Quasi-periodic standing wave solutions of gravity-capillary water waves /

The authors prove the existence and the linear stability of small amplitude time quasi-periodic standing wave solutions (i.e. periodic and even in the space variable x) of a 2-dimensional ocean with infinite depth under the action of gravity and surface tension. Such an existence result is obtained...

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Bibliographic Details
Call Number:Libro Electrónico
Main Authors: Berti, Massimiliano (Author), Montalto, Riccardo (Author)
Format: Electronic eBook
Language:Inglés
Published: Providence, RI : American Mathematical Society, [2020]
Series:Memoirs of the American Mathematical Society ; no. 1273.
Subjects:
Wave equation.
Differential equations, Partial.
Équations d'onde.
Équations aux dérivées partielles.
Ecuaciones diferenciales
Ecuaciones de onda
Mecánica de fluidos
Differential equations, Partial
Wave equation
Fluid mechanics {For general continuum mechanics, see 74Axx, or other parts of 74-XX} > Incompressible inviscid fluids > Water waves, gravity waves; dispersion and scattering, nonlinear interaction.
Dynamical systems and ergodic theory [See also 26A18, 28Dxx, 34Cxx, 34Dxx, 35Bxx, 46Lxx, 58Jxx, 70-XX] > Infinite-dimensional Hamiltonian systems [See also 35Axx, 35Qxx] > Perturbations, KAM for inf.
Fluid mechanics {For general continuum mechanics, see 74Axx, or other parts of 74-XX} > Incompressible viscous fluids > Capillarity (surface tension) [See also 76B45].
Dynamical systems and ergodic theory [See also 26A18, 28Dxx, 34Cxx, 34Dxx, 35Bxx, 46Lxx, 58Jxx, 70-XX] > Infinite-dimensional Hamiltonian systems [See also 35Axx, 35Qxx] > Bifurcation problems.
Partial differential equations > Pseudodifferential operators and other generalizations of partial differential operators [See also 47G30, 58J40] > Pseudodifferential operators.
Online Access:Texto completo
Table of Contents:
  • Cover
  • Title page
  • Chapter 1. Introduction and main result
  • 1.1. Ideas of proof
  • 1.2. Notation
  • Chapter 2. Functional setting
  • 2.1. Pseudo-differential operators and norms
  • 2.2. ^{ ₀}-tame and ^{ ₀}-modulo-tame operators
  • 2.3. Integral operators and Hilbert transform
  • 2.4. Dirichlet-Neumann operator
  • Chapter 3. Transversality properties of degenerate KAM theory
  • Chapter 4. Nash-Moser theorem and measure estimates
  • 4.1. Nash-Moser Théoréme de conjugaison hypothétique
  • 4.2. Measure estimates
  • Chapter 5. Approximate inverse
  • 5.1. Estimates on the perturbation
  • 5.2. Almost approximate inverse
  • Chapter 6. The linearized operator in the normal directions
  • 6.1. Linearized good unknown of Alinhac
  • 6.2. Symmetrization and space reduction of the highest order
  • 6.3. Complex variables
  • 6.4. Time-reduction of the highest order
  • 6.5. Block-decoupling up to smoothing remainders
  • 6.6. Elimination of order \paₓ: Egorov method
  • 6.7. Space reduction of the order ^{1/2}
  • 6.8. Conclusion: partial reduction of ℒ_{\om}
  • Chapter 7. Almost diagonalization and invertibility of ℒ_{\om}
  • 7.1. Proof of Theorem 7.3
  • 7.2. Almost-invertibility of ℒ_{\om}
  • Chapter 8. The Nash-Moser iteration
  • 8.1. Proof of Theorem 4.1
  • Appendix A. Tame estimates for the flow of pseudo-PDEs
  • Bibliography
  • Back Cover

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