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An existence result for conservation laws having BV spatial flux heterogeneities – without concavity. (English) Zbl 1440.35213
Summary: We prove existence for an initial value problem featuring a conservation law whose flux has a discontinuous spatial dependence. For the type of flux considered here, where the spatial dependence occurs in a very general form, a uniqueness result is known, but the existence question was open until the recent work of B. Piccoli and M. Tournus [SIAM J. Math. Anal. 50, No. 3, 2901–2927 (2018; Zbl 1402.35171)]. Piccoli and Tournus [loc. cit.] proved existence using approximate solutions generated by the wave front tracking algorithm. A concavity assumption plays a simplifying role in their analysis. The main contribution of the present paper is an extension of this existence theorem in the absence of the concavity assumption. We accomplish this via finite difference approximations.

MSC:
35L65 Hyperbolic conservation laws
35L03 Initial value problems for first-order hyperbolic equations
35L60 First-order nonlinear hyperbolic equations
35B44 Blow-up in context of PDEs
35A01 Existence problems for PDEs: global existence, local existence, non-existence
65M06 Finite difference methods for initial value and initial-boundary value problems involving PDEs
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