Solutions to nonlinear hyperbolic Cauchy problems without convexity conditions

Ballou, Donald P.

doi:10.1090/S0002-9947-1970-0435615-3

Solutions to nonlinear hyperbolic Cauchy problems without convexity conditions
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by Donald P. Ballou PDF

Trans. Amer. Math. Soc. 152 (1970), 441-460 Request permission

Abstract:

This paper is concerned with the existence of weak solutions to certain nonlinear hyperbolic Cauchy problems. A condition on the curves of discontinuity is used which guarantees uniqueness in the class of piecewise smooth weak solutions. The method of proof is geometric in nature and is constructive in the manner of A. Douglis and Wu Cho-Chün; that is, for certain types of initial data the method of characteristics is employed to construct piecewise smooth weak solutions. A limiting process is then used to obtain existence for bounded, measurable initial data. The solutions in some cases exhibit interesting, new phenomena. For example, a certain class of initial data having one jump gives rise to a solution having a curving contact discontinuity which does not enter the region of intersecting characteristics.

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