Pitcher Lecture s in the Mathematical Science s

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2 Pitcher Lecture s in the Mathematical Science s Lehigh Universit y Everett Pitcher The Pitcher Lectures are named in honor of Everett Pitcher, distinguished Professor Emeritus of Mathematics of Lehigh University. Professo r Pitche r served the mathematical community as Secretary of the American Mathematica l Society ( ), following service as Associate Secretary ( ). Professor Pitche r was bom on July 18, 1912, in Hanover, New Hampshire. H e received his A.B. from Wester n Reserve University (1932) and both his A.M. (1933) and his Ph.D. (1935) from Harvar d University. H e was a Benjamin Peirc e Instructor ( ), a member of the Institute for Advanced Stud y ( , , and ), and a Guggenheim Fello w ( ). Professor Pitche r joined the Lehigh University faculty i n Durin g World War II he was an army officer a t the Ballistics Research Laboratory, Aberdeen Proving Ground. H e served as Chairman o f the Mathematics Department at Lehigh ( ), where, subsequently, he has served as consultant to the president. Professo r Pitcher was honored by the Mathematical Associatio n of America with the Distinguished Servic e Award (1985), and was a founder and member of the Board of Trustees ( ) of the Society of Industrial and Applied Mathematics.

3 Fritz John Frit/ John was born on June 14, 1910, in Berlin, Germany. H e earned his Ph.D. from Gottingen University i n Durin g the course of his academic career. Professor John has held positions with the University of Kentucky ( ), with the U.S. War Department ( ), and with the Institute for Mathematics and Mechanics (later called the Courant Institut e of Mathematical Sciences) a t New York Universit y ( ), the last three years as Courant Professor. Professor John was a Fulbright Lecturer at Gottingen Universit y (1955), a Sherman Fairchil d Distinguished Schola r at the California Institut e of Technology ( ), and a MacArthur Fellow ( ). H e gave the Josiah Willard Gibbs Lectures at the 81st Annual Meetin g of the American Mathematica l Societ y (1975). Hi s numerous awards and honors include the Humboldt Senio r U.S. Scientist Awar d ( ), the G.D. Birkhoff Priz e in Applied Mathematics (1973), and the Steele Prize, for cumulative mathematical work, from th e American Mathematical Societ y (1982). Professo r John's research interest s include partial differential equations, nonlinear elasticity, analysis, and geometry.

4 University LECTURE Series Volume 2 Nonlinear Wav e Equation s Formation o f Singularitie s Fritz Joh n American Mathematica l Societ y Providence, Rhode Islan d

5 Pitcher Lecture s in the Mathematica l Science s held a t Lehig h Universit y April Mathematics Subject Classification (1985 Revision). Primar y 35L67, 35L70; Secondary 73G05. Library of Congress Cataloging-in-Publication Dat a John, Fritz, Nonlinear wav e equations, formation o f singularities/frit z John. p. cm. (University lectur e series, ; 2) At head o f title: Pitche r lecture s in the mathematical sciences, Lehigh University. "Revised note s of the Seventh Annua l Pitche r Lecture s delivered a t Lehig h University i n Apri l 1989"-Pref. Includes bibliographical references. ISBN (alk. paper ) 1. Nonlinear wav e equations-numerical solutions. 2. Singularitie s (Mathematics). I. Lehigh University. II. Title. III. Series. QA927.J ,.353 dc20 CI P Copying an d reprinting. Individua l reader s o f thi s publication, an d nonprofi t librarie s actin g fo r them, ar e permitte d t o mak e fai r us e o f th e material, suc h a s to cop y a n articl e fo r us e i n teachin g o r research. Permissio n i s granted t o quot e brie f passage s fro m thi s publicatio n i n reviews, provide d th e customary acknowledgment o f the source is given. Republication, systemati c copying, or multiple reproduction o f any material i n this publication (in - cluding abstracts ) i s permitte d onl y unde r licens e fro m th e America n Mathematica l Society. Request s for suc h permissio n shoul d b e addresse d t o the Manage r o f Editoria l Services, American Mathematica l Society, P.O. Box 6248, Providence, Rhode Island Copyright b y the American Mathematica l Society. Al l rights reserved. Printed i n the United State s of Americ a The America n Mathematica l Societ y retain s al l right s except thos e granted t o the Unite d State s Government. The paper use d i n this book i s acid-free an d fall s within th e guidelines established t o ensure permanenc e an d This publication wa s typeset usin g AMS'T^X, the America n Mathematica l Society' s TE X macr o system

