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E-raamat: Quasi-periodic Standing Wave Solutions of Gravity-Capillary Water Waves

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Berti and Montalto prove the existence and linear stability of small amplitude time quasi-periodic standing wave solutions (that is, periodic and even in the space variable x) of a two-dimensional ocean with infinite depth under the action of gravity and surface tension. They obtain such an existence result for all the values of the surface tension belonging to a Borel set of asymptotically full Lebesgue measure. They cover the functional setting, tranversality properties of degenerate KAM theory, approximate inverse, the linearized operator in the normal directions, the almost digitalization and invertibility of Lw, and the Nash-Moser iteration. Annotation ©2020 Ringgold, Inc., Portland, OR (protoview.com)
Chapter 1 Introduction and main result
1(20)
1.1 Ideas of proof
9(8)
1.2 Notation
17(4)
Chapter 2 Functional setting
21(30)
2.1 Pseudo-differential operators and norms
24(9)
2.2 Dko-tame and Dko-modulo-tame operators
33(9)
2.3 Integral operators and Hilbert transform
42(3)
2.4 Dirichlet-Neumann operator
45(6)
Chapter 3 Transversality properties of degenerate KAM theory
51(6)
Chapter 4 Nash-Moser theorem and measure estimates
57(8)
4.1 Nash-Moser Theoreme de conjugaison hypothetique
58(2)
4.2 Measure estimates
60(5)
Chapter 5 Approximate inverse
65(14)
5.1 Estimates on the perturbation P
65(1)
5.2 Almost approximate inverse
66(13)
Chapter 6 The linearized operator in the normal directions
79(38)
6.1 Linearized good unknown of Alinhac
81(1)
6.2 Symmetrization and space reduction of the highest order
82(5)
6.3 Complex variables
87(1)
6.4 Time-reduction of the highest order
88(2)
6.5 Block-decoupling up to smoothing remainders
90(6)
6.6 Elimination of order Δx: Egorov method
96(13)
6.7 Space reduction of the order Δx
109(3)
6.8 Conclusion: partial reduction of &Pound;ω
112(5)
Chapter 7 Almost diagonalization and invertibility of &Pound;ω
117(16)
7.1 Proof of Theorem 7.3
122(9)
7.2 Alrnost-invertibility of &Pound;ω
131(2)
Chapter 8 The Nash-Moser iteration
133(8)
8.1 Proof of Theorem 4.1
138(3)
Appendix A Tame estimates for the flow of pseudo-PDEs 141(28)
Bibliography 169
Massimiliano Berti, Scuola Internazionale Superiore di Studi Avanzati (SISSA), Trieste, Italy

Riccardo Montalto, University of Zurich, Switzerland
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