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Almost global solutions of capillary-gravity water waves equations on the circle

Record Type:	Language materials, printed : Monograph/item
Title/Author:	Almost global solutions of capillary-gravity water waves equations on the circle/ by Massimiliano Berti, Jean-Marc Delort.
Author:	Berti, Massimiliano.
other author:	Delort, Jean-Marc.
Published:	Cham :Springer International Publishing : : 2018.,
Description:	x, 269 p. :ill., digital ; : 24 cm.;
Contained By:	Springer eBooks
Subject:	Cauchy problem. -
Online resource:	https://doi.org/10.1007/978-3-319-99486-4
ISBN:	9783319994864

Almost global solutions of capillary-gravity water waves equations on the circle
Berti, Massimiliano.

Almost global solutions of capillary-gravity water waves equations on the circle[electronic resource] /by Massimiliano Berti, Jean-Marc Delort. - Cham :Springer International Publishing :2018. - x, 269 p. :ill., digital ;24 cm. - Lecture notes of the Unione Matematica Italiana,241862-9113 ;. - Lecture notes of the Unione Matematica Italiana ;16..

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.

ISBN: 9783319994864

Standard No.: 10.1007/978-3-319-99486-4doiSubjects--Topical Terms:

785971
Cauchy problem.

LC Class. No.: QA377 / .B478 2018

Dewey Class. No.: 515.35

Almost global solutions of capillary-gravity water waves equations on the circle
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520 $a 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.
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