Table Of ContentSmall-amplitude steady water waves with
critical layers: non-symmetric waves
7
1
0 Vladimir Kozlova, Evgeniy Lokharua
2
aDepartment of Mathematics, Linköping University, S–581 83 Linköping
n
a
E-mail: [email protected] / V. Kozlov; [email protected] / E. Lokharu
J
8
Abstract
1
Theproblemfortwo-dimensionalsteadywaterwaveswithvorticityis
]
h considered. Using methods of spatial dynamics, we reduce the problem
p to a finite dimensional Hamiltonian system. As an application, we prove
-
h the existence of non-symmetric steady water waves when the number of
t roots of the dispersion equation is greater than 1.
a
m
[ 1 Introduction
1
v
In the present paper we consider the problem for two-dimensional steady water
1
9 waves with vorticity allowing critical layers. Following ideas of [5] and [10],
9 we formulate the problem as an evolutionary equation for which the role of
4
time is played by the spatial coordinate. This approach called spatial dynamics
0
. turnedtobeusefulfortheexistencetheoryofsmall-amplitudesteadywaves(see
1
0 [5,14,6,11,13]). UsingcentermanifoldreductiontechniqueduetoMielke[12],
7
we reduce the problem to a finite-dimensional system of ordinary differential
1
: equations. We show that this system is Hamiltonian by applying results from
v
i [10] (see also [4]). As an application, we find a large class of small-amplitude
X
solutions containing non-symmetric steady waves in the case when the number
r
a of roots of the dispersion equation is greater than 1.
2 Statement of the Problem
Let an open channel of uniform rectangular cross-section be bounded below
by a horizontal rigid bottom and let water occupying the channel be bounded
above by a free surface not touching the bottom. In appropriate Cartesian
coordinates (x,y), the bottom coincides with the x-axis and gravity acts in the
1
negative y-direction. We use the non-dimensional variables proposed by Keady
and Norbury [7] (see also Appendix A in [9] for details of scaling); namely,
lengths and velocities are scaled to (Q2/g)1/3 and (Qg)1/3 respectively. Here
Q and g are the dimensional quantities for the rate of flow and the gravity
acceleration respectively, whereas (Q2/g)1/3 is the depth of the critical uniform
stream in the irrotational case.
The steady water motion is supposed to be two-dimensional and rotational;
thesurfacetensionisneglectedonthefreesurfaceofthewater, wherethepres-
sure is constant. These assumptions and the fact that water is incompressible
allow us to seek the velocity field in the form (ψ ,−ψ ), where ψ(x,y) is re-
y x
ferred to as the stream function. The vorticity distribution ω is supposed to be
a prescribed smooth function depending only on the values of ψ.
Wechoosetheframeofreferencesothatthevelocityfieldistime-independent
aswellastheunknownfree-surfaceprofile. Thelatterisassumedtobethegraph
of Y =η(X), X ∈R, where η is a positive continuous function, and so the lon-
gitudinal section of the water domain is D = {x ∈ R, 0 < Y < η(X)}. The
following free-boundary problem for ψ and η which describes all kinds of waves
has long been known (cf. [7]):
ψ +ψ +ω(ψ)=0, (X,Y)∈D; (2.1)
XX YY
ψ(X,0)=0, X ∈R; (2.2)
ψ(X,η(X))=1, X ∈R; (2.3)
|∇ψ(X,η(X))|2+2η(X)=3r, X ∈R. (2.4)
Incondition(2.4)(Bernoulli’sequation),r isaconstantconsideredastheprob-
lem’s parameter and referred to as Bernoulli’s constant/the total head.
2.1 Stream solutions
By a stream (shear-flow) solution we mean a pair (ψ,η)=(u(Y), d), d=const,
solving problem (2.1)–(2.4) which reduces to the following one:
u(cid:48)(cid:48)+ω(u)=0 on (0,d), u(0)=0, u(d)=1, |u(cid:48)(d)|2+2d=3r. (2.5)
Here, the prime symbol denotes differentiation with respect to Y. A detailed
studyofthesesolutionsincludingthosethatdescribeflowswithcounter-currents
is given in [8]. Here we present a description of such solutions borrowed from
[8]. They are parametrised by s=u(cid:48)(0) which satisfies the inequality
(cid:114) (cid:90) τ
s>s = 2 max Ω(τ), whereΩ(τ)= ω(t)dt.
