t, x).
For obvious reasons, we assume heneforth that the density
3
ρ > 0.
y=0
t
Taking
to represent the loation of the undisturbed water surfae we assume for any xed time
that
Z
η(t, x) dx = 0.
(2.1)
S
Restriting our investigations to steady travelling wave solutions of the governing equations
enables us to express the problem in a moving referene frame where the ow is steady i.e.
time independent, thereby simplifying the water wave problem.
c, alled
(x − ct, y):
that there exists a positive onstant
(x, y, t)
dependene of the form
η(t, x) = η(x − ct),
the
wave speed,
We do this by supposing
suh that all funtions have an
(ρ, u, v, P )(t, x, y) = (ρ, u, v, P )(x − ct, y).
Under the assumption of onstant temperature and zero visosity, the motion of the uid is
presribed by the steady state two-dimensional Euler equations
ρ(u − c)ux + ρvuy = −Px
ρ(u − c)vx + ρvvy = −Py − gρ
in
in
Ωη ,
Ωη .
(2.2a)
The appropriate boundary onditions for waves propagating on a uid domain lying on an
impermeable at bed for whih surfae tension plays a signiant role are given by
with the onstant
P0

 v
P

v
= (u − c)η ′
= P0 − ση ′′ /(1 + η ′2 )3/2
= 0
on
on
on
being the atmospheri pressure and
σ
y = η(x),
y = η(x),
y = −1,
(2.2b)
the oeient of surfae tension.
The rst kinemati surfae ondition in (2.2b) implies a non-mixing ondition, namely that
the wave surfae onsists of the same uid partiles for all times, while the last kinemati
boundary ondition in (2.2b) ensures that the bottom of the oean is impermeable.
The
ontribution of the surfae tension is felt on the free surfae where it exerts a fore whih is
proportional to the urvature of the surfae at any given point, the onstant of proportionality
being the oeient of surfae tension [21, 24℄.
Letting
V = (u − c, v) denote the veloity eld in the moving frame of referene, the steady
ontinuity equation is given by
ρ divV + V · ∇ρ = 0
in
Ωη ,
whih simplies to
divV = 0
in
Ωη
(2.2)
when we make the supposition that
V · ∇ρ = 0,
a ondition whih ensures that the density
ρ
is non-diusive [25, 31, 38℄. This assumption is
√
√
∇ · ( ρ(u − c), ρv) = 0, and this relation
the assoiated pseudo-streamfuntion ψ by
√
√
∂x ψ = − ρv and ∂y ψ = ρ(u − c)
in Ωη .
(2.3)
very important in our analysis, sine it means that
enables us to dene
(2.2d)
4
D. HENRY AND B.V. MATIOC
The level sets of this funtion are indeed the streamlines of the steady ow: for suppose
the path of a uid partile in the steady ow is parametrised by (x(s), y(s)), where x′ (s) =
u(x, y) − c, y ′ (s) = v(x, y) then
ψ ′ (x(s), y(s)) = ψx x′ (s) + ψy y ′ (s) = 0.
Therefore the funtion ψ is onstant along eah streamline. We an see diretly that ψ is
dened for ρ > 0 by the expression
Z y
√
ψ(x, y) := λ +
ρ(x, s)(u(x, s) − c) ds,
(x, y) ∈ Ωη .
−1
Sine ψ is onstant on the free surfae we have the freedom to hoose λ suh that ψ = 0 on
y = η(x). We then have
Z η(x)
√
λ=
ρ(x, s)(c − u(x, s)) ds,
−1
where λ is a onstant alled the mass ux. We note that a further assumption whih is often
invoked is that there are no stagnation points within the uid domain, a ondition we an
express as
√
∂y ψ = ρ(u − c) < 0.
(2.4)
This ondition is known to be physially plausible for waves that are not near breaking (see
for instane the disussions in [5,21,24℄). For uid motions where (2.4) holds we observe that
(2.4) implies that λ > 0. Mathematially, ondition (2.4) is very useful beause it enables
us to reexpress the water wave problem in terms of new (q, p) variables whih transform the
unknown uid domain Ωη into a xed retangular domain Ωλ := S × (−λ, 0). Furthermore,
streamlines in Ωη are mapped to straight horizontal lines in the new domain Ωλ , and we see
below that this provides us with a onvenient method of analysing how ertain quantities
hange either along xed streamlines or as we vary the streamlines. The mapping is dened
by the Dubreil-Jaotin [12℄ semi-hodograph transformation
(q, p) := H(x, y) := (x, −ψ(x, y)),
(x, y) ∈ Ωη .
(2.5)
If (2.4) holds true then the mapping H : Ωη → Ωλ , is an isomorphism whih allows us
to reformulate the problem in terms of the pseudostream funtion. We note that the nondiusive ondition on the density ρ, (2.2d), implies that the density funtion is onstant on
eah streamline, sine
(u − c)ρx + vρy
v
∂q ρ ◦ H −1 ◦ H = ρx + ρy
=
= 0.
u−c
u−c
This means that we an formulate the density as a funtion of p, ρ = ρ(p) where ρ(p) =
ρ ◦ H −1 (q, p) for all p ∈ [−λ, 0], and we all this new version ρ of the density funtion the
streamline density funtion. Furthermore, Bernoulli's theorem states that the quantity
(u − c)2 + v 2
(2.6)
+ gρy
2
E ◦ H −1 = 0 for xed p. Partiularly, when p = 0,
E := P + ρ
is onstant along streamlines, that is ∂q
we obtain
η ′′
|∇ψ|2
−σ
+ gρ(0)y = Q
2
(1 + η ′2 )3/2
on y = η(x),
