Formation and propagation of singularities for 2×2 quasilinear hyperbolic systems

Kong, De-xing

doi:10.1090/S0002-9947-02-02982-3

Formation and propagation of singularities for $2\times 2$ quasilinear hyperbolic systems
HTML articles powered by AMS MathViewer

by De-xing Kong

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

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

Published electronically: April 2, 2002

PDF | Request permission

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 $2\times 2$ quasilinear hyperbolic systems, provide a complete description of the solution close to blow-up points, and investigate the formation and propagation of singularities for $2\times 2$ systems of hyperbolic conservation laws.

References

Serge Alinhac, Blowup for nonlinear hyperbolic equations, Progress in Nonlinear Differential Equations and their Applications, vol. 17, Birkhäuser Boston, Inc., Boston, MA, 1995. MR 1339762, DOI 10.1007/978-1-4612-2578-2
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
Peter H. Chang, On the breakdown phenomena of solutions of quasilinear wave equations, Michigan Math. J. 23 (1976), no. 3, 277–287 (1977). MR 460910
Shu-xing Chen and Li-ming Dong, Formation of shocks for p-system with general smooth initial data, to appear.
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), Lecture Notes in Math., Vol. 468, Springer, Berlin, 1975, pp. 108–134. MR 0606765
Lars Hörmander, Lectures on nonlinear hyperbolic differential equations, Mathématiques & Applications (Berlin) [Mathematics & Applications], vol. 26, Springer-Verlag, Berlin, 1997. MR 1466700
Gray Jennings, Piecewise smooth solutions of a single conservation law exist, Adv. in Math. 33 (1979), no. 2, 192–205. MR 544849, DOI 10.1016/S0001-8708(79)80005-5
Fritz John, Formation of singularities in one-dimensional nonlinear wave propagation, Comm. Pure Appl. Math. 27 (1974), 377–405. MR 369934, DOI 10.1002/cpa.3160270307
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 205520, DOI 10.1002/cpa.3160190404
De-xing Kong, Cauchy problem for quasilinear hyperbolic systems, MSJ Memoirs, vol. 6, Mathematical Society of Japan, Tokyo, 2000. MR 1797837
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, DOI 10.1142/S0252959900000431
P. D. Lax, Hyperbolic systems of conservation laws. II, Comm. Pure Appl. Math. 10 (1957), 537–566. MR 93653, DOI 10.1002/cpa.3160100406
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
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
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
Tai Ping Liu, Uniqueness of weak solutions of the Cauchy problem for general $2\times 2$ conservation laws, J. Differential Equations 20 (1976), no. 2, 369–388. MR 393871, DOI 10.1016/0022-0396(76)90114-5
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 216165, DOI 10.1007/BF00250932
Shizuo Nakane, Formation of shocks for a single conservation law, SIAM J. Math. Anal. 19 (1988), no. 6, 1391–1408. MR 965259, DOI 10.1137/0519102
David G. Schaeffer, A regularity theorem for conservation laws, Advances in Math. 11 (1973), 368–386. MR 326178, DOI 10.1016/0001-8708(73)90018-2
Saunders MacLane, Steinitz field towers for modular fields, Trans. Amer. Math. Soc. 46 (1939), 23–45. MR 17, DOI 10.1090/S0002-9947-1939-0000017-3