Advertisement

Almost Global Solutions of Capillary-Gravity Water Waves Equations on the Circle

  • Massimiliano Berti
  • Jean-Marc Delort

Part of the Lecture Notes of the Unione Matematica Italiana book series (UMILN, volume 24)

Table of contents

  1. Front Matter
    Pages i-x
  2. Massimiliano Berti, Jean-Marc Delort
    Pages 1-25
  3. Massimiliano Berti, Jean-Marc Delort
    Pages 27-30
  4. Massimiliano Berti, Jean-Marc Delort
    Pages 31-91
  5. Massimiliano Berti, Jean-Marc Delort
    Pages 93-112
  6. Massimiliano Berti, Jean-Marc Delort
    Pages 113-155
  7. Massimiliano Berti, Jean-Marc Delort
    Pages 157-216
  8. Massimiliano Berti, Jean-Marc Delort
    Pages 217-252
  9. Massimiliano Berti, Jean-Marc Delort
    Pages 253-262
  10. Back Matter
    Pages 263-269

About this book

Introduction

The goal of this monograph is to prove that any solution of the Cauchy problem for the capillary-gravity water waves equations, in one space dimension, with periodic, even in space, small and smooth enough initial data, is almost globally defined in time on Sobolev spaces, provided the gravity-capillarity parameters are taken outside an exceptional subset of zero measure.

In contrast to the many results known for these equations on the real line, with decaying Cauchy data, one cannot make use of dispersive properties of the linear flow. Instead, a normal forms-based procedure is used, eliminating those contributions to the Sobolev energy that are of lower degree of homogeneity in the solution. Since the water waves equations form a quasi-linear system, the usual normal forms approaches would face the well-known problem of losses of derivatives in the unbounded transformations. To overcome this, after a paralinearization of the capillary-gravity water waves equations, we perform several paradifferential reductions to obtain a diagonal system with constant coefficient symbols, up to smoothing remainders. Then we start with a normal form procedure where the small divisors are compensated by the previous paradifferential regularization. The reversible structure of the water waves equations, and the fact that we seek solutions even in space, guarantees a key cancellation which prevents the growth of the Sobolev norms of the solutions.

Keywords

35A01, 76B15, 35Q35 Capillary-gravity water waves Long-time existence for PDEs Paradifferential calculus Normal forms Small divisors

Authors and affiliations

  • Massimiliano Berti
    • 1
  • Jean-Marc Delort
    • 2
  1. 1.Department of MathematicsInternational School for Advanced Studies SISSATriesteItaly
  2. 2.LAGASorbonne Paris-Cité/University Paris 13VilletaneuseFrance

Bibliographic information

  • DOI https://doi.org/10.1007/978-3-319-99486-4
  • Copyright Information Springer Nature Switzerland AG 2018
  • Publisher Name Springer, Cham
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-3-319-99485-7
  • Online ISBN 978-3-319-99486-4
  • Series Print ISSN 1862-9113
  • Series Online ISSN 1862-9121
  • Buy this book on publisher's site