Discontinuous progressing waves for semilinear systems. (English) Zbl 0598.35069

Rauch, Jeffrey; Reed, Michael C.

Commun. Partial Differ. Equations 10, 1033-1075 (1985).

This paper is concerned with the existence, propagation and interaction of piecewise smooth solutions to semilinear systems of partial differential equations \(P(x,D)u+f(x,u)=0\). Here P is a first order system and the jump discontinuities of the solutions, u, occur across a simply characteristic surface \(\Sigma\). Transport equations along rays describe the propagation of the jumps. The interaction problem is harder. When P is a \(2\times 2\) system the interaction of such waves and the production of a pair of waves from piecewise smooth discontinuous Cauchy data is studied. Related recent results are reported by M. Beals and G. Métivier [Duke Math. J. 53, 125-138 (1986)] and by the authors in the Proceedings of the College de France Seminar, 1985-1986.

MSC:

35L67	Shocks and singularities for hyperbolic equations
35L60	First-order nonlinear hyperbolic equations
47F05	General theory of partial differential operators

Keywords:

piecewise smooth solutions; semilinear systems; jump discontinuities; discontinuous Cauchy data

References:

[1]	Bers L., Interscience (1964)
[2]	Bony J.M., Sem. Goulaouic-Meyer-Schwartz 22 (1979)
[3]	Bony J.M., Sem. Goulaouic-Meyer- Schwart 2 (1981)
[4]	Bony J.M., Seminaire Goulaouic-Meyer-Schwartz 10 (1983)
[5]	Courant, R., Interscience (1962)
[6]	Courant R., Proc. Nat. Acad. Sci. 42 pp 872– (1956) · Zbl 0072.30803
[7]	Hormander, L. 1983.The Analysis of Linear Partial Differential Operators I, 16Berlin: Springer-Verlag.
[8]	Majda A., Mem. AMS 275 (1982)
[9]	Melrose R., Lecture at Notre Dame Conference on Microlocal Analysis
[10]	Melrose, R. and N. Ritter, ”Interaction of Nonlinear Progressing Waves,” Annals of Math., to appear. · Zbl 0575.35063
[11]	Micheli L., Onthe Existence of Anomalous Singularities caused by the Two by Two Interaction of Progressing Waves
[12]	Micheli L., Annals of Math 1 pp 531– (1980)
[13]	Rauch J., Comm. Math Phys 81 pp 203– (1981) · Zbl 0468.35064
[14]	Rauch J., Comm. P.D.E. 7 pp 1117– (1982) · Zbl 0502.35060
[15]	Rauch, J. and M. Reed, ”Striated Solutions of Semilinear, Two Speed Wave Equations,” Indiana Journal, to appear · Zbl 0537.35057
[16]	Ritter N., MIT Thesis (1984)