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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 } =1n i , 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 } =1n i .

  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.

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References

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Communicated by J. Glimm

Research partially supported by NSF Grant # MCS-79-01857

Research partially supported by NSF Grant # MCS-78-02179

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Rauch, J., Reed, M. Jump discontinuities of semilinear, strictly hyperbolic systems in two variables: Creation and propagation. Commun.Math. Phys. 81, 203–227 (1981). https://doi.org/10.1007/BF01208895

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  • DOI: https://doi.org/10.1007/BF01208895

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