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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Solutions to nonlinear hyperbolic Cauchy problems without convexity conditions


Author: Donald P. Ballou
Journal: Trans. Amer. Math. Soc. 152 (1970), 441-460
MSC: Primary 35L45
DOI: https://doi.org/10.1090/S0002-9947-1970-0435615-3
MathSciNet review: 0435615
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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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DOI: https://doi.org/10.1090/S0002-9947-1970-0435615-3
Keywords: Hyperbolic partial differential equations, nonlinear Cauchy problems, global weak solutions, shocks, contact discontinuities, rarefaction waves, method of characteristics, convexity conditions
Article copyright: © Copyright 1970 American Mathematical Society