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Existence and nonexistence of TV bounds for scalar conservation laws with discontinuous flux
Author(s) -
Ghoshal Shyam Sundar,
Dutta Rajib,
Veerappa Gowda G. D.
Publication year - 2011
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/cpa.20346
Subject(s) - conservation law , counterexample , mathematics , bounded variation , bounded function , scalar (mathematics) , norm (philosophy) , regular polygon , contraction (grammar) , semigroup , variation (astronomy) , mathematical analysis , law , discrete mathematics , geometry , physics , medicine , political science , astrophysics
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 (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.