An Introduction to the Theory of Wave Maps and Related Geometric Problems

491 Pages·2017·2.844 MB·English

by  Dan-Andrei Geba , Manoussos G Grillakis

An Introduction to the
Theory of Wave Maps
and Related
Geometric Problems

9752hc_ 9789814713900_tp.indd 1 20/6/16 9:53 AM
May2,2013 14:6 BC:8831-ProbabilityandStatisticalTheory PST˙ws
TThhiiss ppaaggee iinntteennttiioonnaallllyy lleefftt bbllaannkk
An Introduction to the
Theory of Wave Maps
and Related
Geometric Problems
Dan-Andrei Geba
University of Rochester, USA
Manoussos G Grillakis
University of Maryland, College Park, USA

World Scientifc
NEW JERSEY • LONDON • SINGAPORE • BEIJING • SHANGHAI • HONG KONG • TAIPEI • CHENNAI • TOKYO
9752hc_ 9789814713900_tp.indd 2 20/6/16 9:53 AM
Published by
World Scientific Publishing Co. Pte. Ltd.
5 Toh Tuck Link, Singapore 596224
USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601
UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE
Library of Congress Cataloging-in-Publication Data
Names: Geba, Dan-Andrei, 1973– | Grillakis, Manoussos G.
Title: An introduction to the theory of wave maps and related geometric problems /
by Dan-Andrei Geba (University of Rochester, USA),
Manoussos G. Grillakis (University of Maryland, College Park, USA).
Description: New Jersey : World Scientific, 2016. | Includes bibliographical references and index.
Identifiers: LCCN 2016026890 | ISBN 9789814713900 (hardcover : alk. paper)
Subjects: LCSH: Nonlinear wave equations. | Differential equations, Hyperbolic. |
Solitons. | Geometry, Differential.
Classification: LCC QA927 .G37 2016 | DDC 516.9--dc23
LC record available at https://lccn.loc.gov/2016026890
British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library.
Copyright © 2017 by World Scientific Publishing Co. Pte. Ltd.
All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means,
electronic or mechanical, including photocopying, recording or any information storage and retrieval
system now known or to be invented, without written permission from the publisher.
For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance
Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy
is not required from the publisher.

Printed in Singapore
EH - An Introduction to the Theory of Wave Maps.indd 1 13-07-16 9:23:29 AM
July27,2016 15:57 WorldScientificBook-9inx6in Book-final pagev
Dan-Andrei Geba dedicates this work to his family, mentors,
and friends, to whom he owes everything.
Manoussos Grillakis dedicates this book to his parents, Marika
and Georgios.

