Small divisor problem in the theory of three-dimensional water gravity waves

1 MB·English

by  Gérard Iooss

Small divisor problem in the theory of
6
0
three-dimensional water gravity waves
0
2

n G´erard Iooss†, Pavel Plotnikov‡
a
J IUF, INLNUMR 6618 CNRS- UNSA,1361 rte desLucioles, 06560 Valbonne, France
†
3 Russian academy of Sciences, Lavryentyevpr. 15, Novosibirsk 630090, Russia
2 ‡
[email protected], [email protected]

]
P February 2, 2008
A
.
h Abstract
t
a We consider doubly-periodic travelling waves at the surface of an in-
m
finitely deep perfect fluid,only subjected to gravity g and resulting from
[ the nonlinear interaction of two simply periodic travelling waves making

an angle 2θ between them.
1 2
Denotingbyµ=gL/c thedimensionlessbifurcationparameter(Listhe
v
wavelengthalongthedirectionofthetravellingwaveandcisthevelocity
1
5 of the wave), bifurcation occurs for µ = cosθ. For non-resonant cases,
5 we first give a large family of formal three-dimensional gravity travelling
1 waves,intheformofanexpansioninpowersoftheamplitudesoftwoba-
0 sictravellingwaves. ”Diamondwaves”areaparticularcaseofsuchwaves,
6 when they are symmetric with respect to thedirection of propagation.
0 The main object of the paper is the proof of existence of such symmet-
h/ ric waves having the above mentioned asymptotic expansion. Dueto the
t occurence of small divisors, the main difficulty is the inversion of the
a
linearized operator at a non trivial point, for applying the Nash Moser
m
theorem. Thisoperator isthesumof asecondorderdifferentiation along
: acertaindirection,andanintegro-differentialoperatoroffirstorder,both
v
dependingperiodically of coordinates. It is shown that for almost all an-
i
X gles θ, the 3-dimensional travelling waves bifurcate for a set of ”good”
r values of the bifurcation parameter having asymptotically a full measure
a near thebifurcation curvein theparameter plane (θ,µ).
Contents
1 Introduction 3
1.1 Presentation and history of the problem . . . . . . . . . . . . . . 3
1.2 Formulation of the problem . . . . . . . . . . . . . . . . . . . . . 5
1.3 Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.4 Mathematical background . . . . . . . . . . . . . . . . . . . . . . 11
1.5 Structure of the paper . . . . . . . . . . . . . . . . . . . . . . . . 11
1
2 Formal solutions 16
2.1 Differential of . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
η
G
2.2 Linearized equations at the origin and dispersion relation . . . . 16
2.3 Formal computation of 3-dimensional waves in the simple case . 18
2.4 Geometric pattern of diamond waves . . . . . . . . . . . . . . . . 21
3 Linearized operator 23
3.1 Linearized system in (ψ,η)=0 . . . . . . . . . . . . . . . . . . . 23
6
3.2 Pseudodifferential operators and diffeomorphism of the torus . . 25
3.3 Main orders of the diffeomorphism and coefficient ν . . . . . . . 33
4 Small divisors. Estimate of L resolvent 34
−
4.1 Proof of Theorem 4.10 . . . . . . . . . . . . . . . . . . . . . . . . 41
5 Descent method-Inversion of the linearized operator 50
5.1 Descent method. . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
5.2 Proof of Theorem 5.1. . . . . . . . . . . . . . . . . . . . . . . . . 60
5.3 Verification of assumptions of Theorem 5.1 . . . . . . . . . . . . 65
5.4 Inversion of . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
L
6 Nonlinear problem. Proof of Theorem 1.3 70
A Analytical study of 74
η
G
A.1 Computation of the differential of . . . . . . . . . . . . . . . . 74
η
G
A.2 Second order Taylor expansion of in η =0 . . . . . . . . . . . 76
η
G
B Formal computation of 3-dimensional waves 77
B.1 Formal Fredholm alternative . . . . . . . . . . . . . . . . . . . . 78
B.2 Bifurcation equation . . . . . . . . . . . . . . . . . . . . . . . . . 79
C Proof of Lemma 3.6 83
D Proofs of Lemmas 3.7 and 3.8 84
E Distribution of numbers ω n2 87
0
{ }
F Pseudodifferential operators 92
G Dirichlet-Neuman operator 100
H Proof of Lemma 5.8 111