0
0≤τ≤1 0
2
In order to describe all solutions to (2.5) it is convenient to introduce the fol-
lowing quantities:
• Denote by τ (s) (τ (s)) the smallest positive root (the largest negative
+ −
root) of the equation 2Ω(τ)=s2.
• Let U(Y;s), s>0 be the solution of the Cauchy problem
U(cid:48)(cid:48)+ω(U)=0, U(0;s)=0, U(cid:48)(0;s)=s. (2.6)
Then the maximal interval of monotonicity of the function U is (y (s),y (s)),
− +
where
(cid:90) τ± dτ
y (s)= .
± (cid:112)
s2−Ω(τ)
0
We note that if ω(τ (s))=0 then y (s)=±∞
± ±
• If s>s then
0
(cid:90) 1 dτ
d(s)=
(cid:112)
s2−Ω(τ)
0
is the smallest positive value of d such that U(d;s)=1. Therefore the function
U(·;s) satisfies the first three equations in (2.5) on the interval (0,d(s)).
Geometrical meaning of the quantities τ is the following:
±
τ (s)=supU(z;s) and τ (s)=infU(z;s).
+ −
z z
Now we are in position to describe solutions to problem (2.5):
a) (u(Y),d)=(U(Y;s),d(s)), where d=d(+), j =0,1,...,
j
d(+)(s)=d(s)+2k(y (s)−y (s)),
2k + −
d(+) (s)=d(s)+2(y (s)−d(s))+2k(y (s)−y (s)), k =0,1,...
2k+1 + + −
and
1(cid:16) (cid:17)
r =R(+)(s)= s2−2Ω(1)+2d(+) .
j 3 j
The corresponding solution U has the following properties: U(cid:48)(0,s)<0 and U(cid:48)
has j zeros on the interval (0,d(s)).
b) (u(Y),d)=(U(Y −2y (s);s),d(s)), where d=d(−), j =1,2,...,
− j
1(cid:16) (cid:17)
d(−)(s)=d(+)(s)−2y (s), r =R(−)(s)= s2−2Ω(1)+2d(−) .
j j − j 3 j
The corresponding solution U satisfies: U(cid:48)(0,s) > 0 and U(cid:48) has j zeros inside
the interval (0,d(s)).
3
In what follows we assume that the vorticity function ω depends on a pa-
rameter λ(cid:48) =(λ ,...,λ )∈Rm−1, ω(u)=ω(u;λ(cid:48)), and that this parameter
1 m−1
is taken from a neighborhood of some λ(cid:48) and put
(cid:63)
ω (p)=ω(p;λ(cid:48)),
(cid:63) (cid:63)
Since a stream solution (u,d) depends on both the vorticity ω (which depends
on λ(cid:48)) and s, it will be convenient to use the notation λ for the parameter s
0
so that we have now only one vector parameter λ = (λ ,λ(cid:48)) = (λ ,...,λ ),
0 0 m−1
which is taken from a certain neighborhood
Λ={λ∈Rm : |λ−λ |<ε}, λ =(s ,λ(cid:48)),
∗ (cid:63) (cid:63) (cid:63)
where the parameters λ(cid:48) and s are fixed throughout the paper. Thus, both
(cid:63) (cid:63)
the vorticity ω and the stream solution (u,d) depend on the same parameter
λ∈Λ. Furthermore, we put
u (Y)=u(Y;λ ):=U(Y;s ), d =d(s ),
(cid:63) ∗ (cid:63) (cid:63) (cid:63)
where(u(Y;λ),d(λ))isthestreamsolutioncorrespondingtothevorticityω(p;λ)
and such that u(cid:48)(0)=λ . The Bernoulli constant for u is given by r =r(λ),
0 (cid:63) (cid:63)
where
1(cid:16) (cid:90) 1 (cid:17)
r(λ)= λ2−2 ω(p;λ)dp+2d(λ) .
