Nonlinear Wave Equations

397 Pages·2017·5.526 MB·English

by  Tatsien Li,Yi Zhou (auth.)

Series in Contemporary Mathematics 2
Tatsien Li · Yi Zhou
Nonlinear Wave
Equations
Series in Contemporary Mathematics
Volume 2
Editor-in-chief
Tatsien Li, Shanghai, China
Series editors
Philippe CIARLET, Heidelberg, Germany
Jean-Michel Coron, Paris, France
Weinan E, Princeton, USA
Jianshu Li, Heidelberg, Germany
Jun Li, Stanford, USA
Tatsien Li, Shanghai, China
Fanghua Lin, New York, USA
Zhiming Ma, Heidelberg, Germany
Andrew Majda, New York, USA
Cédric Villani, Lyon, France
Ya-Xiang Yuan, Beijing, China
Xiaonan Ma, Tianjin, China
Series in Contemporary Mathematics (SCM), featuring high-quality mathematical
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Tatsien Li Yi Zhou
(cid:129)
Nonlinear Wave Equations
Volume 2
Translated by Yachun Li
123
TatsienLi YiZhou
Schoolof Mathematical Sciences Schoolof Mathematical Sciences
FudanUniversity FudanUniversity
Shanghai Shanghai
China China
ISSN 2364-009X ISSN 2364-0103 (electronic)
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Preface
Nonlinear wave equations belong to a typical category of nonlinear evolutionary
equations that is of great theoretical significance and practical value. Research on
the global existence and the blow-up phenomenon of classical solutions to the
Cauchy problem with small initial data for nonlinear wave equations relates to the
asymptotic stability of the null solution or the stabilization of the corresponding
control system of such equations, and is a highly meaningful and challenging
subjectofstudy.TheresearchinthisfieldwasinitiatedbyProf.F.Johninthelate
1970s and early 1980s when he gave some examples to reveal the blow-up phe-
nomenonofsolutionstononlinearwaveequations.Lateron,Profs.S.Klainerman,
D. Christodoulou, and L. Hörmander together with Prof. F. John, as well as some
otherProfs.likeM.Kovalyov,H.Lindblad,G.Ponce,J.Shatah,T.C.Sideris,gave
variousresultsabouttheglobalexistenceandthelowerboundestimatesoflife-span
of classical solutions in different space dimensions and with different powers of
nonlineartermsontheright-handside,whichformedafrontierresearchdirectionof
great significance and attraction. Although the results obtained by these mathe-
maticianswereprofound,theydidnotcoverallthepossibleimportantsituationsat
the time, and there remained a lack about the sharpness of the established lower
boundestimatesoflife-spanforclassicalsolutions.Thewholeresearchisstill,soto
speak, in theinitial stages of development. On theother hand, there is a great deal
of diversity in the methods adopted by these mathematicians, each having its own
characteristics and some being rather complex. So there does not seem to be a
unified and easy approach to tackling such type of problems.
During my stay in France, I visited Heriot-Watt University of Great Britain in
1980 and encountered Prof. F. John who at the time happened to visit that uni-
versity,too.SoIgotthechancetoconsulthimfacetoface,whichturnedouttobea
mostinstructiveexperience.Atthebeginningof1981whenIpaidavisittoCourant
Institute of Mathematical Science in the USA, I met Prof. S. Klainerman and had
careful discussions with him. He presented me with the preprint of his long article
close to 60 pages. All these aroused my concern and interest in nonlinear wave
equationsandpromptedourstudiesinthisfield.Someofmydoctorstudentsinthe
earlier years, including Yunmei Chen, Xin Yu, and Yi Zhou, chose this as the
v
vi Preface
subject of their doctoral dissertations and made valuable contributions. Thanks to
their participation and efforts, especially Yi Zhou’s long persistent hard work, we
havemanagedtocarryonwiththisresearchsubjectinFudanUniversityuptonow
and obtained fruitful results. Our limited accomplishment in this area can be gen-
erally summarizedinto twoaspects.One isto haveestablished thecomplete lower
bound estimates of life-span (including the result of global existence) for classical
solutionstotheCauchyproblemofnonlinearwaveequationswithsmallinitialdata
inallpossiblespacedimensionsandwithallpossiblepowersofnonlineartermson
the right-hand side, and the estimates are the best ones that are unlikely to be
improved, that is to say, we in principle draw the conclusion for research in this
regard. The other is to have proposed the unified and straightforward approach to
handling such problems, that is, the global iteration method, which applies the
simple contraction mapping principle and requires roughly the same amount of
work done to prove the local existence of classical solutions.