ON STRATIFIED CAPILLARY-GRAVITY WATER WAVES
5
for some onstant Q ∈ R, where Q is known as the hydrauli head of the ow. Moreover, a
diret alulation shows that
∆ψ − gyρ′ ◦ H = −∂p (E ◦ H −1 ) ◦ H
in Ωη .
This is the Long-Yih
equation for steady stratied water waves [25, 31, 38℄, and we infer that
sine ∂q E ◦ H −1 = 0 there exists a funtion β = β(p), alled Bernoulli's funtion, suh
that −∂p (E ◦ H −1 ) = β in Ωλ .
Summarising, ψ solves the following problem
∆ψ = f (y, ψ)
ψ = 0
ψ = λ
|∇ψ|2
η ′′
−σ
+ gρ(0)y = Q
2
(1 + η ′2 )3/2
in Ωη ,
on y = η(x),
on y = −1,
(2.7a)
(2.7b)
(2.7)
on y = η(x),
(2.7d)
where for onveniene we dene f : [−1, 1] × R → R to be the funtion
f (y, ψ) := gyρ′ (−ψ) + β(−ψ),
(y, ψ) ∈ [−1, 1] × R.
√
More expliitly, we have f = ρ[uy − vx ] − ρ′ (u − c)2 + v 2 /2, and in the ase of a homoge√
neous uid (i.e. onstant density ρ = ρ) then f (y, ψ) = β(−ψ) = ργ(−ψ), with γ denoting
the vortiity funtion of the homogeneous uid, and we get the usual governing equations for
apillary-gravity ows with vortiity, [33, 37℄. We also note, from the divergene struture of
the urvature operator, that, xing the volume of uid to be equal to that when the uid
interfae is at rest η = 0, f. (2.1), the head Q an be determined from (2.7d):
Z
|∇ψ|2
Q=
(2.7e)
(x, η(x)) dx,
2
S
R
if we normalise the integral suh that S 1 dx = 1. We observe that if we are given funtions
ψ, ρ(−ψ), β(−ψ) whih solve the Long-Yih equation (2.7a), then the orresponding funtions
(u, v, P ) we obtain from the relations (2.3), (2.6) respetively, will satisfy equations (2.2a)
and (2.2), see [31℄. This is irrespetive of whether the non-stagnation ondition (2.4) holds,
and therefore in the following we make no suh assumption on the uid ow when studying
the system (2.7).
3.
Laminar flow solutions to system (2.7)
The stratied apillary-gravity water wave problem (2.7) is an over-determined semilinear
Dirihlet system, with an additional boundary ondition to be satised on the wave surfae,
whih determines the eet of surfae tension, given by equations (2.7d) and (2.7e). In
Theorem 3.3 we prove that the semilinear system (2.7a)(2.7) is well-posed. In partiular,
this Theorem is used to establish the existene of laminar-ow solutions to the full system
(2.7). We then introdue an operator formulation of system (2.7) in Setion 4 whih allows
us to address the question onerning whether non-trivial (that is, non-laminar) solutions
exist whih also satisfy the additional boundary onditions (2.7d)-(2.7e). We do this by
means of the Crandall-Rabinowitz loal bifuration theorem, whih owing to its importane
in subsequent developments we now state:
6
D. HENRY AND B.V. MATIOC
( [10℄). Let X, Y be Banah spaes and let F ∈ C k (R × X, Y ) with k ≥ 2
Theorem 3.1
satisfy:
(a) F(γ, 0) = 0 for all γ ∈ R;
(b) The Fréhet derivative ∂x F(γ ∗ , 0) is a Fredholm operator of index zero with a onedimensional kernel:
Ker(∂x F(γ ∗ , 0)) = {sx0 : s ∈ R, 0 6= x0 ∈ X};
() The tranversality ondition holds:
∂γx F(γ ∗ , 0)[(1, x0 )] 6∈ Im(∂x F(γ ∗ , 0)).
Then γ ∗ is a bifuration point in the sense that there exists ǫ0 > 0 and a branh of solutions
(γ, x) = {(Γ(s), sχ(s)) : s ∈ R, |s| < ǫ0 } ⊂ R × X,
with F(γ, x) = 0, Γ(0) = γ ∗ , χ(0) = x0 , and the maps
s 7→ Γ(s) ∈ R,
s 7→ sχ(s) ∈ X,
are of lass C k−1 on (−ǫ0 , ǫ0 ). Furthermore there exists an open set U0 ⊂ R × X with
(γ ∗ , 0) ∈ U0 and
{(γ, x) ∈ U0 : F(γ, x) = 0, x 6= 0} = {(Γ(s), sχ(s)) : 0 < |s| < ǫ0 }.
Throughout the remainder of this paper we shall assume that the following onditions
hold:
(A1) ρ > 0;
(A2) ρ ∈ C 4− (R), β ∈ C 3− (R), and β, ρ′ ∈ BC 2 (R);
(A3) ∂ψ f (y, ψ) ≥ 0 for all (y, ψ) ∈ [−1, 1] × R.
Here, BC 2 (R) is the subspae of C 2 (R) whih ontains only funtions with bounded derivatives up to order 2, while, given m ∈ N with m ≥ 1,
C m− (R) := {h ∈ C m−1 (R) : h(m−1) is Lipshitz ontinuous}.
Remark 3.2. Though it is possible to onsider uids having streamline density funtion whih
is not monotone, the assumption (A3) restrits our onsiderations to uids having a noninreasing Bernoulli funtion. Furthermore, we mention that for the partiular ase of uids
with onstant density then ondition (A3) is equivalent to the ondition γ ′ (−ψ)≤0 for the
vortiity funtion. In this ontext, we note that this type of ondition appears in studying
(linear) stability properties of homogeneous ows with a free surfae, f. the disussion in [6℄.
Let thus α ∈ (0, 1) be xed for the remainder of this paper, and given m ∈ N and η > −1,
m+α (Ω ) denote the subspae of C m+α (Ω ) onsisting only of funtions whih are
we let Cper
η
η
2π−periodi in the x−variable.
Assume that (A1)-(A3) are satised. Given η ∈ C 2+α (S) with |η| < 1 and
λ ∈ R, the semilinear Dirihlet problem

 ∆ψ = f (y, ψ) in Ωη ,
ψ = 0
on y = η(x),
(3.1)