3 0
0
In what follows we will assume that
u(cid:48)(d )(cid:54)=0, d (cid:54)=r .
(cid:63) (cid:63) (cid:63) (cid:63)
We will always assume that the function ω(p;λ(cid:48)) is of class Cν+1 for a certain
ν = 1,2,.... Since we consider only d±(s) < ∞ then ω(τ (s)) (cid:54)= 0 when it is
j ±
relevant and so the functions τ (s), y (s), which depend also on λ(cid:48), are also
± ±
Cν+1 smooth for λ∈Λ, provided ε is sufficiently small. Moreover all functions
U, d, R± are of the same smoothness.
j
2.2 Dispersion equation
In what follows an important role will play a dispersion equation. It’s impor-
tance for the existence of small amplitude Stokes waves was demonstrated in
[2] for unidirectional flows and in [9] for general flows. Let (u,d) be a stream
solutionof (2.1)–(2.4)withu(cid:48)(d)(cid:54)=0forsomefixedBernoulli’sconstantr. The
dispersion equation is defined as the following eigenvalue problem:
−ϕ −ω(cid:48)(u)ϕ=µϕ on (0,d), ϕ(0)=0, ϕ (d)=κϕ(d). (2.7)
zz z
4
Here
κ=1/k2−ω(1)/k, k =u(cid:48)(d). (2.8)
Thespectrumof (2.7)consistsofcountablesetofsimplerealeigenvalues{µ }∞
j j=1
ordered so that µ < µ for all j < l. Furthermore, only a finite number of
j l
eigenvalues may be negative. The normalized eigenfunction corresponding to
an eigenvalue µ will be denoted by ϕ . Thus, the set of all eigenfunctions
j j
{ϕ }∞ forms an orthonormal basis in L2(0,d).
j j=1
Note that the Sturm-Liouville problem (2.7) depend on the triple (ω,u,d).
Thus, let µ (λ) be the j-th eigenvalue of the problem (2.7) for the triple
j
(ω(p;λ),u(Y;λ),d(λ)), λ∈Λ. Furthermore, we define
1 ω (1)
k =u(cid:48) (d ;λ ), κ = − (cid:63)
(cid:63) Y (cid:63) (cid:63) (cid:63) k2 k
(cid:63) (cid:63)
and
µ∗ =µ (λ ).
j j ∗
In what follows we will assume that
µ∗ ≤0, µ >0 (2.9)
N N+1
for some N > 1 which will be fixed throughout the paper. Because all the
eigenvalues of (2.7) are simple, then the functions µ (λ) and ϕ (z,λ) are of
j j
class Cν+1 in Λ and [0,d(λ)]×Λ respectively.
3 Reduction to a finite dimensional system
3.1 First order system
We start by rectifying the domain D. For this purpose, we change the coordi-
nates to
d
x=X, z =Y ,
η(X)
whilethehorizontalcoordinateremainsunchanged. Thus, thedomainDtrans-
formsintothestripS =IR×(0,d). Next,weintroduceanewunknownfunction
Φ(x,z) on S¯ by
(cid:16) z (cid:17)
Φ(x,z)=ψ x, η(x) .
d
A direct calculation shows that problem (2.1)-(2.3) rewrites as
(cid:20) zη (cid:21) zη (cid:20) zη (cid:21) (cid:18)d(cid:19)2
Φ − xΦ − x Φ − xΦ + Φ +ω(Φ)=0; (3.1)
x η z η x η z η zz
x z
Φ(x,0)=0, Φ(x,d)=1, x∈IR; (3.2)
5
while the Bernoulli’s equation (2.4) becomes
η2 (cid:18)3r−2η(cid:19)
Φ2 = . (3.3)
z d2 1+η2
x
The next step is to write (3.1) as a first order system. For this purpose, we
introduce a new variable
(cid:20) (cid:21)
η(x) zη (x)
Ψ(x,z)= Φ (x,z)− x Φ (x,z) .
