Lecture Notes of the Unione Matematica Italiana

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

Authors: Berti, Massimiliano, Delort, Jean-Marc

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  • Winner of the 2017 Book Prize of the Unione Matematica Italiana
  • Contains new results not presented elsewhere
  • Includes general tools of paradifferential calculus useful in other contexts
  • Suggests a new strategy for other problems
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About this book

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.

Table of contents (8 chapters)

Table of contents (8 chapters)

Buy this book

eBook $39.99
price for USA in USD
  • ISBN 978-3-319-99486-4
  • Digitally watermarked, DRM-free
  • Included format: PDF, EPUB
  • ebooks can be used on all reading devices
  • Immediate eBook download after purchase
Softcover $54.99
price for USA in USD
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Bibliographic Information

Bibliographic Information
Book Title
Almost Global Solutions of Capillary-Gravity Water Waves Equations on the Circle
Authors
Series Title
Lecture Notes of the Unione Matematica Italiana
Series Volume
24
Copyright
2018
Publisher
Springer International Publishing
Copyright Holder
Springer Nature Switzerland AG
eBook ISBN
978-3-319-99486-4
DOI
10.1007/978-3-319-99486-4
Softcover ISBN
978-3-319-99485-7
Series ISSN
1862-9113
Edition Number
1
Number of Pages
X, 269
Number of Illustrations
3 b/w illustrations
Topics