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Book Cover
Author Ionescu, Alexandru D

Title Global Regularity for 2D Water Waves with Surface Tension
Published Providence : American Mathematical Society, 2019


Description 1 online resource (136 pages)
Series Memoirs of the American Mathematical Society Ser. ; v. 256
Memoirs of the American Mathematical Society Ser
Contents Cover; Title page; Chapter 1. Introduction; 1.1. Free boundary Euler equations and water waves; 1.2. The main results; 1.3. Main ideas of the proof; 1.4. Paralinearization and the Dirichlet-Neumann operator; 1.5. Energy estimates and quartic energy inequalities; 1.6. Compatible vector-field structures; 1.7. Decay and modified scattering; 1.8. Organization; Chapter 2. Preliminaries; 2.1. Notation and basic lemmas; 2.2. The main proposition; Chapter 3. Derivation of the main scalar equation; 3.1. Symmetrization of the equations; 3.2. Higher order derivatives and weights
Chapter 4. Energy estimates I: high Sobolev estimates4.1. The higher order energy functional; 4.2. Analysis of the symbols and proof of Lemma 4.2; 4.3. Proof of Lemma 4.3; Chapter 5. Energy estimates II: low frequencies; 5.1. The basic low frequency energy; 5.2. The cubic low frequency energy; 5.3. Analysis of the symbols and proof of Lemma 5.2; 5.4. Proof of Lemma 5.3; Chapter 6. Energy estimates III: Weighted estimates for high frequencies; 6.1. The weighted energy functionals; 6.2. Analysis of the symbols and proof of Lemma 6.2; 6.3. Proof of Lemma 6.3
Chapter 7. Energy estimates IV: Weighted estimates for low frequencies7.1. The cubic low frequency weighted energy; 7.2. Analysis of the symbols and proof of Lemma 7.2; 7.3. Proof of Lemma 7.3; Chapter 8. Decay estimates; 8.1. Set up; 8.2. The "semilinear" normal form transformation; 8.3. The profile; 8.4. The -norm and proof of Proposition 8.1; 8.5. The equation for and proof of Proposition 8.5; Chapter 9. Proof of Lemma 8.6; 9.1. Proof of (9.6); 9.2. Proof of (9.7); 9.3. Proof of (9.8); Chapter 10. Modified scattering; Appendix A. Analysis of symbols; A.1. Notation
A.2. Quadratic symbolsAppendix B. The Dirichlet-Neumann operator; B.1. The perturbed Hilbert transform and proof of Proposition B.1; B.2. Proof of Lemma B.3; Appendix C. Elliptic bounds; C.1. The spaces _{, }; C.2. Linear, quadratic, and cubic bounds; C.3. Semilinear expansions; Bibliography; Back Cover
Summary The authors consider the full irrotational 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 the authors' analysis is to develop a sufficiently robust method (the ""quasilinear I-method"") which allows the authors to deal with strong singularities arising from time resonances in the applications of the normal form method (the so-called ""division problem""). As a result, they are able to consider a suit
Notes Print version record
Subject Waves -- Mathematical models
Water waves -- Mathematical models
Water waves -- Mathematical models
Waves -- Mathematical models
Form Electronic book
Author Pusateri, Fabio
ISBN 9781470449179