d x η(x) z
Thus,writing (3.1)intermsofΨandΦandusingthedefinitionofΨtoexpress
Φ , we obtain
x
d z
Φ = Ψ+ η Φ , (3.4)
x η η x z
1 d η
Ψ = η (zΨ) − Φ − ω(Φ). (3.5)
x η x z η zz d
Using the identity η = −Ψ(x,d)/Φ (x,d) which follows from the definition of
x z
Ψ by letting z =d, the Bernoulli’s equation (3.3) becomes
Ψ2(x,d)+Φ2(x,d)=P(η(x)), (3.6)
z
where the function P(t) is defined by
P(t)=t2[3r−2t]/d2.
If Ψ and Φ−u are small then the function η can be uniquely resolved from
the equation (3.6) so that η is close to d, provided d (cid:54)= r. Having this in
mind and that η =−Ψ(x,d)/Φ (x,d) equations (3.4), (3.5) together with the
x z
boundary conditions (3.2) can be considered as first order system with respect
to the variables Φ and Ψ only. In this case the latter functions are consider as
continuous functions with values in W2,2(0,d) and W1,2(0,d) whose derivatives
1
with respect to x belong to W1,2(0,d) and L (0,d) respectively. W2,2(0,d)
1 2 γ
consists of functions w from W2,2(0,d) satisfying w(0) = 0 and w(d) = γ. In
[10] it was shown that the system (3.4)-(3.6) admits Hamiltonian structure (see
also [4]).
3.2 Linearization around the stream solution
We put
Φ=Φ¯ +u, Ψ=Ψ¯, η =ζ+d. (3.7)
6
Using that P(cid:48)(d)/2k = k/d−1/k, equations (3.4) and (3.5) together with the
boundary conditions (3.2) and (3.6) lead to
zu ζ
Φ¯ =Ψ¯ + z x +N¯ (3.8)
x d 1
2ζω(u)
Ψ¯ =−Φ¯ − −ω(cid:48)(u)Φ¯ +N¯ (3.9)
x zz d 2
Φ¯(x,0)=Φ¯(x,d)=Ψ¯(x,0)=0 (3.10)
(cid:18) (cid:19)
k 1
Φ¯ (x,d)− − ζ =N¯ , (3.11)
z d k 3
where the nonlinear operators N¯ ,j =1,2,3, are given by
j
ζΨ¯ (cid:20)dΦ −ζu (cid:21)
N¯ (Ψ¯,Φ¯ ;ζ,ζ )=− +zζ z z (3.12)
1 z x d+ζ x d(d+ζ)
and
N¯ (Ψ¯,Ψ¯ ,Φ¯,Φ¯ ;ζ,ζ )=
2 z zz x
ζ ζ(Φ¯ −ω(cid:48)(u)Φ¯) ζ2
d+xζ(zΨ¯)z+ zzd+ζ − d(d+ζ)uzz (3.13)
ζ+d
− (ω(u+Φ¯)−ω(u)−ω(cid:48)(u)Φ¯),
d
while the nonlinear part in the Bernoulli’s equation is defined by
N¯ (Ψ¯,Φ¯ ;ζ)= 1 (cid:2)−(Ψ¯2+Φ¯2) +(P(d+ζ)−P(d)−P(cid:48)(d)ζ)(cid:3).
