Formation and propagation of singularities for quasilinear hyperbolic systems

Author:
De-xing Kong

Journal:
Trans. Amer. Math. Soc. **354** (2002), 3155-3179

MSC (2000):
Primary 35L45, 35L67; Secondary 35L65, 76L05

Published electronically:
April 2, 2002

MathSciNet review:
1897395

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Employing the method of characteristic coordinates and the singularity theory of smooth mappings, in this paper we analyze the long-term behaviour of smooth solutions of general quasilinear hyperbolic systems, provide a complete description of the solution close to blow-up points, and investigate the formation and propagation of singularities for systems of hyperbolic conservation laws.

**[1]**Serge Alinhac,*Blowup for nonlinear hyperbolic equations*, Progress in Nonlinear Differential Equations and their Applications, 17, Birkhäuser Boston, Inc., Boston, MA, 1995. MR**1339762****[2]**Robert Bryant, Phillip Griffiths, and Lucas Hsu,*Toward a geometry of differential equations*, Geometry, topology, & physics, Conf. Proc. Lecture Notes Geom. Topology, IV, Int. Press, Cambridge, MA, 1995, pp. 1–76. MR**1358612****[3]**Peter H. Chang,*On the breakdown phenomena of solutions of quasilinear wave equations*, Michigan Math. J.**23**(1976), no. 3, 277–287 (1977). MR**0460910****[4]**Shu-xing Chen and Li-ming Dong,*Formation of shocks for p-system with general smooth initial data*, to appear.**[5]**John Guckenheimer,*Solving a single conservation law*, Dynamical systems—Warwick 1974 (Proc. Sympos. Appl. Topology and Dynamical Systems, Univ. Warwick, Coventry, 1973/1974; presented to E. C. Zeeman on his fiftieth birthday), Springer, Berlin, 1975, pp. 108–134. Lecture Notes in Math., Vol. 468. MR**0606765****[6]**Lars Hörmander,*Lectures on nonlinear hyperbolic differential equations*, Mathématiques & Applications (Berlin) [Mathematics & Applications], vol. 26, Springer-Verlag, Berlin, 1997. MR**1466700****[7]**Gray Jennings,*Piecewise smooth solutions of a single conservation law exist*, Adv. in Math.**33**(1979), no. 2, 192–205. MR**544849**, 10.1016/S0001-8708(79)80005-5**[8]**Fritz John,*Formation of singularities in one-dimensional nonlinear wave propagation*, Comm. Pure Appl. Math.**27**(1974), 377–405. MR**0369934****[9]**Joseph B. Keller and Lu Ting,*Periodic vibrations of systems governed by nonlinear partial differential equations*, Comm. Pure Appl. Math.**19**(1966), 371–420. MR**0205520****[10]**De-xing Kong,*Cauchy problem for quasilinear hyperbolic systems*, MSJ Memoirs, vol. 6, Mathematical Society of Japan, Tokyo, 2000. MR**1797837****[11]**Dexing Kong,*Life-span of classical solutions to quasilinear hyperbolic systems with slow decay initial data*, Chinese Ann. Math. Ser. B**21**(2000), no. 4, 413–440. MR**1801773**, 10.1142/S0252959900000431**[12]**P. D. Lax,*Hyperbolic systems of conservation laws. II*, Comm. Pure Appl. Math.**10**(1957), 537–566. MR**0093653****[13]**M.-P. Lebaud,*Description de la formation d’un choc dans le 𝑝-système*, J. Math. Pures Appl. (9)**73**(1994), no. 6, 523–565 (French, with French summary). MR**1309163****[14]**Ta Tsien Li,*Global classical solutions for quasilinear hyperbolic systems*, RAM: Research in Applied Mathematics, vol. 32, Masson, Paris; John Wiley & Sons, Ltd., Chichester, 1994. MR**1291392****[15]**Ta Tsien Li and Wen Ci Yu,*Boundary value problems for quasilinear hyperbolic systems*, Duke University Mathematics Series, V, Duke University, Mathematics Department, Durham, NC, 1985. MR**823237****[16]**Tai Ping Liu,*Uniqueness of weak solutions of the Cauchy problem for general 2×2 conservation laws*, J. Differential Equations**20**(1976), no. 2, 369–388. MR**0393871****[17]**R. C. MacCamy and V. J. Mizel,*Existence and nonexistence in the large of solutions of quasilinear wave equations*, Arch. Rational Mech. Anal.**25**(1967), 299–320. MR**0216165****[18]**Shizuo Nakane,*Formation of shocks for a single conservation law*, SIAM J. Math. Anal.**19**(1988), no. 6, 1391–1408. MR**965259**, 10.1137/0519102**[19]**David G. Schaeffer,*A regularity theorem for conservation laws*, Advances in Math.**11**(1973), 368–386. MR**0326178****[20]**Hassler Whitney,*On singularities of mappings of euclidean spaces. I. Mappings of the plane into the plane*, Ann. of Math. (2)**62**(1955), 374–410. MR**0073980**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC (2000):
35L45,
35L67,
35L65,
76L05

Retrieve articles in all journals with MSC (2000): 35L45, 35L67, 35L65, 76L05

Additional Information

**De-xing Kong**

Affiliation:
Department of Applied Mathematics, Shanghai Jiao Tong University, Shanghai 200030, China

Email:
dkong@mail.sjtu.edu.cn

DOI:
https://doi.org/10.1090/S0002-9947-02-02982-3

Keywords:
Quasilinear hyperbolic system,
smooth solution,
blow-up 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