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1 A0-A NEW YORK( UNIV NY COURANT INST OF MATHEMATICAL SCIENCE-S F/6 20/11 RESEARCH IN C ONTI NUUM MECANICS- C-30 SIIDNOV 81 F JOHN N5001V76C-130NL CHNCS( UNLASIIE

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3 _EVELV New York University A rnate unnesly in Mhe public serv ice " L Couran Institute of Mathematical Sciences 251 Mercer eret New York, N.Y Telephone: (212) Novembe Director M,, 1.2 Office of Naval Research 1981 Branch Office (N 2879) ~ 666 Summer Street Boston, N-A E L Dear Sirs: Enclosed please find Professor Fritz John's Final Technical Report for research conducted under ONR contract N C We trust this is satisfactory and urge you to contact me if anything further is required. VR/sg cc: Prof. John Ms. A. Greenberg Hr. R. Alvarado Mr. J. Blaire Mr. G. Bellisari ONR-NY DDC, VA (12) NRL (Code 2627) (6) LU ONR (Code 1021P), VA (6) Sincerel, VtoRizz Financial Administrator

4 SECURITY CLPSSFi'ATION OF TNI% 041;t (II.,j noare EntIesd) REOTDCJENAINPG READ INSTRUCTIONS REPORTDOCUMENTATION PAGE BEFOREC0O61PLETINGFORM I. REPORT %umide 2 GOVT ACCESSION NO. 3. RECP-rPT'3 CATALOG NUMBER 4.TTE adsutte S. TYPE OF REPORT 6PER1OOCOVERED 2 Research in Continuum Mechanics. Final - 4/l/76-5/31/81 6. PERFORMING ORG. REPORT NUMBER 7. AUTmOR(s) S. CONI PACT OR GRANT NUMBER(s) (C Fritz4Joh / N C_--301 ~.PERFORMING ORGANIZATION NAME AND ADDRESS 0.PPOGPAM ELEMENT. PROJECT. TASK Courant Institute of Mathematical Scienceo APEA A WORK UNIT NUMBERS New York University, 251 Mercer Street* NR / New York, N. Y It. CON 7ROLLING OFFICE NAME AND ADDRESS 12. REPORT DATE Of fice of Naval Research(j B00 North Quincy Street (Code 1021P') Arlington, VA MONITORING _AGENCY WAME & ADDRESSrif different from Controllind Olffice) 15. SECURITY CLASS. (of this report) Office of Naval Research- (5th Floor) 71S Broadway, Resident Representative New York. N. Y Unclassified 16. DISTRIBUTION STATEIENT (of this Report) Ii _' Approval for public release -distribution 150. DECLASSIFICATION,'OOWNGRAOINC., SCH EDULE unlimited I?. ISTRUU?4N.SX$6M4 #ffe~~btrec-wnt,5~h *oe 4 ift I m Report) 18. SUPPLEMENTARY NOTES (see attached) It. KEY WORDS (Continue on revere. aide it necessary and ldentif; by block number) Asymptotic behavior of solutions, nonlinear equations and systems, higher order, wave equation, nonlinear wave propagation, blow-up o solutions, instability, hyperbolicity, systems of differential equations. 20. ASS TRACT (Continue on,e~erae side It necessary and Identify by block numbe) VF. John worked on two different topics during the Period coveredb the contract. The first one concerned the formation of singularities in nonlinear waves in several dimensions; the other one algebraic constraints implied by hyperbolicity of systems of partial differential equations with constant coefficients. DD I JA'4"7s 1473 EDITION OF I NOV 6 1 IOBSOLETE - / Unlassified,~,. / SCURITY CLASSIFICATION OF T04IS PAGE (whesn Dom.

5 RESEARCH IN CONTINUUM MECHANICS CONTRACT NO. N C-0301 The final report consists of the progress reports of 1976 through In addition', Fritz John continued his work on the blow-up of solutions of hyperbolic equations. The main results follow. F. John worked on two different topics during the period covered by the contract. The first one concerned the formation of singularities in nonlinear waves in several dimensions; the other one algebraic constraints implied by hyperbolicity of systems of partial differential equations with constant coefficients. For the formation of singularities the number n of space dimensions is all important. For n = 1 solutions of genuinely nonlinear wave equations with initial data of compact support blow-up after a finite time T. Here T is of order 1/c for initial data of "amplitude" c. F. J. showed (see [11) that for n 1 3 T is at least of order "2. For equations of the type utt - Au - F(u) with F(o) - F'(o) - 0, F"(o) > c > 0 the lifespan T is exactly of order 1 /ez (see [21, [3)). S. Klainerman had shown that for equations of the type utt - Au - FCu',u") no blow-up occurs for sufficiently small e if n > S. In (4) F. J. showed that Klainerman's theorem does not hold for n- 3 ; in the example utt - Au - 2 ututt a solution'with initial data of compact support and amplitude e can blow-up for arbitrarily small e. The equations fii considered play a role in nonlinear elasticity, and the

6 -2- results indicate instability of certain elastic media. For a single linear homogeneous partial differential equation with constant coefficients conditions for hyperbolicity take the form of inequalities satisfied by the coefficients. It is shown in [51, [61, [71 that for systems of equations hyperbolicity conditions can take the form of algebraic identities that must hold for the coefficients. F. J. studied in particular the case of three second order equations in four independent variables; it turns out that in significant cases the number of identities assuring hyperbolicity equals the number of double points of the characteristic surface. A consequence is that hyperbolic systems are very rare near some of the classical hyperbolic systems of mathematical physics. j~v;, l l:ll1% -Codes - I I A t,,'-0 D::: t! $; e

7 t List of Publications IA1 [I] Delayed singularity formation for solutions of nonlinear partial differential equations in higher dimensions, Proc. Nat. Acad. Sci. USA, Vol. 73, 2, pp , February [2] Blow-up of solutions of nonlinear wave equations in three space dimensions, Proc. Natl. Acad. Sci. USA, Vol. 76, 4, pp , April [3] Blow-up of solutions of nonlinear wave equations in three space dimensions, Manuscripta Mathematica 28, pp , [4) Blow-up for quasi-linear wave equations in three space dimensions, Comm. Pure Appl. Math., Vol. 34, pp , January [5] Restrictions on the coefficients of hyperbolic systems of partial differential equations, Proc. Natl. Acad. Sci. USA, Vol. 74, 10, pp. 41S0-4151, October ] Algebraic conditions for hyperbolicity of systems of partial differential equations, Comm. Pure Appl. Math., Vol. 31, pp , March [7) Addendum to: Algebraic conditions for hyperbolicity of systems of partial differential equations, Comm. Pure Appl. Math., Vol. 31, pp , November '

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