3 z 2k z z=d
Next we use the following change of variables (see [3], Sect.3):
zu ζ
Φˆ =Φ¯ − z , Ψˆ =Ψ¯. (3.14)
d
Then the relations Φ¯(x,d)=0 and η =−Ψ(x,d)/Φ (x,d) imply
x z
Ψˆ(x,d)
ζ =−(1/k)Φˆ(x,d), ζ =− . (3.15)
x (cid:104) (cid:105)
k+Φˆ (x,d)− 1 − ω(1) Φˆ(x,d)
z d k
for all x ∈ IR. This allows to rewrite (3.8)-(3.11) as a system of the first order
only in terms of the functions Ψˆ and Φˆ:
Φˆ =Ψˆ +Nˆ (3.16)
x 1
Ψˆ =−Φˆ −ω(cid:48)(u)Φˆ +Nˆ (3.17)
x zz 2
Φˆ(x,0)=Ψˆ(x,0)=0 (3.18)
Φˆ (x,d)−κΦˆ(x,d)=Nˆ , (3.19)
z 3
7
where κ is given by (2.8). The nonlinear operators Nˆ are naturally defined by
j
ζΨ¯ dΦˆ +zζu
Nˆ (Ψˆ,Φˆ,Φˆ )=− +zζ z zz;
1 z d+ζ x d(d+ζ)
ζ ζ(Φˆ −ω(cid:48)(u)Φˆ −2ζzω(cid:48)(u)u /d)
Nˆ (Ψˆ,Φˆ,Φˆ ,Φˆ )= x (zΨˆ) + zz z (3.20)
2 z zz d+ζ z d+ζ
ζ+d(cid:16) (cid:16) zu ζ(cid:17) (cid:16) zu ζ(cid:17)(cid:17)
− ω u+Φˆ + z −ω(u)−ω(cid:48)(u) Φˆ + z ,
d d d
1 (cid:20) (cid:16) (cid:16) (zu ) ζ(cid:17)2(cid:17) (cid:21)
Nˆ (Ψˆ,Φˆ,Φˆ )= − Ψˆ2+ Φˆ + z z +(P(d+ζ)−P(d)−P(cid:48)(d)ζ) .
3 z 2k z d z=d
where ζ =ζ(Φˆ) and ζ =ζ (Ψˆ,Φˆ,Φˆ ) are defined by (3.15).
x x z
Next, by using implicit function theorem we solve equation (3.19) with re-
spect to Φ and obtain
z
Φˆ (x,d)−κΦˆ(x,d)=F (Ψˆ,Φˆ), (3.21)
z 3
where F is analytic function satisfying
3
F (ξ ,ξ )=O(|ξ|2).
3 1 2
Moreover, one can verify, for example by solving (3.19) with the help of fixed
point iterations, that
F (Ψˆ,Φˆ)=Nˆ (Ψˆ,Φˆ,κΦˆ)(x,d)+F (Ψˆ,Φˆ), (3.22)
3 3 31
where
F (ξ ,ξ )=O(|ξ|3).
31 1 2
3.3 Spectral decomposition
Let
H2 ={w ∈H2(0,d):w(0)=0}, H1 ={v ∈H1(0,d):v(0)=0}.
d d
Also let
X ={w ∈H2 :w (d)−κw(d)=0}.
λ d z
Weremindthatdandκmaydependonλ. ThespaceX representsthedomain
λ
oftheoperator−∂2−ω(cid:48)(u)withtheboundaryconditions(2.7)andthestandard
z
norm in H2(0,d) is equivalent to the graph norm.