I Fluid particles dynamics 114
2
1 Introduction
1.1 Presentation and history of the problem
Weconsidersmall-amplitudethree-dimensionaldoublyperiodictravellinggrav-
ity waves on the free surface of a perfect fluid. These unforced waves appear
in literature as steady 3-dimensional water waves, since they are steady in a
suitable moving frame. The fluid layer is supposed to be infinitely deep, and
the flow is irrotational only subjected to gravity. The bifurcation parameter is
the horizontal phase velocity, the infinite depth case being not essentially dif-
ferent from the finite depth case, except for very degenerate situations that we
do not consider here. The essential difficulty here, with respect to the existing
literature is that we assume the absence of surface tension. Indeed the surface
tension plays a major role in all existing proofs for three-dimensional travelling
gravity-capillarywaves,andwhenthesurfacetensionisverysmall,whichisthe
caseinmanyusualsituations,thisimplies areduceddomainofvalidityofthese
results.
In 1847 Stokes [40] gave a nonlinear theory of two-dimensional travelling
gravity waves, computing the flow up to the cubic order of the amplitude of
the waves, and the first mathematical proofs for such periodic two-dimensional
waves are due to Nekrasov [30], Levi-Civita [28] and Struik [41] about 80 years
ago. Mathematicalprogressesonthestudyofthree-dimensional doublyperiodic
water waves came much later. In particular, to our knowledge, first formal
expansions in powers of the amplitude of three-dimensional travelling waves
can be found in papers [16] and [39]. One can find many references and results
of researches on this subject in the review paper of Dias and Kharif [14] (see
section 6). The work of Reeder and Shinbrot (1981)[36] represents a big step
forward. These authors consider symmetric diamond patterns, resulting from
(horizontal) wave vectors belonging to a lattice Γ (dual to the spatial lattice
′
Γ of the doubly periodic pattern) spanned by two wave vectors K and K
1 2
with the same length, the velocity of the wave being in the direction of the
bissectrix of these two wave vectors, taken as the x horizontal axis. We give
1
atFigure1two examplesofpatternsfor these waves(see the detailedcomment
about these pictures at the end of subsection 2.4). These waves also appear
in litterature as ”shortcrested waves”(see Roberts and Schwartz[37], Bridges,
Dias,Menasce[5]foranextensivediscussiononvarioussituationsandnumerical
computations). If we denote by θ the angle between K and the x axis,
1 1
−
ReederandShinbrotprovedthatbifurcationtodiamondwavesoccursprovided
theangleθ isnottoocloseto0ortoπ/2,andprovidedthatthesurface tension
is not too small. In additiontheir resultis only valid outside a ”bad”set in the
parameter space, corresponding to resonances, a quite small set indeed. This
means that if one considers the dispersion relation ∆(K,c) = 0, where K and
c R2 are respectively a wave vector and the velocity of the travelling wave,
∈
then there is no resonance if for the critical value of the velocity c there are
0
only the four solutions K , K of the dispersion equation, for K Γ (i.e.
1 2 ′
± ± ∈
for K being any integer linear combination of K and K ). The fact that the
1 2
3
surface tensionis supposed not to be too small is essentialfor being able to use
Lyapunov-Schmidttechnique, and the authors mention a small divisor problem
if there is no surface tension, as computed for example in [37]. Notice that the
existenceofspatiallybi-periodicgravity waterwaveswasprovedbyPlotnikovin
[32], [31] in the case of finite depth and for fixed rational values of gL/c2tanθ,
where g,L,c are respectively the acceleration of gravity,the wave length in the
direction of propagation, and the velocity of the wave. Indeed, such a special
choice of parameters avoids resonances and the small divisor problem, because
the pseudo-inverse of the linearized operator is bounded.
