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

144 Pages·2009·0.735 MB·English

by  Gerard Iooss, Pavel I. Plotnikov

M
EMOIRS
of the
American Mathematical Society
Number 940
Small Divisor Problem in the
Theory of Three-Dimensional
Water Gravity Waves
Gérard Iooss
Pavel I. Plotnikov
(cid:1)(cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:7)(cid:7)(cid:8)(cid:5)(cid:5)(cid:9)(cid:5)(cid:5)(cid:10)(cid:11)(cid:3)(cid:2)(cid:12)(cid:13)(cid:5)(cid:6)(cid:7)(cid:7)(cid:5)(cid:5)(cid:9)(cid:5)(cid:5)(cid:14)(cid:2)(cid:12)(cid:15)(cid:13)(cid:16)(cid:5)(cid:8)(cid:17)(cid:7)(cid:5)(cid:18)(cid:19)(cid:20)(cid:19)(cid:21)(cid:22)(cid:5)(cid:11)(cid:19)(cid:5)(cid:23)(cid:5)(cid:24)(cid:2)(cid:12)(cid:15)(cid:13)(cid:16)(cid:25)(cid:26)(cid:5)(cid:5)(cid:9)(cid:5)(cid:5)(cid:27)(cid:28)(cid:28)(cid:14)(cid:5)(cid:7)(cid:7)(cid:23)(cid:29)(cid:30)(cid:8)(cid:6)(cid:23)(cid:23)
American Mathematical Society
Small Divisor Problem in the
Theory of Three-Dimensional
Water Gravity Waves
This page intentionally left blank
M
EMOIRS
of the
American Mathematical Society
Number 940
Small Divisor Problem in the
Theory of Three-Dimensional
Water Gravity Waves
Gérard Iooss
Pavel I. Plotnikov
(cid:1)(cid:2)(cid:3)(cid:4)(cid:5)(cid:6)(cid:7)(cid:7)(cid:8)(cid:5)(cid:5)(cid:9)(cid:5)(cid:5)(cid:10)(cid:11)(cid:3)(cid:2)(cid:12)(cid:13)(cid:5)(cid:6)(cid:7)(cid:7)(cid:5)(cid:5)(cid:9)(cid:5)(cid:5)(cid:14)(cid:2)(cid:12)(cid:15)(cid:13)(cid:16)(cid:5)(cid:8)(cid:17)(cid:7)(cid:5)(cid:18)(cid:19)(cid:20)(cid:19)(cid:21)(cid:22)(cid:5)(cid:11)(cid:19)(cid:5)(cid:23)(cid:5)(cid:24)(cid:2)(cid:12)(cid:15)(cid:13)(cid:16)(cid:25)(cid:26)(cid:5)(cid:5)(cid:9)(cid:5)(cid:5)(cid:27)(cid:28)(cid:28)(cid:14)(cid:5)(cid:7)(cid:7)(cid:23)(cid:29)(cid:30)(cid:8)(cid:6)(cid:23)(cid:23)
American Mathematical Society
Providence, Rhode Island
2000MathematicsSubjectClassification.
Primary76B15,47J15,35S15,76B07.
Library of Congress Cataloging-in-Publication Data
Iooss,G´erard.
Small divisor problem in the theory of three-dimensional watergravity waves / G´erard Iooss
andPavelI.Plotnikov.
p.cm. —(MemoirsoftheAmericanMathematicalSociety,ISSN0065-9266;no.940)
“Volume200,number940(fifthof6numbers).”
Includesbibliographicalreferencesandindex.
ISBN978-0-8218-4382-6(alk.paper)
1.Waterwaves. 2.Gravity waves. 3.Multiphase flow—Mathematicalmodels. 4.Pseudo-
differential operators. 5. Boundary value problems. 6. Bifurcation theory. 7. Small divisors.
I.Plotnikov,PavelI. II.Title.
QA922.L66 2009
532(cid:1).593—dc22 2009008894
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Contents
Chapter 1. Introduction 1
1.1. Presentation and History of the Problem 1
1.2. Formulation of the Problem 4
1.3. Results 7
1.4. Mathematical background 9
1.5. Structure of the paper 9
Chapter 2. Formal Solutions 15
2.1. Differential of G 15
η
2.2. Linearized equations at the origin and dispersion relation 15
2.3. Formal computation of 3-dimensional waves 17
2.4. Geometric pattern of diamond waves 20
Chapter 3. Linearized Operator 23
3.1. Linearized system in (ψ,η) (cid:2)= 0 23
3.2. Pseudodifferential operators and diffeomorphism of the torus 25
3.3. Main orders of the diffeomorphism and coefficient ν 33
Chapter 4. Small Divisors. Estimate of L− Resolvent 35
4.1. Proof of Theorem 4.10 41
Chapter 5. Descent Method-Inversion of the Linearized Operator 51
5.1. Descent method 52
5.2. Proof of Theorem 5.1 61
5.3. Verification of assumptions of Theorem 5.1 66
5.4. Inversion of L 68
Chapter 6. Nonlinear Problem. Proof of Theorem 1.3 71
Appendix A. Analytical study of G 75
η
A.1. Computation of the differential of G 75
η
A.2. Second order Taylor expansion of G in η = 0 77
η
Appendix B. Formal computation of 3-dimensional waves 79
B.1. Formal Fredholm alternative 79
B.2. Bifurcation equation 81
Appendix C. Proof of Lemma 3.6 87
v
vi CONTENTS
Appendix D. Proofs of Lemmas 3.7 and 3.8 89
Appendix E. Distribution of Numbers {ω n2} 93
0
Appendix F. Pseudodifferential Operators 99