We introduce the orthogonal projectors
N (cid:90) d
(cid:88)
Pφ= αjϕj, αj = φϕjdz, P(cid:101) =id−P,
j=1 0
8
and represent the functions Ψˆ and Φˆ as
N N
Φˆ =PΦˆ +P˜Φˆ =(cid:88)α (x)ϕ +φ˜, Ψˆ =PΨˆ +P˜Ψˆ =(cid:88)β (x)ϕ +ψ˜. (3.23)
j j j j
j=1 j=1
Multiplying (3.16)-(3.17) by ϕ , j = 1,...,N, and integrating over (0,d),
j
we obtain
α(cid:48) =β +F (3.24)
j j 1j
β(cid:48) =µ α +F ; (3.25)
j j j 2j
where
(cid:90) d
F = Nˆ (Ψˆ,Φˆ,Φˆ )ϕ dz,
1j 1 z j
0
and
(cid:90) d
F = Nˆ (Ψˆ,Φˆ,Φˆ ,Φˆ )ϕ dz−F (Ψˆ,Φˆ)ϕ (d),
2j 2 z zz j 3 j
0
Subtractingthesumofequations(3.24)and(3.25)multipliedbuϕ from(3.16)
j
and (3.17) respectively, we obtain
φ(cid:101)x =ψ(cid:101)+P(cid:101)(Nˆ1) (3.26)
N
ψ(cid:101)x =−φ(cid:101)zz−ω(cid:48)(u)φ(cid:101)+P(cid:101)(Nˆ2)+(cid:88)F3(Ψˆ,Φˆ)(d)ϕj(d)ϕj. (3.27)
j=1
The boundary conditions (3.18) and (3.21) take the form
φ(cid:101)(x,0)=ψ(cid:101)(x,0)=0, φ(cid:101)z(x,d)−κφ(cid:101)(x,d)=F3(Ψˆ,Φˆ). (3.28)
In what follows we are going to apply a center manifold reduction theorem
due to Mielke and for this purpose we need to fix the domain of the operators
in (3.26)-(3.28) independently of the parameter λ ∈ Λ. To reduce the problem
to the fixed strip R×(0,d ), we apply the following change of variables:
∗
dy dy
y =d∗z/d and φ(cid:101)(x,y)=φ(cid:101)(x, d ) ψ(cid:101)(x,y)=ψ(cid:101)(x, d ). (3.29)
∗ ∗
Weusethesamenotationsforthefunctionsφ(cid:101)andψ(cid:101)innewandoldcoordinates
and hope that this will not cause misunderstanding. Then the system (3.26)–
(3.28) transforms to
φ(cid:101)x =ψ(cid:101)+P(cid:101)(Nˆ1),
ψ(cid:101)x =−dd2∗2φ(cid:101)yy−ω(cid:48)(u)φ(cid:101)+P(cid:101)(Nˆ2)+(cid:88)N F3(Ψˆ,Φˆ)(d)ϕj(d)ϕj (3.30)
j=1
9
and
dκ d
φ(cid:101)(x,0)=ψ(cid:101)(x,0)=0, φ(cid:101)y(x,d∗)− d φ(cid:101)(x,d∗)= d F3(Ψˆ,Φˆ). (3.31)
∗ ∗
Moreover, the projector P in new coordinates becomes
d (cid:88)N (cid:90) d∗
Pu(y)=Pλu(y)=d u(τ)ϕj(τd/d∗)dτ ϕj(yd/d∗) (3.32)
∗ j=1 0
and the variables φ(cid:101)z and φ(cid:101)zz must be replaced by d∗φ(cid:101)y/d and d2∗φ(cid:101)y/d2 respec-
tively.
3.4 Main result
We put
Hλn ={ψ(cid:101)∈Hdn∗ : Pλψ(cid:101)=0}, n=0,1,2.
Theorem 3.1. There exists neighborhoods W, W and W of the origins in
2 1
R2N, H2 and in H1 respectively and smooth vector functions
λ λ
h : W ×Λ→W , g : W ×Λ→W
2 1
with the following properties:
1). Functions h and g are of the class Cν provided the vorticity function
ω(y;λ(cid:48)) is of the class Cν+1. Moreover,
||h;H2 ||+||g;H1 ||=O(|α|2+|β|2).
d∗ d∗
2). We introduce the system
α(cid:48) =β +f (α,β;λ)
j j 1j
β(cid:48) =µ2α +f (α,β;λ), j =1,...,N. (3.33)
j j j 2j
Here
f (α,β;λ)=F (Ψˆ,Φˆ,Φˆ ) and f (α,β;λ)=F (Ψˆ,Φˆ,Φˆ ,Φˆ ),
1j 1,j z 2j 2,j z zz
where F and F are the same as in system (3.24), (3.25) and
1,j 2,j
N N
Φˆ(α,β)(z)=(cid:88)α ϕ (z)+h(zd /d;α,β), Ψˆ(α,β)(z)=(cid:88)β ϕ (z)+g(zd /d;α,β).
j j ∗ j j ∗
j=1 j=1
(3.34)
10
