Global existence of small amplitude solutions for a model quadratic quasilinear coupled wave-Klein-Gordon system in two space dimension, with mildly decaying Cauchy data / A. Stingo.

Format:

Book

Author/Creator:

Stingo, A., 1991- author.

Series:

Memoirs of the American Mathematical Society ; no.1441.

Memoirs of the American Mathematical Society, 0065-9266 ; number 1441

Language:

English

Subjects (All):

Geometric analysis.

Physical Description:

v, 256 pages : illustrations ; 26 cm.

Place of Publication:

Providence, RI : American Mathematical Society, [2023]

Summary:

The aim of this monograph is to study the global existence of solutions to a coupled wave-Klein-Gordon system in space dimension two when initial data are small smooth and mildly decaying at infinity. Some physical models strictly related to general relativity have shown the importance of studying such systems but very few results are known at present in low space dimension. We study here a model two-dimensional system, in which the nonlinearity writes in terms of "null forms", and show the global existence of small solutions. Our goal is to prove some energy estimates on the solution when a certain number of Klainerman vector fields is acting on it, and some optimal uniform estimates. The former ones are obtained using systematically quasilinear normal forms, in their para-differential version; the latter ones are recovered by deducing a new coupled system of a transport equation and an ordinary differential equation from the starting PDE system by means of a semiclassical micro-local analysis of the problem. We expect the strategy developed here to be robust enough to enable us, in the future, to treat the case of the most general nonlinearities

Contents:

Chapter 1. Introduction

Chapter 2. Main theorem and preliminary results

2.1. Statement of the main theorem

2.2. Preliminary Results

Chapter 3. Energy estimates

3.1. Paralinearization and symmetrization

3.2. Normal forms and energy estimates

Chapter 4. Uniform estimates

4.1. Semilinear normal forms

4.2. From PDEs to ODEs

4.3. Analysis of the transport equation and end of the proof

Appendix A.

Appendix B.

B.1. Some preliminary lemmas

B.2. First range of estimates

B.3. Last range of estimates

B.4. The sharp decay estimate of the Klein-Gordon solution with a Klainerman vector field

Bibliography

Index.

Notes:

"October 2023, volume 290, number 1441 (first of 5 numbers)."

Includes bibliographical references (pages 253-254) and index.

Other Format:

Online version: Stingo, A. Global Existence of Small Amplitude Solutions for a Model Quadratic Quasilinear Coupled Wave-Klein-Gordon System in Two Space Dimension, with Mildly Decaying Cauchy Data

ISBN:

1470459922

9781470459925

OCLC:

1411843839