Communications in Mathematical Physics

, Volume 81, Issue 2, pp 203–227

Jump discontinuities of semilinear, strictly hyperbolic systems in two variables: Creation and propagation

  • Jeffrey Rauch
  • Michael Reed
Article

DOI: 10.1007/BF01208895

Cite this article as:
Rauch, J. & Reed, M. Commun.Math. Phys. (1981) 81: 203. doi:10.1007/BF01208895

Abstract

The creation and propagation of jump discontinuities in the solutions of semilinear strictly hyperbolic systems is studied in the case where the initial data has a discrete set, {xi}i=1n, of jump discontinuities. LetS be the smallest closed set which satisfies:
  1. (i)

    S is a union of forward characteristics.

     
  2. (ii)

    S contains all the forward characteristics from the points {xi}i=1n.

     
  3. (iii)

    if two forward characteristics inS intersect, then all forward characteristics from the point of intersection lie inS.

     

We prove that the singular support of the solution lies inS. We derive a sum law which gives a lower bound on the smoothness of the solution across forward characteristics from an intersection point. We prove a sufficient condition which guarantees that in many cases the lower bound is also an upper bound.

Copyright information

© Springer-Verlag 1981

Authors and Affiliations

  • Jeffrey Rauch
    • 1
  • Michael Reed
    • 2
  1. 1.Department of MathematicsUniversity of MichiganAnn ArborUSA
  2. 2.Department of MathematicsDuke UniversityDurhamUSA