Formation and propagation of singularities for quasilinear hyperbolic systems
Author:
Dexing Kong
Journal:
Trans. Amer. Math. Soc. 354 (2002), 31553179
MSC (2000):
Primary 35L45, 35L67; Secondary 35L65, 76L05
Published electronically:
April 2, 2002
MathSciNet review:
1897395
Fulltext PDF Free Access
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Abstract: Employing the method of characteristic coordinates and the singularity theory of smooth mappings, in this paper we analyze the longterm behaviour of smooth solutions of general quasilinear hyperbolic systems, provide a complete description of the solution close to blowup points, and investigate the formation and propagation of singularities for systems of hyperbolic conservation laws.
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 [2]
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 [3]
 P. H. Chang, On the breakdown phenomena of solutions of quasilinear wave equations, Michigan Math. J. 23 (1976), 277287. MR 57:901
 [4]
 Shuxing Chen and Liming Dong, Formation of shocks for psystem with general smooth initial data, to appear.
 [5]
 J. Guckenheimer, Solving a single conservation law, Lect. Notes Math. 468, SpringerVerlag, 1975, pp. 108134. MR 58:29336
 [6]
 L. Hörmander, Lectures on Nonlinear Hyperbolic Differential Equations, Mathématiques et Applications 26, Springer, 1997. MR 98e:35103
 [7]
 G. Jennings, Piecewise smooth solutions of a single conservation law exist, Adv. in Math. 33 (1979), 192205. MR 80j:35067
 [8]
 F. John, Formation of singularities in onedimensional nonlinear wave propagation, Comm. Pure Appl. Math. 27 (1974), 377405. MR 51:6163
 [9]
 J. B. Keller and L. Ting, Periodic vibration of systems governed by nonlinear partial differential equations, Comm. Pure Appl. Math. 19 (1966), 371420. MR 34:5347
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 Dexing Kong, Cauchy Problem for Quasilinear Hyperbolic Systems, MSJ Memoirs 6, The Mathematical Society of Japan, Tokyo, 2000. MR 2002b:35127
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 Dexing Kong, Lifespan of classical solutions to quasilinear hyperbolic systems with slow decay initial data, Chinese Ann. of Math. 21B (2000), 413440. MR 2001j:35186
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 P. D. Lax, Hyperbolic systems of conservation laws , Comm. Pure Appl. Math. 10 (1957), 537566. MR 20:176
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 M. P. Lebaud, Description de le formation d'un choc dans le système, J. Math. Pures Appl. 73 (1994), 523565. MR 96a:35115
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 [17]
 R. C. MacCamy and V. J. Mizel, Existence and nonexistence in the large of solutions of quasilinear wave equations, Arch. Rat. Mech. Anal. 25 (1967), 299320. MR 35:7000
 [18]
 S. Nakane, Formation of shocks for a single conservation law, SIAM J. Math. Anal. 19 (1988), 13911408. MR 89k:35142
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 D. G. Schaeffer, A regularity theorem for conservation laws, Adv. in Math. 11 (1973), 368386. MR 48:4523
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 H. Whitney, On singularities of mappings of Euclidean space. I: Mappings of the plane into the plane, Ann. of Math. 62 (1955), 374410. MR 17:518d
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Additional Information
Dexing Kong
Affiliation:
Department of Applied Mathematics, Shanghai Jiao Tong University, Shanghai 200030, China
Email:
dkong@mail.sjtu.edu.cn
DOI:
http://dx.doi.org/10.1090/S0002994702029823
PII:
S 00029947(02)029823
Keywords:
Quasilinear hyperbolic system,
smooth solution,
blowup of cusp type,
shock,
weak discontinuity
Received by editor(s):
May 24, 2000
Received by editor(s) in revised form:
May 4, 2001
Published electronically:
April 2, 2002
Additional Notes:
The author was supported in part by the National Science Foundation of China under Grant # 10001024 and the Special Funds for Major State Basic Research Projects of China.
Article copyright:
© Copyright 2002
American Mathematical Society
