Existence and nonexistence of TV bounds for scalar conservation laws with discontinuous flux

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 L1‐contraction semigroup (see [2]). One of the main unsettled questions concerning conservation law with discontinuous flux is boundednes...

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Bibliographic Details
Published in:Communications on pure and applied mathematics 2011-01, Vol.64 (1), p.84-115
Main Authors: Adimurthi, Ghoshal, Shyam Sundar, Dutta, Rajib, Veerappa Gowda, G. D.
Format: Article
Language:English
Subjects:
Applied mathematics
Entropy
Exact sciences and technology
General mathematics
General, history and biography
Group theory
Group theory and generalizations
Mathematical analysis
Mathematical models
Mathematics
Partial differential equations
Sciences and techniques of general use
Theoretical mathematics
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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 L1‐contraction semigroup (see [2]). 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 [6]. 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. © 2010 Wiley Periodicals, Inc.
ISSN:0010-3640
1097-0312
DOI:10.1002/cpa.20346