 | Iooss G. Small divisor problem in the theory of three-dimensional water gravity waves / Iooss G., Plotnikov P.I. - Providence: American Mathematical Society, 2009. - vii, 128 p.: ill. - (Memoirs of the American Mathematical Society; Vol.200, N 940). - Bibliogr.: p.127-128. - ISBN 978-0-8218-4296-6; ISSN 0065-9266
|
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 Gη ........................................ 15
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) ≠ 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  -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  ............................................ 68
Chapter 6. Nonlinear Problem. Proof of Theorem 1.3 ............. 71
Appendix A. Analytical study of Gη ............................. 75
A.l. Computation of the differential of Gη ..................... 75
A.2. Second order Taylor expansion of Gη in η = 0 .............. 77
Appendix B. Formal computation of 3-dimensional waves .......... 79
B.l. Formal Fredholm alternative ............................... 79
B.2. Bifurcation equation ...................................... 81
Appendix C. Proof of Lemma 3.6 ................................. 87
Appendix D. Proofs of Lemmas 3.7 and 3.8 ....................... 89
Appendix E. Distribution of Numbers {W0n2} ..................... 93
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
|
|