on y = −1
ψ = λ
Theorem 3.3.
ON STRATIFIED CAPILLARY-GRAVITY WATER WAVES
7
2+α (Ω ).
possesses a unique solution ψ ∈ Cper
η
We note that when η = 0 and Ω := Ω0 then, for any λ ∈ R, Theorem 3.3 ensures the
2+α (Ω) of problem (3.1), whih in this ase
existene and uniqueness of a solution ψλ ∈ Cper
depends only upon y . Therefore, the pair (η, ψ) := (0, ψλ ) is a solution of (2.7) for all λ ∈ R.
Moreover, we obtain that the funtion ψλ veries the following identity
ψλ (y) = −λy − y
Z
0
−1
Z
t
−1
f (s, ψλ (s)) ds dt −
Partiularly, by dierentiation we have
ψλ′ (y)
= −λ −
Z
0
−1
Z
Z
y
t
f (s, ψλ (s)) ds dt +
−1
0Z t
Z
f (s, ψλ (s)) ds dt,
−1
y ∈ [−1, 0]. (3.2)
y
f (s, ψλ (s)) ds,
−1
y ∈ [−1, 0].
(3.3)
2+α (Ω ) are two solutions
Proof of Theorem 3.3. We prove the uniqueness rst. If ψ1 , ψ2 ∈ Cper
η
of (3.1), then ψ := ψ1 − ψ2 is a solution of the equation
in Ωη
satisfying ψ = 0 on ∂Ωη . Using the mean value theorem, we have:
∆ψ = f (y, ψ1 ) − f (y, ψ2 )
f (y, ψ1 ) − f (y, ψ2 ) = c(x, y)ψ
with
c(x, y) :=
Z
0
1
∂ψ f (y, ψ2 + t(ψ1 − ψ2 )) dt.
Invoking (A3), the weak ellipti maximum priniple yields the desired uniqueness result.
α (Ω ),
In order to prove the existene part, we use a xed point argument. Given ϕ ∈ Cper
η
2+α (Ω ) the unique solution of the linear Dirihlet problem
we denote by T ϕ ∈ Cper
η

 ∆ψ = f (y, ϕ) in Ωη ,
on y = η(x),
ψ = 0

ψ = λ
on y = −1.
(3.4)
2+α (Ω ) ֒→ C α (Ω )
By (A2), the operator T is well-dened and ontinuous. Moreover, Cper
η
η
per
α
α
is a ompat embedding, and T : Cper (Ωη ) → Cper (Ωη ) is therefore ompletely ontinuous.
Sine any solution of (3.1) is a xed point of T , Leray-Shauder's theorem [16, Theorem 10.3℄
yields the following result.
If there exists a positive onstant M > 0 suh that
kψkC α (Ωη ) ≤ M
(3.5)
per
2+α (Ω ) and κ ∈ [0, 1] satisfying ψ = κT ψ, then T possesses a xed
for all ψ ∈ Cper
η
point.
That ψ is a solution of ψ = κT ψ, may be re-expressed by saying that ψ is a solution of the
Dirihlet problem:

 ∆ψ = κf (y, ψ) in Ωη ,
ψ = 0
on y = η(x),
(3.6)