6 Contents Preface vi Introduction 1 1. Equations in One Space Variable 3 i 2. Blow-U p in Higher Dimensions Longtime Existenc e fo r Solutions of Nonlinear Wave Equations with Small Initial Dat a 3 9 Appendix 1. Uniqueness fo r Nonlinear Wav e Equations 5 3 Appendix 2. Klainerman's Inequalit y (Adapte d fro m Klainerma n [4] ) 5 7 Bibliography 6 2 V

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8 Preface This boo k represent s revise d note s o f th e Sevent h Annua l Pitche r Lec - tures delivered a t Lehig h Universit y i n April I t wa s particularly grati - fying fo r m e to be invited by the Selection Committe e to give these lectures, in vie w o f m y long-standin g friendshi p wit h Everet t Pitche r an d m y appre - ciation o f hi s service s t o th e mathematica l community. Preparatio n o f th e Lectures ha s bee n mad e possibl e i n par t b y Nationa l Scienc e Foundatio n Grant DMS The lecture s dea l wit h som e aspect s o f th e phenomeno n o f "blow-up. " More precisely, they show how solutions of nonlinear wave equations ("finit e amplitude waves" ) hav e a tendenc y t o becom e singula r afte r a finite time, even i f they are perfectly regula r to begin with. I f and whe n blow-up occur s depends ver y muc h o n th e precis e natur e o f th e nonlinearit y an d o n th e number of spac e dimensions. Fritz Joh n

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10 Bibliography L. A. CAFFARELL I an d AVNE R FRIEDMA N I. The blow-up boundary for nonlinear wave equations. Trans. Amer. Math. Soc (1986), A. CAUCH Y 1. Memoire sur I 'integration des equations lineaires aux differentielles partielles et a coefficients constants. Oeuvres Completes, 2e serie v. I, pp D. CHRISTODOULO U 1. Global solutions of nonlinear hyperbolic equations for small data. Preprin t (1985 ) R. COURAN T an d K. O. FRIEDRICH S 1. Supersonic flow and shock waves. Interscience Springer Verlag. R. COURAN T an d D. HILBER T 1. Methods of mathematical physics. Vol II, Interscience Publ K. O. FRIEDRICH S 1. Symmetric hyperbolic linear differential equations. Comm. Pure Appl. Math. 7 (1954), Conservation equations and the laws of motion in classical physics. Comm. Pure Appl. Math. 31 (1978), R. GLASSE Y 1. Existence in the large for DM = F(u) in two space dimensions. Math Z. 178 (1981), Finite time blow-up for solutions of nonlinear wave equations. Math Z. 177 (1981), J. GLIM M 1. Solutions in the large for nonlinear hyperbolic systems of equations. Comm. Pur e Appl. Math. 18(1965), L. GARDIN G 1. Linear hyperbolic partial differential equations with constant coefficients. Act a Math. 8 5 (1950), M. GRILLAKI S 1. Regularity and asymptotic behavior of the wave equation with a critical nonlinearity. Preprint. L. HORMANDE R 1. The lifespan of classical solutions of nonlinear hyperbolic equations. Springer Lecture Notes in Math (1986),

