Skip to Main Content

AMS eBook CollectionsOne of the world's most respected mathematical collections, available in digital format for your library or institution

Global Regularity for 2d Water Waves with Surface Tension

About this Title

Alexandru D. Ionescu, Department of Mathematics, Princeton University and Fabio Pusateri, Department of Mathematics, Princeton University

Publication: Memoirs of the American Mathematical Society
Publication Year: 2018; Volume 256, Number 1227
ISBNs: 978-1-4704-3103-7 (print); 978-1-4704-4917-9 (online)
Published electronically: September 21, 2018
MSC: Primary 76B15.; Secondary 35Q35

View full volume PDF

View other years and numbers:

Table of Contents


  • 1. Introduction
  • 2. Preliminaries
  • 3. Derivation of the main scalar equation
  • 4. Energy estimates I: high Sobolev estimates
  • 5. Energy estimates II: low frequencies
  • 6. Energy estimates III: Weighted estimates for high frequencies
  • 7. Energy estimates IV: Weighted estimates for low frequencies
  • 8. Decay estimates
  • 9. Proof of Lemma
  • 10. Modified scattering
  • A. Analysis of symbols
  • B. The Dirichlet-Neumann operator
  • C. Elliptic bounds


We 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 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.

References [Enhancements On Off] (What's this?)