on y = −1.
ψ = κλ
We prove (3.5) in three steps.
Step 1 First, we bound the supremum bound of ψ independently of κ. Indeed, using apriori
8
D. HENRY AND B.V. MATIOC
bounds for inhomogeneous Dirihlet problems, f. [16, Theorem 3.7℄, we obtain, by taking
into aount that Ωη ⊂ [|y| < 1], the following estimate
(3.7)
kψkCper (Ωη ) ≤ sup |ψ| + e2 sup |κf (y, ψ)| ≤ C1 ,
Ωη
∂Ωη
where C1 := |λ| + e2 gkρ′ kBC(R) + kβkBC(R) .
Step 2 Using tehnis from [16℄, the previous step and the fat that η ∈ C 2+α (S), so that in
partiular Ωη satises a uniform exterior sphere ondition (with radius R > 0), we onstrut
barrier funtions to prove that the gradient of eah solution ψ of (3.6) is bounded on ∂Ωη :
k∇ψkC(∂Ωη ) ≤ C2 ,
(3.8)
where C2 is independent of κ and ψ. Indeed, let z0 = (x0 , y0 ) ∈ ∂Ωη and z1 = (x1 , y1 ) ∈/ Ωη
be suh that the exterior ball B(z1 , R) satises B(z1 , R) ∩ Ωη = {z0 }. We dene the set
Ωa := {z ∈ Ωη : dist(z, B(z1 , R)) < a},
where a > 0 will be xed later on, and let d : Ωa → R be the funtion whih measures
the distane from a point z ∈ Ωa to B(z1 , R), that is d(z) = |z − z1 | − R. Furthermore, let
φ : R≥0 → R be the funtion dened by
φ(d) :=
ln(1 + d/a)
,
µ
d ≥ 0,
with a positive onstant µ. Finally, we onsider the funtion w := φ ◦ d ∈ C ∞ (Ωa ) and set
ψ − κλ, if z0 ∈ [y = η(x)];
u :=
if z0 ∈ [y = −1].
ψ,
With a small to guarantee that Ωa intersets only one boundary omponent of Ωη , we note
that u is a solution of the problem
∆u = κf (y, ψ) in Ωa ,
(3.9)
u = 0
on ∂Ωη ∩ ∂Ωa .
Moreover, by the hain rule, we obtain the following relations
∆w = φ′′ ((∂x d)2 + (∂y d)2 ) + φ′ ∆d,
(∂x d)2 + (∂y d)2 = 1,
and taking into aount that κ|f (y, ψ)| ≤ C1 , we have
∆d(z) = 1/|z − z1 |,
∆w ± κf (y, ψ) ≤ φ′′ ((∂x d)2 + (∂y d)2 ) + φ′ ∆d + C1 ≤ −µφ′2 +
≤−
1
1
+
+ C1
2
4a µ Raµ
1 ′
φ + C1
R
in Ωa .
(3.10)
We hoose now a and µ suh that
in Ωa
and
on ∂Ωa .
(3.11)
Clearly, the seond inequality holds true on ∂Ωa ∩ ∂Ωη , f. (3.6). On the other hand, if
z ∈ ∂Ωa \ ∂Ωη , then neessarily d(z) = a, and therefore
∆w ± κf (y, ψ) ≤ 0
w(z) = φ(a) =
w ≥ |u|
ln 2
.
µ
ON STRATIFIED CAPILLARY-GRAVITY WATER WAVES
9
Taking into aount that |u| ≤ C1 + |λ|, we onlude that (3.11) is satised if a and µ are
suiently small.
Gathering (3.9) and (3.11), we nd that
(
∆(u − w) ≥ 0 in Ωa ,
u − w ≤ 0 on ∂Ωa ,
and
(
∆(u + w) ≤ 0 in Ωa ,
u + w ≥ 0 on ∂Ωa .
We then infer from the ellipti maximum priniple that
w(z0 ) = u(z0 ) = 0
and
− w ≤ u ≤ w in Ωa ,
and, sine u = 0 on ∂Ωa ∩ ∂Ωη , we onlude that
|∇u|(z0 ) = |h∇u|νi|(z0 ) ≤ |h∇w|νi|(z0 ) = |φ′ (0)| = 1/aµ.
This proves (3.8).
Step 3 We nally show that there exists a positive onstant C3 , independent of κ, suh that
(3.12)
for all solutions ψ of (3.4) and all κ ∈ [0, 1]. To this end we observe that partial derivatives
∂x ψ and ∂y ψ are weak solutions of the ellipti equations
∆u − κ∂ψ f (y, ψ)u = 0 and ∆u − κ∂ψ f (y, ψ)u = κ∂y f (y, −ψ),
respetively. The assertion (3.12) follows now from (3.8) by using [16, Theorem 3.7℄ for
∂x ψ and a maximum priniple for non-homogeneous divergene struture inequalities, f.
1 (Ω ) ֒→
[29, Theorem 6.1.5℄, for ∂y ψ. Invoking (3.7) and (3.12), we obtain, in view of Cper
η
α
Cper (Ωη ), the desired result (3.5).
k∇ψkC(Ωη ) ≤ C3
4.
Non-laminar solutions
4.1. The funtional analyti setting. The next aim is to show that there exist non-trivial
(that is, non-laminar) solutions of the stratied water wave problem (2.7), and in order to
ahieve this we must reast the water wave problem (2.7) using an operator formulation.
We will then apply the Crandall-Rabinowitz theorem to prove the loal existene of solutions
bifurating from the laminar ows. We an show the loal existene of non-laminar solutions,
rstly xing the mass-ux onstant λ and then using the oeient of surfae tension σ as a
bifurating parameter, and then seondly xing σ and using λ as a bifurating parameter.
m+α
To this end, we introdue the subspaes Cbe,k
(S), m ∈ N, of C m+α (S) onsisting of even
m+α
funtions whih are 2π/k−periodi and have integral mean zero, and we dene Ce,k
(Ω), as
m+α
the subspae of Cper (Ω) ontaining only even and 2π/k−periodi funtions in the x variable.
The evenness ondition imposes a symmetry on the free-surfae (and on the underlying ow).
In the absene of stratiation and of stagnation points (but allowing for vortiity), one an
show that the mere monotoniity of the free-surfae between trough and rest ensures this
symmetry, f. [3, 4℄. To some extent, this result also applies to stratied ows, f. [36℄. By
onsidering only funtions η with integral mean zero we inorporate the volume onstraint
(2.1) in the spaes we work with.
The bottom of the uid being loated at y = −1, we take the wave prole η from the set
b 2+α (S) : |η| < 1}.
Ad := {η ∈ C
e,k
10
D. HENRY AND B.V. MATIOC
Given η ∈ Ad, we dene the mapping Φη : Ω = S × (−1, 0) → Ωη by the relation
Φη (x, y) := (x, (y + 1)η(x) + y),
(x, y) ∈ Ω.
This mapping has the benet of attening the unknown free-boundary uid domain into a
xed retangle, similar to [14,34℄. The operator Φη is a dieomorphism for all η ∈ Ad, and we
use this property to transform problem (2.7) onto the retangle Ω. Therefore, orresponding
to the semilinear ellipti operator in (2.7a) we introdue the transformed ellipti operator
2+α
α (Ω) with
(Ω) → Ce,k
A : Ad ×Ce,k
−1
A(η, ψ̃) := ∆(ψ̃ ◦ Φ−1
η ) ◦ Φη − f (y, ψ̃ ◦ Φη ) ◦ Φη ,
2+α
(η, ψ̃) ∈ Ad ×Ce,k
(Ω).
To ease notation we deompose A(η, ψ̃) = A0 (η)ψ̃ + b(η, ψ̃), where
A0 (η) := ∂11 −
2(1 + y)η ′
1 + (1 + y)2 η ′2
(1 + η)η ′′ − 2η ′2
∂12 +
∂
−
(1
+
y)
∂2 ,
22
1+η
(1 + η)2
(1 + η)2
(4.1)
b(η, ψ̃) := −f ((1 + y)η + y, ψ).
2+α
Furthermore, orresponding to (2.7d), we dene the boundary operator B : Ad ×Ce,k
(Ω) →
α
Ce,k (S) by the relation
2
tr |∇(ψ̃ ◦ Φ−1
1
η )| ◦ Φη
=
B(η, ψ̃) :=
2
2
1 + η ′2
2η ′
2
2
tr ψ̃1 tr ψ̃2 +
tr ψ̃2 ,
tr ψ̃1 −
1+η
(1 + η)2
with tr denoting the trae operator with respet to S = S × {0}.
Having introdued this notation, we remark that the laminar ow solutions (η, ψ) = (0, ψλ ),
λ ∈ R, orrespond to the trivial solutions (σ, λ, η) = (σ, λ, 0) of the equation
(4.2)
Ψ(σ, λ, η) = 0
α (S) is given by
where Ψ : (0, ∞) × R × Ad → Cbe,k
η ′′
Ψ(σ, λ, η) := B(η, T (λ, η)) − σ
+ gρ(0)η −
(1 + η ′2 )3/2
Z
S
B(η, T (λ, η)) dx,
2+α
and T : R × Ad → Ce,k
(Ω) is the solution operator to the semilinear Dirihlet problem