v
May2,2013 14:6 BC:8831-ProbabilityandStatisticalTheory PST˙ws
TThhiiss ppaaggee iinntteennttiioonnaallllyy lleefftt bbllaannkk
July27,2016 15:57 WorldScientificBook-9inx6in Book-final pagevii
Preface
Thewavemapproblemisoneofthemostbeautifulandchallengingnonlin-
ear hyperbolic problems, which has kept the attention of mathematicians
for more than thirty years now. The study of the problem involves diverse
issues, e.g., well-posedness, regularity, formation of singularities, and sta-
bilityofsolitons, andcombinesintricatetoolsfromanalysis, geometry, and
topology. Moreover, thewavemapsystemhasanaturalformulationasthe
Euler-Lagrange system for a map between manifolds, a special case being
thenonlinearsigmamodel,whichisoneofthefundamentalobjectsinclas-
sical field theory. One of the goals of this book is to offer an up-to-date
and self-contained overview of the main regularity theory for wave maps.
Another goal is to introduce, to a wide mathematical audience, physically
motivatedgeneralizationsofthewavemapsystem(e.g.,theSkyrmemodel),
whichareextremelyinterestingandposechallengingnewquestionsintheir
own right.
The topic of wave maps has experienced an incredible advancement in
thepasttenyears. Thisispreciselythetimepassedfromthemomentwhen
the last monograph (i.e., Tao [171]) which tried to give a state-of-the-art
for this topic appeared. Our book tries to fill this gap by presenting the
most recent developments in the field, e.g., the resolution of the large data
global regularity theory for wave maps. These results are very technical,
being accessible only to experts in their current format. Our goal is to
try to explain them to a wider group, which includes advanced graduate
students, in the hope of stimulating new research ideas. Moreover, this
book is the first one which discusses, from a mathematical point of view,
the time evolution for the models which are extensions of the nonlinear
sigma model: the Skyrme, Faddeev, and Adkins-Nappi theories.
Our book starts by introducing the reader to the physical motivation
vii
July27,2016 15:57 WorldScientificBook-9inx6in Book-final pageviii
viii Wave Maps and Related Geometric Problems
and the mathematical formulation of the wave map problem and its gener-
alizations. This is followed by developing the analytic background needed
in the investigation of these problems, which include Strichartz estimates
and hyperbolic Sobolev spaces. The third chapter is devoted to the study
ofthelocalandsmalldataglobalwell-posednesstheoriesforthewavemap
equation, where one can see the motivation for the emergence of more and
more powerful analytic techniques needed in handling the challenging na-
tureofthese topics. Thischapteralso includesdetailedexpositionsfortwo
results,duetoTao[169]andShatah-Struwe[146],respectively,whichmake
the case for the important role played by the intrinsic geometric aspect of
the wave map problem. Next, we discuss the resolution of the large data
regularity theory for wave maps in the energy-critical case by Sterbenz-
Tataru’s program [160,161]. We focus on the second part (i.e., [161]) of
this work, which provides a complete description of the regimes when a
large data, finite energy wave map blows up and when it is global-in-time.
OurpresentationofthistopicreworksSterbenz-Tataru’sargument,includ-
ingasignificantnumberofrefinementsandextradetails. Thefifthchapter
addresses well-posedness questions for the classical Skyrme model and its
extensions. There, we present Wong’s result [191] on the Skyrme model
using Christodoulou’s regular hyperbolicity framework and Lei-Lin-Zhou’s
global regularity theorem [103] for the 2+1-dimensional Faddeev problem,
which relies on an adaptation of Klainerman’s vector field method. Fol-
lowing this, we discuss equivariant results for all the equations considered
in this book, which include non-concentration of energy, small data global
well-posednessincriticalBesovspaces,andglobalregularityforsufficiently
smooth large data. Finally, we turn our attention to the phenomenon of
collapse for wave maps and examine Rapha¨el-Rodnianski’s result [136] on
the topic. We put forth a novel approach relying on the associated Hodge
system, which makes the structure of the problem more transparent. The
book also includes an appendix detailing the basic differential geometry
concepts needed for following the presentation of the material.
Dan-Andrei Geba and Manoussos Georgios Grillakis
July27,2016 15:57 WorldScientificBook-9inx6in Book-final pageix
Acknowledgments
Dan-Andrei Geba expresses his gratitude to Viorel Barbu, Sergiu Klain-
erman, Igor Rodnianski, and especially Daniel Tataru, his mentors in the
area of partial differential equations, for introducing him to the subject,
for supervising his doctoral dissertation, and for inspiration, support, and
friendship throughout his academic career. He also thanks his colleagues
atUniversityofRochester,particularlySaradaRajeev,forintroducinghim
to the Skyrme model and related problems and for countless hours spent
on explaining ideas from theoretical physics, and Allan Greenleaf and Alex
Iosevich,formanyvaluablediscussions,comments,andprofessionaladvice.
He owes a great deal of appreciation to the students and participants to
his lectures, especially to his doctoral students, Matthew Creek, Daniel da
Silva, and Xiang Zhang. Extra thanks go to Matthew for typing a version
of some course notes that were partly included in this book.
Dan-Andrei Geba acknowledges support from the College of Arts and
Sciences at University of Rochester for allowing him to take a sabbatical
leave in the spring of 2015, which significantly helped in the completion of
thisproject. HeisalsogratefulforsupporttotheNationalScienceFounda-
tion, through Career grant DMS-0747656, and to the Simons Foundation,
through grant # 359727. Finally, he expresses infinite appreciation toward
his family for everything, especially his wife, Dana, for her immense love,
patience, and strong belief in him, his older daughter, Maria, for providing
first-class in-house editorship, and his younger one, Ioana, for motivating
him throughout the writing process by frequently asking “How much do
you have left?”
Manoussos Grillakis would like to thank Constantine Gryllakis, Walter
A. Strauss, Constantine Dafermos and Louis Nirenberg for inspiring him
and his wife Kristi Dobrovolski for her love and support.
ix