InthebookNonlinearEvolution Equations (inChinese)coauthoredbyYunmei
Chen and me and published by Science Press (Beijing) in 1989, we proved the
globalexistenceofclassicalsolutionstotheCauchyproblemwithsmallinitialdata
fornonlinearwaveequationsbytheglobaliterationmethod.Later,inanotherbook
Global Classical Solutions for Nonlinear Evolution Equations coauthored also by
Yunmei Chen and me and published by Longman Scientific & Technical Press in
1992, the method was further employed to make some lower bound estimates of
life-span for classical solutions. However, restricted by the progress of scientific
research at that time, we did not touch upon or get the best results about the
importantsituationsofspacedimensionsn¼2andn¼4,etc.Besides,wefailedto
deal with theories related to the null condition and the sharpness of some lower
bound estimates of life-span. Since the two books both mentioned nonlinear evo-
lution equations including also nonlinear heat equations and nonlinear wave
equations are only a part of them, the length of the discussion was inevitably
limited, which to a certain extent caused the deficiency mentioned above. Around
1995, we basically finished work on the global existence and the lower bound
estimatesoflife-spanofclassicalsolutionstotheCauchyproblemwithsmallinitial
data for nonlinear wave equations, so Yi Zhou and I began to think about a
monograph on nonlinear wave equations. In fact, the Shanghai Scientific and
TechnicalPublishershadinvitedustowriteonthislongbefore,butwithtoomany
errandstogo,wewroteonandofforsometimesevenputoffforquitealongperiod.
Another important reason that hindered us from finishing the book soon was that
someofthelowerboundestimatesoflife-spanwegotthenhadnotbeenprovedto
bethebestonesimmune tofurtherimprovement,soifwewrappedupthebookin
hasteanddelivereditforpublication,itmightneverbeconsummate,whichisafact
we would not be content with. In the recent years, the sharpness of all the lower
boundestimatesofthelife-spanhasbeenobtained,andthuswefelttheurgencyof
finishingthebookasquicklyaspossible.Meanwhile,throughouttheyearswehave
found that some of the previous proofs can be simplified or improved so as to be
presentedinarelativelynewform,whichcanbeseenasanadditionalachievement.
Althoughweweredeterminedtopullourforcestogetherandstartafresh,ittookyet
Preface vii
anothertwotothreeyearsforustocompletethefinalversionofthebookin2014,
asmanyproofsneededtoberewritten.Seeingthebookfinallycomeoutaftersuch
a long track of time, the gratification of the authors can well be imagined.
Thewholebookhasgotfifteenchaptersinall.Thefirstsevenserveasaprelude
forlaterdiscussions,butstillhavetheirownmeaningsandvalues.Amongthelater
eight chapters, five discuss the global existence and the lower bound estimates of
life-span for classical solutions in all possible situations by adopting the global
iterationmethod,includingtheproofofglobalexistenceofclassicalsolutionsunder
the hypothesis of the null condition; two focus on demonstrating the sharpness
of the obtained lower bound estimates of life-span; and the last chapter entails
relevant applications and extensions. Most of the references listed in the bibliog-
raphy are cited in the body part of the book, while a few of them, though not
formally quoted, are more or less related to the content, from which we hope
readerscangetsomenecessaryinformation.Dr.KeWanghasbeenresponsiblefor
thetypewritingandtypesettingofthebookinChinese,whiletheEnglishversionis
to be translated by Prof. Yachun Li and published by the Springer-Verlag. The
authors would like to express their deep-felt gratitude to them all for their hearty
devotion, earnest support, and strong help.
