Overview
- 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
Part of the book series: Lecture Notes of the Unione Matematica Italiana (UMILN, volume 24)
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Table of contents (8 chapters)
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Front Matter
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Back Matter
Keywords
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.
Authors and Affiliations
Bibliographic Information
Book Title: Almost Global Solutions of Capillary-Gravity Water Waves Equations on the Circle
Authors: Massimiliano Berti, Jean-Marc Delort
Series Title: Lecture Notes of the Unione Matematica Italiana
DOI: https://doi.org/10.1007/978-3-319-99486-4
Publisher: Springer Cham
eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)
Copyright Information: Springer Nature Switzerland AG 2018
Softcover ISBN: 978-3-319-99485-7Published: 12 November 2018
eBook ISBN: 978-3-319-99486-4Published: 02 November 2018
Series ISSN: 1862-9113
Series E-ISSN: 1862-9121
Edition Number: 1
Number of Pages: X, 269
Number of Illustrations: 3 b/w illustrations
Topics: Partial Differential Equations, Fourier Analysis, Dynamical Systems and Ergodic Theory, Functional Analysis