 A(η, ψ̃) = 0 in Ω,
ψ̃ = 0 on y = 0,

ψ̃ = λ on y = −1.
(4.3)
That T is well-dened between these spaes follows easily from the weak ellipti maximum
priniple, as in the uniqueness proof of Theorem 3.3. Moreover, problems (2.7) and (4.2) are
equivalent, that is (λ, σ, η) is a solution of (4.2) if and only if (η, ψ := T (λ, η) ◦ Φ−1
η ) is a
solution of (2.7). Following ideas from [15, Theorem 2.3℄, we establish now a preliminary
result onerning the regularity of T .
ON STRATIFIED CAPILLARY-GRAVITY WATER WAVES
Lemma 4.1. We have
Proof.
11
2+α
(Ω)).
T ∈ C 2 (R × Ad, Ce,k
2+α
α (Ω) × (C 2+α (S))2 be the operator dened by
Letting L : R × Ad ×Ce,k
(Ω) → Ce,k
e,k
L(λ, η, ψ̃) := (A(η, ψ̃), tr0 ψ̃, tr− ψ̃ − λ),
where tr− is the trae with respet to S = S × {−1}, the assumption (A2) guarantees that L
is of lass C 2 . Moreover, it holds that
L(λ, η, T (λ, η)) = 0
for all (λ, η) ∈ R × Ad . We show now that the operator ∂ψ̃ L(λ, η, T (λ, η)) is an isomorphism,
2+α
α (Ω) × (C 2+α (S))2 ), for all (λ, η) ∈ R × Ad .
that is ∂ψ̃ L(λ, η, T (λ, η)) ∈ Isom(Ce,k
(Ω), Ce,k
e,k
The impliit funtion theorem leads us then to the laim. It is not diult to see that for all
2+α
ψ̃, w ∈ Ce,k
(Ω), we have
∂ψ̃ L(λ, η, ψ̃)[w] = (A0 (η)w + ∂ψ̃ b(η, ψ̃)w, tr0 w, tr− w),
where
∂ψ̃ b(η, ψ̃) = −∂ψ f ((1 + y)η + y, ψ̃) ≤ 0,
by (A3). Consequently, the operator ∂ψ̃ L(λ, η, ψ̃)[w] satises the weak-maximum priniple
[16℄, and using standard arguments for ellipti operators on the existene and uniqueness of
solutions it follows that ∂ψ̃ L(λ, η, ψ̃)[w] is an isomorphism for all (λ, η, ψ̃). This ompletes
the proof.
Sine B depends analytially on (η, ψ̃), we onlude that
b α (S)).
Ψ ∈ C 2 ((0, ∞) × R × Ad, C
e,k
(4.4)
This is the starting point to our bifuration analysis.
4.2. Charaterisation of ∂η Ψ(σ, λ, 0) as a Fourier multiplier. In order to prove that
loal bifuration ours using the Crandall-Rabinowitz Theorem 3.1 we need to determine
rst the Fréhet derivative of Ψ with respet to η. In this setion, we show that ∂η Ψ(σ, λ, 0) is
in fat a Fourier multiplier, and this allows us to analyse its Fredholm properties in a rather
neat fashion by examining the behaviour of its symbols (µm (σ, λ)), whih we determine below.
This examination of the symbols will be the essene of the tehnial proofs of Theorems 4.3,
4.6 in Setion 4.3. To this end we x (σ, λ) and, realling that T (λ, 0) = ψλ , we obtain
∂η Ψ(σ, λ, 0)[η] =∂η B(0, ψλ )[η] + ∂ψ̃ B(0, ψλ )[∂η T (λ, 0)[η]] − ση ′′ + gρ(0)η
Z −
∂η B(0, ψλ )[η] + ∂ψ̃ B(0, ψλ )[∂η T (λ, 0)[η]] dx
S
2+α
for all η ∈ Cbe,k
(S). Taking into aount that ψλ depends only upon y , we have the following
relations:
∂η B(0, ψλ )[η] = −ψλ′2 (0)η
and ∂ψ̃ B(0, ψλ )[w] = ψλ′ (0) tr w2 .
12
D. HENRY AND B.V. MATIOC
Moreover, dierentiating the equations of (4.3) with respet to η, we see that ∂η T (λ, 0)[η] is
the solution of the Dirihlet problem:
(
A0 (0)w + ∂ψ̃ b(0, ψλ )w = −∂η A0 (0)[η]ψλ − ∂η b(0, ψλ )[η] in Ω,
(4.5)
w = 0
on ∂Ω,
whene
(
∆w − ∂ψ f (y, ψλ )w = (1 + y)ψλ′ η ′′ + [2f (y, ψλ ) + g(1 + y)ρ′ (−ψλ )]η in Ω,
(4.6)
w = 0
on ∂Ω.
We onsider now the Fourier series expansions of η and w:
η=
∞
X
am cos(mkx),
w :=
am wm (y) cos(mkx),
m=1
m=1
and nd that
∞
X
(1 + y)ψλ′ η ′′ + [2f (y, ψλ ) + g(1 + y)ρ′ (−ψλ )]η =
∞
X
am bm (y) cos(mky)
m=1
where
bm (y) := 2f (y, ψλ (y)) + g(1 + y)ρ′ (−ψλ (y)) − (mk)2 (1 + y)ψλ′ (y).
Inserting these expressions into (4.7) and omparing the oeients of cos(mkx) on both
sides of the equations, we onlude that wm is the unique solution of the boundary value
problem
(
′′ − ((mk)2 + ∂ f (y, ψ ))w
wm
m = bm , −1 < y < 0,
ψ
λ
wm (0) = wm (−1) = 0.
(4.7)
Though we an not solve (4.7) expliitly, we will see later on that ellipti maximum priniples
may be use to study the properties of the solution of (4.7). Understanding the behaviour of
the solutions of (3.2) and (4.7) is the key point in our analysis.
Summarising, we nd that the Fréhet derivative of ∂η Ψ(σ, λ, 0) is the Fourier multiplier
∂η Ψ(σ, λ, 0)
∞
X
m=1
am cos(mkx) =
∞
X
µm (σ, λ)am cos(mkx),
(4.8)
m=1
and the symbol (µm (σ, λ))m≥1 is given by the expression
′
µm (σ, λ) := σ(mk)2 + gρ(0) − ψλ′2 (0) + ψλ′ (0)wm
(0).
(4.9)
Clearly, by virtue of Lemma 4.1, µm is of lass C 2 in (σ, λ). It is well-known that ∆ :
C 2+α (S) → C α (S) is the generator of an analyti and ontinuous semigroup in L(C α (S)).
Moreover, ∆ is a Fourier multiplier with symbol (−k2 )k∈Z and the operator σ∆+∂η Ψ(σ, λ, 0) ∈
b 2+α (S), C
b 1+α (S)), f. Lemma 4.1. This shows, in view of Proposition 2.4.1 (i) in
L(C
e,k
e,k
[26℄, that −∂η Ψ(σ, λ, 0) is itself the generator of an analyti strongly ontinuous semigroup
α (S)). Partiularly, we infer from Theorem III.8.29 in [22℄ that the spetrum of
in L(Cbe,k
∂η Ψ(σ, λ, 0) oinides with its point spetrum, that is
σ(∂η Ψ(σ, λ, 0)) := {µm (σ, λ) : m ≥ 1}.
(4.10)
ON STRATIFIED CAPILLARY-GRAVITY WATER WAVES
13
4.3. Main existene results. We now prove our main existene results, rst using the
oeient of surfae tension σ as a bifuration parameter (the γ in the Crandall-Rabinowitz
Theorem 3.1) and seondly using the mass-ux λ as a bifuration parameter. We make the
following additional assumption on the system (2.7):
(A4) 2f (y, ψ) + g(1 + y)ρ′ (−ψ) ≤ 0 for all y ∈ [−1, 0] × (−∞, 0].
Remark 4.2. We note that for homogeneous uids, ρ onstant, ondition (A4) states that
the vortiity funtion is negative, γ(−ψ) ≤ 0. For an analysis of the bearing that suh a
ondition has on the underlying ows f. the disussions in [8, 32℄.
Theorem 4.3. Let ρ and β be given suh that (A1)-(A4) are satised and x k ≥ 1. There
exists Λ− ∈ R suh that for all λ ≤ Λ− , we may nd a sequene (σm )m≥1 ⊂ (0, ∞) dereasing
to zero with the following properties:
(i) Given m ≥ 1, there exist a ontinuously dierentiable urve (σm , ηm ) : (−ε, ε) → Ad
onsisting only of solutions of (4.2), that is Ψ(λ, σm (s), ηm (s)) = 0 for all |s| < ε.
(ii) We have the following asymptoti relations
σm (s) = σ m + O(s), ηm (s) = −s cos(mkx) + O(s2 )
for s → 0.
Moreover, for xed λ ≤ Λ− , all the solutions of (4.2) lose to (λ, σm , 0) are either laminar
ows or belong to the urve (λ, σm , ηm ).
Remark 4.4. If we replae (A4) by
(A4′ ) 2f (y, ψ) + g(1 + y)ρ′ (−ψ) ≥ 0 for all (y, ψ) ∈ [−1, 0] × [0, ∞)
then the onlusion of Theorem 4.3 remains true with the modiation that in this ase we
nd Λ+ ∈ R suh that the assertions (i) and (ii) of the theorem hold for all λ ≥ Λ+ . In this
ase we obtain travelling wave solutions of the original problem (2.2), whih have positive
mass ux. That some of the non-laminar waves obtained in Theorem 4.3 ontain stagnation
points an be see from the disussion in Remark 4.8.
For our seond main result we require, in addition to (A1)-(A4), that
(B1) ∂ψψ f ≥ 0 on [−1, 0] × (−∞, 0];
(B2) 2∂ψ f (y, ψ) − g(1 + y)ρ′′ (−ψ) ≥ 0 for all (y, ψ) ∈ [−1, 0] × (−∞, 0];
(B3) ∂ψ f (y, 0) ≤ 2 for all y ∈ [−1, 0].
Remark 4.5. For the partiular setting of homogeneous ows (ρ onstant), we an see that
(B1) requires that the vortiity funtion γ ′′ (−ψ) ≥ 0, (B2) is simply equivalent to (A3), and
(B3) requires that on the surfae γ ′ (0) ≥ −2.
Let ρ and β be given suh that (A1)-(A4) and (B1)-(B3) are satised and
let σ > 0 be xed. There exists a positive integer K ∈ N and for all k ≥ K a sequene
(λm )m≥1 ⊂ R with λm → −∞ and:
(i) Given m ≥ 1, there exist a ontinuously dierentiable urve (λm , ηm ) : (−ε, ε) → Ad
onsisting only of solutions of (4.2), that is Ψ(λm (s), σm , ηm (s)) = 0 for all |s| < ε.
(ii) We have the following asymptoti relations
λm (s) = λm + O(s), ηm (s) = −s cos(mkx) + O(s2 )
for s → 0.
Theorem 4.6.
14
D. HENRY AND B.V. MATIOC
Moreover, all the solutions of (4.2) lose to (λm , σ, 0) are either laminar ows or belong to
the urve (λm , σ, ηm ).
Remark 4.7. We note that for s suiently small, it follows diretly from Theorem 4.3 (ii)
and Theorem 4.6 (ii) that the resulting non-laminar solutions have a wave surfae prole ηm
whih has minimal period 2π/mk, ηm has a unique rest and trough per period, and ηm is
stritly monotone from rest to trough.
Before proving the theorems, we onsider more losely the funtions ψλ . Sine by Lemma
4.1 the mapping λ 7→ ψλ is of lass C 2 , we dierentiate the equations of (4.3), when η = 0,
with respet to λ and nd that ∂λ ψλ is the solution of the Dirihlet problem:

 ∆u = ∂ψ f (y, ψλ )u in Ω,
on y = 0,
u = 0
(4.11)

u = 1
on y = −1.
Invoking (A3), we obtain that ∂λ ψλ (0) ≤ ∂λ ψλ ≤ ∂λ ψλ (−1), and sine ∂λ ψλ is not onstant
Hopf's priniple ensures that
∂λ ψλ′ (0) < 0 and ∂λ ψλ′ (−1) < 0
for all λ ∈ R.
′
Consequently, λ 7→ ψλ (0) is a dereasing funtion and, realling (A2) and (3.3), we obtain
that and there exists a threshold value Λ ∈ R with
(
ψλ′ (0) > 0, for λ < Λ,
′
ψΛ
(0) = 0 and
(4.12)
ψλ′ (0) < 0, for λ > Λ.
Proof of Theorem 4.3. Fix k ∈ N, k ≥ 1. For eah m ∈ N, m ≥ 1, and λ ∈ R, we let
σ m (λ) =: σ m := −
′ (0)
gρ(0) − ψλ′2 (0) + ψλ′ (0)wm
,
(mk)2
(4.13)
be the solution of the equation µm (λ, ·) = 0.
We rst show that if λ is small, then σ m are all positive. To this end, we laim that
the right hand side bm of (4.7)1 is non-positive for all m ∈ N, provided λ is small enough.
Indeed, realling (3.3), we see that ψλ′ →λ→−∞ ∞ uniformly in y ∈ [−1, 0], so that there
exists Λ− ≤ Λ with the following properties:
ψλ′ ≥ 0
and
gρ(0) ≤ ψλ′2 (0) for all λ ≤ Λ− .
(4.14)
The laim follows now from (A4) and the fat that ψλ ≤ 0 for all λ ≤ Λ− . Let thus λ ≤ Λ−
be xed. The weak ellipti maximum priniple applied to (4.7) ensures that wm attains its
′ (0) < 0 for
minimum on the boundary, that is wm ≥ 0, and Hopf's priniple leads us to wm
all m ∈ N. Consequently, σ m > 0 for all m ≥ 1 and µm (λ, σ m ) = 0.
Next we prove that
(σ m ) is a dereasing sequene. Realling (4.13), it sues to show
′ (0)/(mk)2
is non-dereasing. In order to do this, we onsider the parameter
that wm
m≥1
dependent problem