Owing to the limitation of the authors’ knowledge, there must be mistakes and
careless omissions in the book, so the authors hereby sincerely invite readers to
make frank comments and criticism in any respect.
Shanghai, China Tatsien Li
February 2015
Contents
1 Introduction and Overview. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Past and Present . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.3 Methods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.4 Supplements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.5 Arrangement of Contents. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2 Linear Wave Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.1 Expression of Solutions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.1.1 Expression of Solutions When n(cid:2)3 . . . . . . . . . . . . . . 19
2.1.2 Method of Spherical Means . . . . . . . . . . . . . . . . . . . . 20
2.1.3 Expression of Solutions When nð[1Þ Is Odd. . . . . . . 25
2.1.4 Expression of Solutions When nð(cid:3)2Þ Is Even . . . . . . 26
2.2 Expression of Fundamental Solutions. . . . . . . . . . . . . . . . . . . . 28
2.3 Fourier Transform. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.4 Appendix—The Area of Unit Sphere. . . . . . . . . . . . . . . . . . . . 32
3 Sobolev Type Inequalities with Decay Factor . . . . . . . . . . . . . . . . . 35
3.1 Preliminaries. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.1.1 Commutant Relations . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.1.2 Lp;qðIRnÞ space. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.1.3 Generalized Sobolev Norms . . . . . . . . . . . . . . . . . . . . 39
3.1.4 Commutativity with the Wave Operator. . . . . . . . . . . . 41
3.1.5 Representing Derivatives Under Ordinary
Coordinates by Derivatives Under Polar
Coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.2 Some Variations of Classical Sobolev Embedding
Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3.2.1 Sobolev Embedding Theorems on a Unit Sphere . . . . . 45
3.2.2 Sobolev Embedding Theorems on a Ball . . . . . . . . . . . 46
ix
x Contents
3.2.3 Sobolev Embedding Theorems on an Annulus. . . . . . . 47
3.2.4 Sobolev Embedding Theorems for Decomposed
Dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
3.3 Sobolev Embedding Theorems Based on Binary Partition of
Unity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.3.1 Binary Partition of Unity . . . . . . . . . . . . . . . . . . . . . . 51
3.3.2 Sobolev Embedding Theorems Based on Binary
partition of Unity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.4 Sobolev Type Inequalities with Decay Factor. . . . . . . . . . . . . . 58
3.4.1 Sobolev Type Inequalities with Decay Factor
Inside the Characteristic Cone. . . . . . . . . . . . . . . . . . . 58
3.4.2 Sobolev Type Inequalities with Decay Factor
on the Entire Space . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4 Estimates on Solutions to the Linear Wave Equations . . . . . . . . . . 65
4.1 Estimates on Solutions to the One-Dimensional Linear
Wave Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.2 Generalized Huygens Principle . . . . . . . . . . . . . . . . . . . . . . . . 67
4.3 Estimates on Solutions to the Two-Dimensional Linear
Wave Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
4.4 An L2 Estimate on Solutions to the nð(cid:3)4Þ-Dimensional
Linear Wave Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
4.5 Lp;q Estimates on Solutions to the Linear Wave
Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
4.6 L1–L1 Estimates on Solutions to the Linear Wave
Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
4.6.1 L1–L1 Estimates on Solutions to the Homogeneous
Linear Wave Equation . . . . . . . . . . . . . . . . . . . . . . . . 85
4.6.2 L1–L1 Estimates on Solutions to the Inhomogeneous
Linear Wave Equations. . . . . . . . . . . . . . . . . . . . . . . . 91
4.6.3 L1–L1 Estimates on Solutions to the Linear
Wave Equations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
5 Some Estimates on Product Functions and Composite
Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
5.1 Some Estimates on Product Functions . . . . . . . . . . . . . . . . . . . 113
5.2 Some Estimates on Composite Functions . . . . . . . . . . . . . . . . . 121
5.3 Appendix—A Supplement About the Estimates on Product
Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
6 Cauchy Problem of the Second-Order Linear Hyperbolic
Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133