Figure 1: 3-dim travelling wave, the elevation η(2) is computed with formula
ε
(1.10). Top: θ = 11.3o,τ = 1/5,ε = 0.8µ ; bottom: θ = 26.5o,τ = 1/2,ε =
c
0.6µ . The dashed line is the direction of propagation of the waves. Crests are
c
dark and troughs are grey.
CraigandNicholls(2000)[9]usedthehamiltonianformulationintroducedby
Zakharov [44], in coupling the Lyapunov-Schmidt technique with a variational
methodonthebifurcationequation. Stillinthepresenceofsurfacetension,they
could suppress the restriction of Reeder and Shinbrot on the ”bad” resonance
set in parameter space, but they pay this complementary result in losing the
smoothness of the solutions. Among other results, the other paper by Craig
and Nicholls (2002) [8] gives the principal parts of ”simple” doubly periodic
4
waves (i.e. in the non resonant cases), expanded in Taylor series, taking into
account the two-dimensions of the parameter c. They emphasize the fact that
this expansion is only formal in the absence of surface tension.
Mathematical results of another type are obtained in using ”spatialdynam-
ics”, in which one of the horizontal coordinates (the distinguished direction)
plays the role of a time variable, as was initiated by Kirchga¨ssner [25] and ex-
tensivelyappliedtotwo-dimensionalwaterwaveproblems(seeareviewin[13]).
The advantage of this method is that one does not choose the behavior of the
solutionsinthedirectionofthedistinguishedcoordinate,andsolutionsperiodic
inthis coordinateareaparticularcase,aswellasquasi-periodicorlocalizedso-
lutions (solitary waves). In this frameworkone may a prioriassume periodicity
in a direction transverse to the distinguished direction, and a periodic solu-
tion in the distinguished direction is automatically doubly periodic. The first
mathematicalresultsobtainedbythismethod,containing3-dimensionaldoubly
periodic travelling waves,start with Haragus,Kirchg¨assner,Groves and Mielke
(2001)[19], [17], [21], generalizedby GrovesandHaragus(2003)[18]. They use
a hamiltonian formulation and center manifold reduction. This is essentially
based on the fact that the spectrum of the linearized operator is discrete and
hasonlyafinitenumberofeigenvaluesontheimaginaryaxis. Theseeigenvalues
arerelatedwith the dispersionrelationmentioned above. Here, one component
(or multiples of such a component) of the wave vector K is imposed in a di-
rection transverse to the distinguished one, and there is no restriction for the
component of K in the distinguished direction, which, in solving the dispersion
relation, gives the eigenvalues of the linearized operator on the imaginary axis.
The resonant situations, in the terminology of Craig and Nicholls correspond
hereto morethanone pairofeigenvaluesonthe imaginaryaxis,(inadditionto
theorigin). Inallcasesitisknownthatthe largesteigenvalueonthe imaginary
axis leads to a family of periodic solutions, via the Lyapunov center theorem
(hamiltonian case), so, here again, there is no restriction on the resonant set
in the parameter space at a fixed finite depth. The only restriction with this
formulation is that it is necessary to assume that the depth of the fluid layer is
finite. This ensures that the spectrum of the linearized operator has a spectral
gap near the imaginary axis, which allows to use the center manifold reduction
method. In fact ifwe restrictthe study to periodic solutions ashere, the center
manifold reduction is not necessary, and the infinite depth case might be con-
sideredin using anextensionofthe proofof Lyapunov-Devaneycenter theorem
in the spirit of [23], in this case where 0 belongs to the continuous spectrum.
However,it appears that the number of imaginary eigenvalues becomes infinite
when the surface tension cancels, which prevents the use of center manifold re-
duction in the limiting case we are considering in the present paper, not only
because of the infinite depth.
1.2 Formulation of the problem
Since we are looking for waves travelling with velocity c, let us consider the
system in the moving frame where the waves look steady. Let us denote by ϕ
5
the potential defined by
ϕ=φ c X,
− ·
where φ is the usual velocity potential, X = (x ,x ) is the 2-dim horizontal
1 2
coordinate, x is the vertical coordinate, and the fluid region is
3
Ω= (X,x ): <x <η(X) ,
3 3
{ −∞ }
which is bounded by the free surface Σ defined by
Σ= (X,x ):x =η(X) .
