Quasiperiodic Solutions of the Generalized SQG Equation.

Format:

Book

Author/Creator:

Gómez-Serrano, Javier.

Series:

Annals of Mathematics Studies

Annals of Mathematics Studies ; v.222

Language:

English

Physical Description:

1 online resource (377 pages)

Edition:

1st ed.

Place of Publication:

Princeton : Princeton University Press, 2026.

Summary:

New, broadly applicable, parameter-free techniques for constructing stable quasiperiodic solutions of quasilinear evolution equations This monograph addresses an important problem in mathematical fluid dynamics: constructing stable, long-term solutions to certain quasilinear evolution equations.

Contents:

Cover

Contents

Preface

Acknowledgments

1. Introduction

1.1. Motivation of the problem: From Euler to gSQG0

1.2. Overview of related works

1.3. Main result

1.4. Strategy of the proof and the structure of the monograph

2. Preliminaries and Notations

2.1. Basic notations

2.2. Function spaces and norms

2.3. Linear operators

2.4. Modified fractional Laplacians Λ −1 and Υ −3

2.5. Hamiltonian Structure in 2/0

2.6. Time-reversible Hamiltonians

2.7. Translation invariance

3. Hamiltonian Structure of the gSQG Equations

3.1. Hamiltonian equation in the patch setting

3.2. Expansion of ℋ

3.3. Conservation of momentum, time-reversibility, and M-fold symmetry

4. Weak Birkhoff Normal Form

4.1. Tangential sites and normal sites

4.2. Composition with a time-1 flow

4.3. Weak Birkhoff normal form

5. Action-Angle Variables

5.1. Hypotheses on the tangential sites

6. The Nonlinear Functional Setting

6.1. Regularity of the functional ℱ

6.2. Reversible and 2 /M-translation invariant solutions

6.3. Statement of the main theorem

7. Approximate Inverse

7.1. Linearized system at an invariant torus

8. Linearized Operator in the Normal Directions

8.1. Homogeneous expansion

8.2. Finite-dimensional operators

8.3. Linearized operator in the normal directions

8.4. Structure of the operator of size ( 2)

9. Symplectic Transformations

9.1. Properties of the type (1) flow

9.2. Properties of the type (2) flow

10. Reduction to a Constant Coefficient Operator

10.1. Change of the space variables

10.2. Reparametrization of time

10.3. Egorov method

10.4. Linear Birkhoff normal form

10.5. Lipschitz tame estimates for the remainders

10.6. KAM reducibility and inversion of ℒ

10.7. Invertibility of ℒ : Proof of Proposition 7.14.

11. Nash-Moser Iteration

11.1. Measure of the frequency set

12. Proof of Theorem 6.5

A. Conjugation with the Operators Λ / −1 ,Υ / −3

B. Analysis of the Nonresonance Condition

B.1. Expressions of and → ( )

B.2. Estimates for B1, B2, B3, and ◦ /3,

B.3. Asymptotics of and →D (ξ) for large M

B.4. Choice of tangential sites

Bibliography

Index.

Notes:

Description based on publisher supplied metadata and other sources.

Other Format:

Print version: Gómez-Serrano, Javier Quasiperiodic Solutions of the Generalized SQG Equation

ISBN:

9780691280516

OCLC:

1581072960