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Existence and nonexistence of TV bounds for scalar conservation laws with discontinuous flux. (English) Zbl 1223.35222
Summary: For the scalar conservation laws with discontinuous flux, an infinite family of \((A,B)\)-interface entropies are introduced and each one of them is shown to form an \(L^{1}\)-contraction semigroup. One of the main unsettled questions concerning conservation law with discontinuous flux is boundedness of total variation of the solution. Away from the interface, boundedness of total variation of the solution has been proved in a recent paper [R. Bürger, A. García, K .H. Karlsen and J. D. Towers, J. Eng. Math. 60, No. 3–4, 387–425 (2008; Zbl 1200.76126)]. In this paper, we discuss this particular issue in detail and produce a counterexample to show that the solution, in general, has unbounded total variation near the interface. In fact, this example illustrates that smallness of the BV norm of the initial data is immaterial. We hereby settle the question of determining for which of the aforementioned \((A,B)\) pairs the solution will have bounded total variation in the case of strictly convex fluxes.

MSC:
35L65 Hyperbolic conservation laws
35R05 PDEs with low regular coefficients and/or low regular data
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