Global regularity for 2D water waves with surface tension / Alexandru D. Ionescu, Fabio Pusateri.

Format:

Book

Author/Creator:

Ionescu, Alexandru D., author.

Pusateri, Fabio, author.

Series:

Memoirs of the American Mathematical Society ; no. 1227.

Memoirs of the American Mathematical Society ; Number 1227

Language:

English

Subjects (All):

Waves--Mathematical models.

Waves.

Water waves--Mathematical models.

Water waves.

Physical Description:

1 online resource (v, 123 pages).

Edition:

1st ed.

Place of Publication:

Providence, RI : American Mathematical Society, [2018]

Summary:

The authors consider the full irrotational water waves system with surface tension and no gravity in dimension two (the capillary waves system), and prove global regularity and modified scattering for suitably small and localized perturbations of a flat interface. An important point of the authors' analysis is to develop a sufficiently robust method (the "quasilinear I-method") which allows the authors to deal with strong singularities arising from time resonances in the applications of the normal form method (the so-called "division problem"). As a result, they are able to consider a suitable class of perturbations with finite energy, but no other momentum conditions. Part of the authors' analysis relies on a new treatment of the Dirichlet-Neumann operator in dimension two which is of independent interest. As a consequence, the results in this paper are self-contained.

Contents:

Cover

Title page

Chapter 1. Introduction

1.1. Free boundary Euler equations and water waves

1.2. The main results

1.3. Main ideas of the proof

1.4. Paralinearization and the Dirichlet-Neumann operator

1.5. Energy estimates and quartic energy inequalities

1.6. Compatible vector-field structures

1.7. Decay and modified scattering

1.8. Organization

Chapter 2. Preliminaries

2.1. Notation and basic lemmas

2.2. The main proposition

Chapter 3. Derivation of the main scalar equation

3.1. Symmetrization of the equations

3.2. Higher order derivatives and weights

Chapter 4. Energy estimates I: high Sobolev estimates

4.1. The higher order energy functional

4.2. Analysis of the symbols and proof of Lemma 4.2

4.3. Proof of Lemma 4.3

Chapter 5. Energy estimates II: low frequencies

5.1. The basic low frequency energy

5.2. The cubic low frequency energy

5.3. Analysis of the symbols and proof of Lemma 5.2

5.4. Proof of Lemma 5.3

Chapter 6. Energy estimates III: Weighted estimates for high frequencies

6.1. The weighted energy functionals

6.2. Analysis of the symbols and proof of Lemma 6.2

6.3. Proof of Lemma 6.3

Chapter 7. Energy estimates IV: Weighted estimates for low frequencies

7.1. The cubic low frequency weighted energy

7.2. Analysis of the symbols and proof of Lemma 7.2

7.3. Proof of Lemma 7.3

Chapter 8. Decay estimates

8.1. Set up

8.2. The "semilinear" normal form transformation

8.3. The profile

8.4. The -norm and proof of Proposition 8.1

8.5. The equation for and proof of Proposition 8.5

Chapter 9. Proof of Lemma 8.6

9.1. Proof of (9.6)

9.2. Proof of (9.7)

9.3. Proof of (9.8)

Chapter 10. Modified scattering

Appendix A. Analysis of symbols

A.1. Notation

A.2. Quadratic symbols.

Appendix B. The Dirichlet-Neumann operator

B.1. The perturbed Hilbert transform and proof of Proposition B.1

B.2. Proof of Lemma B.3

Appendix C. Elliptic bounds

C.1. The spaces _{ , }

C.2. Linear, quadratic, and cubic bounds

C.3. Semilinear expansions

Bibliography

Back Cover.

Notes:

Includes bibliographical references.

Description based on print version record.

ISBN:

1-4704-4917-X