, Volume 81, Issue 2, pp 203-227

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

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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, {x i } 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 {x i } 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.