Quasi-periodic standing wave solutions of gravity-capillary water waves / Massimiliano Berti, Riccardo Montalto.

Format:

Book

Author/Creator:

Berti, Massimiliano, author.

Montalto, Riccardo, author.

Series:

Memoirs of the American Mathematical Society ; Volume 263.

Memoirs of the American Mathematical Society ; Volume 263

Language:

English

Subjects (All):

Water waves--Mathematical models.

Water waves.

Wave equation--Numerical solutions.

Wave equation.

Standing waves.

Kolmogorov-Arnold-Moser theory.

Capillarity.

Physical Description:

1 online resource (184 pages).

Edition:

1st ed.

Place of Publication:

Providence, Rhode Island : American Mathematical Society, 2020.

Summary:

"We prove the existence and the linear stability of small amplitude time quasiperiodic standing wave solutions (i.e. periodic and even in the space variable x) of a 2-dimensional ocean with infinite depth under the action of gravity and surface tension. Such an existence result is obtained for all the values of the surface tension belonging to a Borel set of asymptotically full Lebesgue measure"-- Provided by publisher.

Contents:

Cover

Title page

Chapter 1. Introduction and main result

1.1. Ideas of proof

1.2. Notation

Chapter 2. Functional setting

2.1. Pseudo-differential operators and norms

2.2. ^{ ₀}-tame and ^{ ₀}-modulo-tame operators

2.3. Integral operators and Hilbert transform

2.4. Dirichlet-Neumann operator

Chapter 3. Transversality properties of degenerate KAM theory

Chapter 4. Nash-Moser theorem and measure estimates

4.1. Nash-Moser Théoréme de conjugaison hypothétique

4.2. Measure estimates

Chapter 5. Approximate inverse

5.1. Estimates on the perturbation

5.2. Almost approximate inverse

Chapter 6. The linearized operator in the normal directions

6.1. Linearized good unknown of Alinhac

6.2. Symmetrization and space reduction of the highest order

6.3. Complex variables

6.4. Time-reduction of the highest order

6.5. Block-decoupling up to smoothing remainders

6.6. Elimination of order \paₓ: Egorov method

6.7. Space reduction of the order | |^{1/2}

6.8. Conclusion: partial reduction of ℒ_{\om}

Chapter 7. Almost diagonalization and invertibility of ℒ_{\om}

7.1. Proof of Theorem 7.3

7.2. Almost-invertibility of ℒ_{\om}

Chapter 8. The Nash-Moser iteration

8.1. Proof of Theorem 4.1

Appendix A. Tame estimates for the flow of pseudo-PDEs

Bibliography

Back Cover.

Introduction and main result

Functional setting

Transversality properties of degenerate KAM theory

Nash-Moser theorem and measure estimates

Approximate inverse

The linearized operator in the normal directions

Almost diagonalization and invertibility of Lω

The Nash-Moser iteration.

Notes:

Description based on print version record.

Includes bibliographical references.

ISBN:

1-4704-5654-0