Global Regularity for 2D Water Waves with Surface Tension (Memoirs of the American Mathematical Society)

Global Regularity for 2D Water Waves with Surface Tension (Memoirs of the American Mathematical Society)

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  • 製本 Paperback:紙装版/ペーパーバック版／ページ数 119 p.
  • 言語 ENG
  • 商品コード 9781470431037
  • DDC分類 532.05930151

Full Description

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 suitable class of perturbations with finite energy, but no other momentum conditions.

Part of the authors' 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.

Contents

Introduction
Preliminaries
Derivation of the main scalar equation
Energy estimates I: high Sobolev estimates
Energy estimates II: low frequencies
Energy estimates III: weighted estimates for high frequencies
Energy estimates IV: weighted estimates for low frequencies
Decay estimates
Proof of Lemma 8.6
Modified scattering
Appendix A. Analysis of symbols
Appendix B. The Dirichlet-Neumann operator
Appendix C. Elliptic bounds
Bibliography.