3 3
{ }
We also make a scaling in choosing c for the velocity scale,and L for a length
| |
scale (to be chosen later), and we still denote by (X,x ) the new coordinates,
3
and by ϕ,η the unknown functions. Now defining the parameter µ = gL (the
c2
Froude number is c ) where g denotes the acceleration of gravity, and u the
√gL
unit vector in the direction of c, the system reads
∆ϕ = 0 in Ω, (1.1)
∂ϕ
η (u+ ϕ) = 0 on Σ, (1.2)
X X
∇ · ∇ − ∂x
3
( ϕ)2
u ϕ+ ∇ +µη = 0 on Σ, (1.3)
X
·∇ 2
ϕ 0 as x .
3
∇ → →−∞
Hilbert spaces of periodic functions. We specialize our study to spa-
tially periodic 3-dimensional travelling waves, i.e. the solutions η and ϕ are
bi-periodic in X. This means that there are two independent wave vectors
K ,K R2 generating a lattice
1 2
∈
Γ = K =n K +n K :n Z ,
′ 1 1 2 2 j
{ ∈ }
and a dual lattice Γ of periods in R2 such that
Γ= λ=m λ +m λ :m Z,λ K =2πδ .
1 1 2 2 j j l jl
{ ∈ · }
The Fourier expansions of η and ϕ are in terms of eiKX, where K Γ and
· ′
K λ=2nπ,n Z,forλ Γ.The situation we consider in the further∈analysis,
· ∈ ∈
is with a lattice Γ generated by the symmetric wave vectors K =(1,τ), K =
′ 1 2
(1, τ). In such a case the functions on R2/Γ are 2π periodic in x , 2π/τ
1
− − −
periodic in x , and invariantunder the shift (x ,x ) (x +π,x +π/τ) (and
2 1 2 1 2
7→
conversely). We define the Fourier coefficients of a bi- periodic function u on
such lattice Γ by
τ
√τ
u(k)= u(X)exp( ik X)dX.
2π − ·
Z[0,2π]×[0,2π/τ]
b
6
Form 0wedenotebyHm(R2/Γ)theSobolevspaceofbi-periodicfunctionsof
X R2≥/Γwhicharesquareintegrableonaperiod,withtheirpartialderivatives
∈
up to order m, and we can choose the norm as
1/2
u = (1+ k )2m u(k)2 .
m
|| || | | | | !
k Γ′
X∈
Operator equations. Now, we reduce thbe above system for (ϕ,η) to a
system of two scalar equations in choosing the new unknown function
ψ(X)=ϕ(X,η(X)),
and we define the Dirichlet-Neumann operator by
η
G
dϕ
ψ = 1+( η)2 (1.4)
Gη ∇X dn|x3=η(X)
p∂ϕ
= η ϕ
∂x |x3=η(X)−∇X ·∇X
3
wherenisnormaltoΣ,exteriortoΩ,andϕisthesolutionoftheη dependent
−
Dirichlet problem
∆ϕ = 0, x <η(X)
3
ϕ = ψ, x =η(X),
3
ϕ 0 as x .
3
∇ → →−∞
Notice that this definition of follows [27] and insures the selfadjointness and
η
positivityofthislinearoperatGorinL2(R2/Γ)(seeAppendixA.1). Ourdefinition
differs from another usual way of defining the Dirichlet - Neumann operator
without the square root in factor in (1.4). Now we have the identity(1.4) and
the system to solve reads
(U,µ,u)=0, =( , ), (1.5)
1 2
F F F F
where U =(ψ,η), and
(U,µ,u) = : (ψ) u η, (1.6)
1 η X
F G − ·∇
( ψ)2
(U,µ,u) = :u ψ+µη+ ∇ + (1.7)
2 X
F ·∇ 2
1
η ( ψ+u) 2.