11 BIBLIOGRAPHY On global existence of solutions of nonlinear hyperbolic equations in R l+ *. Institu t Mittag- Leffler. Repor t No. 9 (1985). 3. The lifespan of classical solutions of nonlinear hyperbolic equations. Institut Mittag-Leffler, Report No. 5 (1985), rev. version. T. J. R. HUGHES, T. KAT O an d J. E. MARSDE N 1. Well posed quasi-linear second-order hyperbolic systems with applications to nonlinear elastodynamics and general relativity. Arch. Rational Mech. Anal ( ), F. JOH N 1. Formation of singularities in one-dimensional nonlinear wave propagation. Comm. Pur e Appl. Math. 27 (1974), Partial differential equations. 4th ed. Springer Verlag, Nonadmissible data for differential equations with constant coefficients. Comm. Pur e Appl. Math. 10(1957), Delayed singularity formation in solutions of nonlinear wave equations in higher dimensions. Comm. Pure Appl. Math. 2 9 (1976), Finite amplitude waves in a homogeneous isotropic elastic solid. Comm. Pur e Appl. Math. 30(1977), Existence for large times of strict solutions of nonlinear wave equations in three space dimensions for small initial data. Comm. Pure Appl. Math. 40 (1987), Blow-up of radial solutions ofuu = c 2 (u t )&u in three space dimensions. Mat. Apl. Comput. V(1985), Blow-up for quasi-linear wave equations in three space dimensions. Comm. Pure Appl. Math. 34(1981), Blow-up of solutions of nonlinear wave equations in three space dimensions. Manuscript a Mathematica 1 8 (1979), Almost global existence of elastic waves of finite amplitude arising from small initial disturbances. Comm. Pure Appl. Math. 41 (1988), Formation of singularities in elastic waves. Lecture Notes in Physics, 195, Springer Verlag, , Solutions of quasi-linear wave equations with small initial data. The third phase. Lecture Notes in Mathematic s 1402, Springer Verlag, C. Carasso (ed.), , T. KAT O * 1. The Cauchy problem for quasilinear symmetric hyperbolic systems. Arch. Rationa l Mech. Anal. 58(1975), S. KLAINERMA N 1. The null condition and global existence in nonlinear wave equations. Lectures in Appl. Math. 23 (1986), American Math. Soc Uniform decay estimates and the Lorentz invariance of the classical wave equation. Comm. Pure Appl. Math. 3 7 (1985), Weighted L and L 1 estimates for solutions to the classical wave equation in three space dimensions. Comm. Pur e Appl. Math. 37, 1984), Remarks on the global Sovolev inequalities in the Minkowski space /? n+l. Comm. Pure Appl. Math. 3 7 (1984), P. D. LA X 1. Hyperbolic systems of conservation laws and the mathematical theory of shock waves. Re - gional Conference Serie s i n App. Math., SIAM (1973). 2. Nonlinear hyperbolic equations. Comm. Pur e Appl. Math. 6 (1953), H. LINDBLA D 1. Blow-up of solutions ofdu = \u\ p with small initial data. Univ. of Lun d and Lun d Inst, of Technology. LT H (1988 ) On the lifespan of solutions of nonlinear wave equations with small initial data. T o appear in Comm. Pur e Appl. Math.

12 64 BIBLIOGRAPHY J. RAUC H 1. The u 5 -Klein-Gordon equation, in Nonlinear PDE's and applications, ed. Brezis, Lions ; Pitman Researc h Notes in Math. 53, B. RlEMAN N 1. Ueber die Fortpjlanzung ebener Luftwellen von endlicher Schwingungsweite. Abh. Konigl. Ges. d. Wissenschaften z u Gottingen (1860), J. SHATA H 1. Ora l communication. W. A. STRAUS S 1. Nonlinear wave equations. CBM S Regional Conf. Ser. in Math., Amer. Math. Soc., Providence, RI, M. STRUV E 1. Globally regular solutions to the u 5 Klein-Gordon equation. Preprint.

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