2f (y, ψλ ) + g(1 + y)ρ′ (−ψλ )
− (1 + y)ψλ′ , −1 < y < 0,
χ
uχ (0) = uχ (−1) = 0.
u′′χ − (χ + ∂ψ f (y, ψλ ))uχ =
(4.15)
ON STRATIFIED CAPILLARY-GRAVITY WATER WAVES
15
whih is obtained by replaing (mk)2 in (4.7) by χ ∈ (0, ∞) and dividing by this variable
the equations of (4.7). Partiularly, wm = (mk)2 u(mk)2 , and the arguments presented above
show that uχ ≥ 0 for all χ > 0. Clearly, the solution uχ of (4.15) depends smoothly upon χ.
Dierentiating the equations in (4.15) with respet to χ yields that vχ := ∂χ uχ is the solution
of



vχ′′ − (χ + ∂ψ f (y, ψλ ))vχ = uχ −
2f (y, ψλ ) + g(1 + y)ρ′ (−ψλ )
≥ 0, −1 < y < 0,
χ2
vχ (0) = vχ (−1) = 0.
(4.16)
that
=
> 0 for all χ ≥ 1.
Whene, vχ ≤ 0 and Hopf's priniple implies
′ (0)/(mk)2
Partiularly, the sequene wm
is
inreasing.
m≥1
We have thus shown that µm (σ p , λ) = 0 if and only if m = p ≥ 1. Consequently, we have
vχ′ (0)
Ker ∂η Ψ(σ m , λ, 0) = span{cos(mkx)},
∂χ (u′χ (0))
b α (S), (4.17)
Im ∂η Ψ(σ m , λ, 0) ⊕ {cos(mkx)} = C
e,k
and so ∂η Ψ(σ m , λ, 0) is a Fredholm operator of index zero. Moreover, in virtue of (4.9), the
transversality ondition (3.1) of Theorem 3.1 is fullled too
∂ση Ψ(σ m , λ, 0)[cos(mkx)] = 2(mk)2 cos(mkx) ∈
/ Im ∂η Ψ(σ m , λ, 0).
(4.18)
sup kwm /(mk)2 kC([−1,0]) < ∞.
(4.19)
Gathering (4.4), (4.17), and (4.18) we may apply Theorem 3.1 to equation (4.2) and obtain
the desired existene result.
We nish by proving that (σ m ) onverges to zero. We apply the estimate from Theorem
3.7 in [16℄ to the Dirihlet problem (4.15), when χ = (mk)2 , and obtain that
m
Letting um := wm /(mk)2 , we re-write (4.7) as follows
u′′m − (mk)2 um = Bm (y), −1 < y < 0,
um (0) = um (−1) = 0,
(4.20)
where Bm (y) := (∂ψ f (y, ψλ (y))wm (y) + bm (y))/(mk)2 is, by (4.19), a bounded sequene in
C([0, 1]). The general solution of (4.20) is
sinh(mky)
um (y) =
sinh(mk)
and partiularly
u′m (0) =
mk
sinh(mk)
Z
Z
0
−1
0
−1
sinh(mk(s + 1))
Bm (s) ds +
mk
Z
y
0
sinh(mk(y − s))
Bm (s) ds,
mk
sinh(mk(s + 1))
cosh(mk)
Bm (s) ds ≤ sup kBm kC([−1,0])
.
mk
mk
sinh(mk)
m
′ (0)/(mk)2 →
Consequently, u′m (0) = wm
m→∞ 0. This proves our laim.
We prove now our seond main existene result, Theorem 4.6. Though it is independent
of Theorem 4.3, the assertions of the latter are satised and we shall make use of some of the
relations derived in its proof.
16
D. HENRY AND B.V. MATIOC
Proof of Theorem 4.6. Let σ > 0 be xed, and let Λ and Λ− be the onstants dened by
′ (0) < 0
(4.12) and (4.14), respetively. Given m ∈ N, m ≥ 1, and λ ≤ Λ− , it holds that wm
2
and, by (3.3) and (A2), we get limλ→−∞ µm (σ, λ) = −∞ and µm (σ, Λ) = σ(mk) +gρ(0) > 0.
Therefore,
λm (σ) =: λm := min{λ < Λ : µm (σ, λ) = 0}
is a well-dened onstant for all m ∈ N. In order to apply Theorem 3.1 we have to make sure
that µm (σ, λn ) 6= 0 for all m 6= n and that ∂λ µm (σ, λm ) 6= 0 for all m ≥ 1.
We rst laim that λm →m→∞ −∞. To prove this, we assume that (λm )m has a bounded
subsequene (whih we denote again by (λm )m ). From the arguments in Theorem 4.3, where
we established that (σ m ) onverges to zero, we dedue, due to the boundedness of (λm )m ,
′ (0)/(mk)2 → 0. Our laim follows by dividing (4.9) by (mk)2 and letting m → ∞.
that wm
This shows (also for k = 1) that indeed λm →m→∞ −∞. Whene, we may hoose K ∈ N
suh that if k ≥ K, then λm ≤ Λ− for all m ≥ 1.
We prove now that if k ≥ K and m ≥ 1, then λm is the unique point within (−∞, Λ− ]
suh that µm (σ, λ) = 0. To this end, we dierentiate µm with respet to λ and observe that,
for λ ≤ Λ− ,
′
′
∂λ µm (σ, λ) = −2ψλ′ (0)∂λ ψλ′ (0) + ∂λ ψλ′ (0)wm
(0) + ψλ′ (0)∂λ wm
(0) > 0
′ (0) ≥ 0. We show now that this is indeed the ase. Dierentiating both
provided ∂λ wm
equations of (4.7) with respet to λ, yields that u := ∂λ wm is the solution of the following
boundary value problem
 ′′
u − ((mk)2 + ∂ψ f (y, ψλ (y)))u = ∂ψψ f (y, ψλ (y)))∂λ ψλ wm