−2(1+( η)2){∇X · ∇X }
X
∇
Let us define the 2-components function space
Hm(R2/Γ)=Hm(R2/Γ) Hm(R2/Γ)
0 ×
We denote the norm of U in Hm(R2/Γ) by
U = ψ + η ,
m Hm Hm
|| || || || || ||
7
where Hm means functions with 0 average, and U = (ψ,η). The 0 average
0
condition comes from the fact that the value ψ of the potential is defined up to
an additive constant (easily checked in equations (1.6), (1.7)). Moreover, the
average of the right hand side of (1.6) is 0 as it can be easily checked (this is
proved for instance in [8]). We have the following
Lemma 1.1 For any fixed m 3, the mapping
≥
(U,µ,u) (U,µ,u) is C :Hm(R2/Γ) R S Hm 1(R2/Γ)
∞ 1 −
7→F × × →
in the neighborhood of 0 R S . Moreover (,µ,u) is equivariant under
1
{ }× × F ·
translations of the plane:
v (U,µ,u)= ( vU,µ,u)
T F F T
where
vU(X)=U(X +v).
T
Inaddition, thereisM >0,suchthatfor U M and µ M , satisfies
3 3 3 3
|| || ≤ | |≤ F
for any m 3 the ”tame” estimate
≥
(U,µ,u) c (M ) U , (1.8)
m 1 m 3 m
||F || − ≤ || ||
where c only depends on m and M .
m 3
Proof. The C smoothness of (ψ,η) (ψ) : Hm(R2/Γ) Hm 1(R2/Γ)
∞ η −
7→ G →
comesfromthestudyoftheDirichlet-Neumannoperator,see(A.1,A.2),andthe
propertiesof elliptic operators. This resultis provedinparticular by Craigand
Nichollsin[9],andbyD.Lannesin[27]. Notice thatHs(R2/Γ)is analgebrafor
s > 1. Notice that it is proved by Craig et al [10] that the mapping (ψ,η)
(ψ):Hm(R2/Γ) Cm(R2/Γ) Hm 1(R2/Γ)isanalyticandtheauthorsgi7→ve
η −
G × →
the explicit Taylor expansion near 0, with the same type of ”tame” estimates
thatwe shalluseinthefollowingsections. We chooseheretostaywith(ψ,η)
Hm(R2/Γ) and we just use the C smoothness of the mapping, in addition t∈o
∞
the tame estimates (see [27]).
The equivariance of under translations of the plane is obvious.
F
We refer to [27]for the proofof the following ”tame”estimate, validfor any
k 1(heresimplerthanin[27]sincewehaveperiodicfunctionsandsincethere
≥
is no bottom wall)
(ψ) c ( η ) η ψ + ψ , (1.9)
η k k 3 k+1 3 k+1
||G || ≤ || || {|| || || || || || }
necessary to get estimate (1.8).
1.3 Results
We are now in a position to formulate the main result of this paper on the
existence of non-linear diamond waves satisfying operator equation (1.5). We
8
findanexplicitsolutionto(1.5)inthe vicinityofanapproximatesolutionU(N)
ε
whichexistenceisstatedinthefollowinglemmarestrictedto”diamondwaves”,
i.e to solutions belonging to the important subspace (still with Γ generated by
′
(1, τ))
±
Hk = U =(ψ,η) Hk(R2/Γ):ψ odd in x , even in x , η even in x and in x .
(S) { ∈ 1 2 1 2}
For these solutions the unit vector u = (1,0) is fixed (see a more general
0
statement at Theorem 2.3, with non necessarily symmetric formal solutions).