Appendix G. Dirichlet-Neumann Operator 107
Appendix H. Proof of Lemma 5.8 119
Appendix I. Fluid particles dynamics 123
Bibliography 127
Abstract
We consider doubly-periodic travelling waves at the surface of an in-
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.
Denoting by µ = gL/c2 the dimensionless bifurcation parameter ( L is the
wavelengthalongthe directionofthetravellingwaveandcisthevelocityof
the wave), bifurcation occurs for µ =cosθ. For non-resonant cases, we first
give a large family of formal three-dimensional gravity travelling waves, in
the form of an expansion in powers of the amplitudes of two basic travelling
waves. ”Diamond waves” are a particular case of such waves, when they are
symmetric with respect to the direction of propagation.
The main object of the paper is the proof of existence of such symmetric
waves having the above mentioned asymptotic expansion. Due to the oc-
curenceof small divisors,themaindifficultyistheinversionofthelinearized
operator at a non trivial point, for applying the Nash Moser theorem. This
operator is the sum of a second order differentiation along a certain di-
rection, and an integro-differential operator of first order, both depending
periodically of coordinates. It is shown that for almost all angles θ, the
3-dimensional travelling waves bifurcate for a set of ”good” values of the
bifurcation parameter having asymptotically a full measure near the bifur-
cation curve in the parameter plane (θ,µ).
Received by the editor November 10, 2005.
2000 Mathematics Subject Classification. 76B15;47J15; 35S15; 76B07.
Key words and phrases. nonlinear water waves; small divisors; bifurcation theory;
pseudodifferential operators; traveling gravitywaves; short crested waves.
vii
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CHAPTER 1
Introduction
1.1. Presentation and History of the Problem
We consider small-amplitude three-dimensional doubly periodic travel-
ling gravity 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
bifurcationparameteristhehorizontalphasevelocity,theinfinitedepthcase
being not essentially different from the finite depth case, except for very de-
generate 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
proofsforthree-dimensionaltravellinggravity-capillarywaves,andwhenthe
surfacetensionisvery small, which isthe case inmany usual situations,this
implies a reduced domain of validity of these results.
In 1847 Stokes [Sto] gave a nonlinear theory of two-dimensional trav-
elling gravity waves, computing the flow up to the cubic order of the am-
plitude of the waves, and the first mathematical proofs for such periodic
two-dimensionalwaves are due to Nekrasov [N], Levi-Civita[Le] and Struik
[Str] about 80 years ago. Mathematical progress on the study of three-
dimensional doubly periodic water waves came much later. Inparticular, to
our knowledge, first formal expansions in powers of the amplitude of three-
dimensional travelling waves can be found in papers [Fu] and [Sr]. One can
find many references and results of research on this subject in the review
paper of Dias and Kharif [DiK] (see section 6). The work of Reeder and
Shinbrot (1981)[ReSh] represents a big step forward. These authors con-
sider symmetric diamond patterns, resulting from (horizontal) wave vectors
(cid:2)
belonging to a lattice Γ (dual to the spatial lattice Γ of the doubly periodic
pattern) spanned by two wave vectors K and K with the same length, the
1 2
velocityofthewavebeinginthedirectionofthebissectrixofthesetwowave
vectors, taken as the x horizontal axis. We give in Figure 1 two examples
1
of patterns for these waves (see the detailed comment about these pictures
at the end of subsection 2.4). These waves also appear in the literature
as ”short crested waves” (see Roberts and Schwartz [RoSc], Bridges, Dias,
Menasce [BDM] for an extensive discussion on various situations and nu-
mericalcomputations). Ifwedenotebyθ theanglebetweenK andthex −
1 1
1