+ (2∂ψ f (y, ψλ (y)) − g(1 + y)ρ′′ (−ψλ )) ∂λ ψλ

−(mk)2 (1 + y)∂λ ψλ′ (y),
−1 < y < 0,




u(0) = u(−1) = 0.
(4.21)
It is not diult to see that
∂λ ψλ′ (y)
= −1 −
Z
0
−1
Z
t
∂ψ f (s, ψλ (s))∂λ ψλ (s) ds dt +
−1
Z
y
∂ψ f (s, ψλ (s))∂λ ψλ (s) ds,
−1
and gathering (A3), (B1), (B3), ψλ ≤ 0, and ∂λ ψλ ∈ [0, 1] we onlude that
∂λ ψλ′ (y)
≤
∂λ ψλ′ (0)
≤ −1 +
Z
= −1 +
0
−1
Z
0
t
Z
0
−1
Z
0
∂ψ f (s, ψλ (s))∂λ ψλ (s) ds dt
t
∂ψ f (s, 0) ds dt ≤ 0.
Sine λ ≤ Λ− , we onlude from (B1) and (B2) that the right-hand side of (4.21)1 is positive.
′ (0) > 0. This
Consequently, u ≤ 0 and Hopf's priniple implies that u′ (0) > 0, that is ∂λ wm
proves that λm is the unique solution of µm (σ, ·) = 0 in (−∞, Λ− ], and shows moreover that
∂λ µm (σ, λm ) > 0 for all m ≥ 1.
Finally, we prove that if k ≥ K, then λn 6= λm for all m 6= n. Let thus m > n ≥ 1.
We assume by ontradition that λn = λm =: λ. Then, sine 0 = µm (σ, λ) = µn (σ, λ), we
ON STRATIFIED CAPILLARY-GRAVITY WATER WAVES
17
onlude
′ (0) ψλ′ (0)wn′ (0)
ψλ′ (0)wm
2
= (nk) σ +
> 0.
(mk) σ +
(mk)2
(nk)2
′ (0)/(mk)2 ) is inreasing and we have
But, for xed λ ≤ Λ− , the sequene (wm
2
σ+
′ (0)
ψλ′ (0)wm
ψλ′ (0)wn′ (0)
>
σ
+
> 0.
(mk)2
(nk)2
This shows that in fat µm (σ, λ) > µn (σ, λ), whih is a ontradition. Therefore
Ker ∂η Ψ(σ, λm , 0) = span{cos(mkx)},
b α (S), (4.22)
Im ∂η Ψ(σ, λm , 0) ⊕ {cos(mkx)} = C
e,k
implying that ∂η Ψ(σ, λm , 0) is a Fredholm operator of index zero. Summarising, if k ≥ K,
then µm (σ, λn ) = 0 if and only if m = n, and ∂λ µm (σ, λm ) > 0 for all m ≥ 1. Thus, we may
apply Theorem 3.1 and obtain the desired result.
Remark 4.8. It is possible to hoose the onstant Λ− in Theorem 4.3 to guarantee that some of
the stratied waves obtained therein possess stagnation points. We reall that the stagnation
points are points within the uid domain where the gradient of the stream funtion vanishes.
Indeed, assume additionally to (A1)-(A4) that we are in the homogeneous ase ρ′ = 0 and
that f > 0. Then, f (y, ψ) = β(−ψ), and ρ does not appear in system (3.1). Furthermore,
sine ∂λ ψλ′ (−1) is stritly dereasing, we may hoose the onstant Λ− in Theorem 4.3 to be
the threshold value where ψλ′ (−1) hanges sign. Indeed, sine ψλ′′ = f > 0, we dedue that
ψλ′ (y) > ψλ′ (−1) ≥ 0 for all λ ≤ Λ− and y ∈ (−1, 0]. If ρ is small enough, we then nd that
(4.14) is satised, and therefore we an make this partiular hoie. Consequently, if λ = Λ− ,
the bifuration branhes (σm (s), ηm (s)) obtained in Theorem 4.3 ontain all laminar ows
whih have stagnation points on the bottom. Not only that but, for small s, the wave with
prole η := ηm (s) ontinues to display this feature.
Indeed, using the expansion Theorem 4.3 (ii), we have
ψ̃ := T (Λ− , η) = T (Λ− , 0) − s∂η T (Λ, 0)[cos(mkx)] + O(s2 ).
Sine T (Λ− , 0) = ψΛ− , we onlude from the arguments presented in Paragraph 4.2 that
′
ψ̃y (x, −1) = −swm
(−1) cos(mkx) + O(s2 ),
′ (−1) > 0
with wm the solution of (4.7) when λ = Λ− . Sine by Hopf's maximum priniple wm
(see the arguments in the proof of Theorem 4.3) we onlude, similarly as in [14, 34℄ that
there exists a unique ξ ∈ (0, π/(mk)) suh that ψ̃y (x, −1) > 0 in (π/(mk) − ξ, π/(mk) + ξ)
and ψ̃y (x, −1) < 0 in [0, π/(mk) − ξ) ∪ (π/(mk) + ξ, 2π/(mk)]. Realling that ψ̃ = Λ− on
y = −1, we onlude that (π/(mk) − ξ, −1) and (π/(mk) + ξ, −1) are both stagnation points
for the ow. Furthermore, sine ψ̃y (x, 0) = ψΛ′ − (0) + O(s) > 0 and ψ̃yy = ψΛ′′ − + O(s) > 0,
we dedue that there exists a point y0 ∈ (−1, 0) suh that
ψ̃y (0, y0 ) = 0.
Moreover, on the line x = 0 the evenness of ψ̃ implies that ψ̃x = 0. Whene ∇ψ̃(0, y0 ) = 0,
and transforming (0, y0 ) under the dieomorphism Φη , we onlude that
(0, (y0 + 1)η(0) + y0 )
is a further stagnation point for the ow whih is loated inside the uid domain.
18
D. HENRY AND B.V. MATIOC
A similar argumet may be used to show that some of the bifuration urves obtained when
′
(A4 ) is satised ontain stagnation points.
Aknowledgement. The authors would like to thank the organisers of the programme Non-
linear Water Waves, Erwin Shrödinger Institute (Vienna), AprilJune 2011, where work on
this paper was undertaken.
DH would like to aknowledge the support of the Royal Irish
Aademy.
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[25℄
R. R. Long:
Shool of Mathematial Sienes, Dublin City University, Glasnevin, Dublin 9, Ireland.
E-mail address : david.henrydu.ie
Institut für Angewandte Mathematik, Leibniz Universität Hannover, Welfengarten 1, 30167
Hannover, Germany.
E-mail address : matioifam.uni-hannover.de