Lemma 1.2 LetN 3beanarbitrarypositive numberandthecriticalvalueof
≥
parameterµ (τ)=(1+τ2) 1/2 is suchthat thedispersion equation n2+τ2m2 =
c −
µ 2n4 has only the solution (n,m) = (1,1) in the circle m2+n2 N2. Then
−c ≤
approximate 3-dimensional diamond waves are given by
U(N) =(ψ,η)(N) = εpU(p) Hk , for any k, (1.10)
ε ε ∈ (S)
1 p N
≤X≤
1
U(1) =(sinx cosτx ,− cosx cosτx ), µ(N) =µ +µ˜, µ˜ =µ ε2+O(ε4),
1 2 µ 1 2 ε c 1
c
where
1 1 3 µ 9
c
µ = +2+ ,
1 4µ3 − 2µ2 − 4µ 2 − 4(2 µ )
c c c − c
(cid:0) (cid:1)
and where for any k,
(U(N),µ(N),u )=εN+1Q ,
F ε ε 0 ε
Q uniformly bounded in Hk , with respect to ε. There is one critical value τ
ε (S) c
of τ such that µ (τ )=0, and µ (τ)<0 for τ <τ , µ (τ)>0 for τ >τ .
1 c 1 c 1 c
Proof. The lemma is a particular case of the general Theorem 2.3 in the
symmetric case.
The followingtheoremon existence of3D-diamondwavesis the mainresult
of the paper (notice that τ =tanθ)
Theorem 1.3 Letuschoosearbitraryintegersl 23,N 3andarealnumber
≥ ≥
δ <1. Assume that
τ (δ,1/δ), µc =(1+τ2)−1/2.
∈
Then there is a set N of full measure in (0,1) with the following property. If
µ N and τ = τ , then there exists a positive ε = ε (µ ,N,l,δ) and a set
c c 0 0 c
∈ 6
= (µ ,N,l,δ) so that
c
E E
2
lim sds=1,
ε 0ε2
→ Z
(0,ε)
E∩
9
and for every µ = µ(N) with ε , equation (1.5) has a ”diamond wave” type
ε
∈ E
solution U = U(N) +εNW with W Hl . Moreover, W : Hl is a
ε ε ε ∈ (S) E → (S)
Lipschitz function cancelling at ε=0, and for τ <τ (resp. τ >τ ), and when
c c
ε varies in , the parameter µ=µ(N) runs over a measurable set of the interval
ε
E
(µ(N),µ ) (resp. (µ ,µ(N)) of asymptotically full measure near µ .
ε0 c c ε0 c
We can roughly express our result in considering the two-dimensional pa-
rameter plane (τ,µ) where τ =tanθ, 2θ being the angle between the two basic
wavevectorsofsamelengthgeneratingthetwo-dimensionallatticeΓ dualofthe
′
latticeΓofperiodsforthewaves. Thecriticalvalueµ (τ)ofµ(=gL/c2),where
c
µ (τ)=(1+τ2) 1/2 =cosθ,correspondstothesolutionsofthedispersionrela-
c −
tion we consider here (in particular 3-dimensional diamond waves propagating
in the direction of the bisectrix of the wave vectors). We show that for τ < τ
c
( 2.48) the bifurcating (diamond) waves of size O(µ µ (τ)1/2) occur for
c
≈ | − |
µ < µ (τ), while for τ > τ it occurs for µ > µ (τ). We prove that bifurcation
c c c
of these 3-dimensional waves occurs on half lines τ = const of the plane, with
their origin on the critical curve, for ”good” values of τ (which appear to be
nearly all values of τ). Moreover, we prove that on each half line, these waves
exist for ”good” values of µ, this set of ”good” values being asymptotically of
full measure at the bifurcation point µ=µ (τ) (see Figure 2).
c
Figure 2: Small sectors where 3-dimensional waves bifurcate. Their vertices lie
on the critical curve µ=µ (τ). The good set of points is asymptotically of full
c
measure at the vertex on each half line (see the detail above). In the paper we
only give the proof for each half line τ =const (dashed line on the figure)
Another way to describe our result is in terms of a bifurcation from a non
isolated eigenvalue in the spectrum of the linearized operator at the origin.
Indeed, for our critical values (τ,µ (τ)) of the parameter, the differential at
c
the origin is a selfadjoint operator with in general a non isolated 0 eigenvalue
(see Theorem 4.1). Our result means that from each point (τ,µ (τ)) where τ
c
is chosen in a full measure set of (0, ), a branch of solutions bifurcates in the
∞
10