"November 2018, volume 256, number 1227 (third of 6 numbers)." Includes bibliographical references (pages 121-123).
Contents:
Elliptic bounds. Introduction -- Appendix C. Preliminaries -- Chapter 3. Derivation of the main scalar equation -- Chapter 4. Energy estimates I: high Sobolev estimates -- Chapter 5. Energy estimates II: low frequencies -- Chapter 6. Energy estimates III: Weighted estimates for high frequencies -- Chapter 7. Energy estimates IV: Weighted estimates for low frequencies -- Chapter 8. Decay estimates -- Chapter 9. Proof of Lemma 8.6 -- Chapter 10. Modified scattering -- Appendix A. Analysis of symbols -- Appendix B. The Dirichlet-Neumann operator -- Appendix C. Elliptic bounds.
Summary:
"We consider the full irrotatioal water waves system with surface tension and no gravity in dimension two (the capillary waves system), and prove global regularity and modified scattering for suitably small and localized perturbations of a flat interface. An important point of our analysis is to develop a sufficiently robust method, based on energy estimates and dispersive analysis, which allows us to deal simultaneously with strong singularities arising from time resonances in the applications of the normal form method and with nonlinear scattering. As a result, we are able to consider a suitable class of perturbations with finite energy, but no other momentum conditions. Part of our analysis relies on a new treatment of the Dirichlet-Neumann operator in dimension two which is of independent interest. As a consequence, the results in this paper are self-contained."--Page v.
Series:
Memoirs of the American Mathematical Society, 0065-9266 ; number 1227
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