# zbMATH — the first resource for mathematics

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
##### Keywords:
interface entropies; counterexample
Full Text:
##### References:
 [1] Adimurthi, Godunov-type methods for conservation laws with a flux function discontinuous in space, SIAM J. Numer. Anal. 42 (1) pp 179– (2004) · Zbl 1081.65082 [2] Adimurthi, Optimal entropy solutions for conservation laws with discontinuous flux-functions, J. Hyperbolic Differ. Equ. 2 (4) pp 783– (2005) · Zbl 1093.35045 [3] Adimurthi, Convergence of Godunov type methods for a conservation law with a spatially varying discontinuous flux function, Math. Comp. 76 (259) pp 1219– (2007) · Zbl 1116.35084 [4] Adimurthi, Explicit Hopf-Lax type formulas for Hamilton-Jacobi equations and conservation laws with discontinuous coefficients, J. Differential Equations 241 (1) pp 1– (2007) · Zbl 1128.35067 [5] Adimurthi, Conservation law with discontinuous flux, J. Math. Kyoto Univ. 43 (1) pp 27– (2003) · Zbl 1063.35114 [6] Bürger, A family of numerical schemes for kinematic flows with discontinuous flux, J. Engrg. Math. 60 (3-4) pp 387– (2008) [7] Bürger, A front tracking approach to a model of continuous sedimentation in ideal clarifier-thickener units, Nonlinear Anal. Real World Appl. 4 (3) pp 457– (2003) · Zbl 1013.35052 [8] Bürger, A relaxation scheme for continuous sedimentation in ideal clarifier-thickener units, Comput. Math. Appl. 50 (7) pp 993– (2005) · Zbl 1122.76063 [9] Bürger, Monotone difference approximations for the simulation of clarifier-thickener units, Comput. Vis. Sci. 6 (2-3) pp 83– (2004) · Zbl 1299.76283 · doi:10.1007/s00791-003-0112-1 [10] Bürger, Well-posedness in BVt and convergence of a difference scheme for continuous sedimentation in ideal clarifier-thickener units, Numer. Math. 97 (1) pp 25– (2004) · Zbl 1053.76047 [11] Bürger, A model of continuous sedimentation of flocculated suspensions in clarifier-thickener units, SIAM J. Appl. Math. 65 (3) pp 882– (2005) · Zbl 1089.76061 [12] Bürger, An Engquist-Osher-type scheme for conservation laws with discontinuous flux adapted to flux connections, SIAM J. Numer. Anal. 47 (3) pp 1684– (2009) · Zbl 1201.35022 [13] Diehl , S. Conservation laws with applications to continuous sedimentation 1995 [14] Diehl, Dynamic and steady-state behavior of continuous sedimentation, SIAM J. Appl. Math. 57 (4) pp 991– (1997) · Zbl 0889.35062 [15] Diehl, Operating charts for continuous sedimentation. II. Step responses, J. Engrg. Math. 53 (2) pp 139– (2005) · Zbl 1086.76069 [16] Diehl, A uniqueness condition for nonlinear convection-diffusion equations with discontinuous coefficients, J. Hyperbolic Differ. Equ. 6 pp 127– (2009) · Zbl 1180.35305 [17] Evans, Partial differential equations (1998) [18] Gimse, Riemann problems with a discontinuous flux function, Third Inter-national Conference on Hyperbolic Problems, Vol. I, II (Uppsala, 1990) pp 488– (1991) · Zbl 0789.35102 [19] Gimse, Solution of the Cauchy problem for a conservation law with a discontinuous flux function, SIAM J. Math. Anal. 23 (3) pp 635– (1992) · Zbl 0776.35034 [20] Godlewski, Hyperbolic systems of conservation laws (1991) · Zbl 0768.35059 [21] Holden, Front tracking for hyperbolic conservation laws (2002) · Zbl 1006.35002 · doi:10.1007/978-3-642-56139-9 [22] Hong, A bound on the total variation of the conserved quantities for solutions of a general resonant nonlinear balance law, SIAM J. Appl. Math. 64 (3) pp 819– (2004) · Zbl 1063.35115 [23] Isaacson, Nonlinear resonance in systems of conservation laws, SIAM J. Appl. Math. 52 (5) pp 1260– (1992) · Zbl 0794.35100 [24] Joseph, Explicit formula for the solution of convex conservation laws with boundary condition, Duke Math. J. 62 (2) pp 401– (1991) [25] Kaasschieter, Solving the Buckley-Leverret equation with gravity in a heterogeneous porous media, Comput. Geosci. 3 (1) pp 23– (1999) · Zbl 0952.76085 [26] Karlsen, On a nonlinear degenerate parabolic transportdiffusion equation with a discontinuous coefficient, Electron. J. Differential Equations 2002 (93) pp 23– (2002) [27] Karlsen, L1 stability for entropy solutions of nonlinear degenerate parabolic convection-diffusion equations with discontinuous coefficients, Skr. K. Nor. Vidensk. Selsk. 2003 (3) pp 1– (2003) · Zbl 1036.35104 [28] Klingenberg, Convex conservation laws with discontinuous coefficients. Existence, uniqueness and asymptotic behavior, Comm. Partial Differential Equations 20 (11-12) pp 1959– (1995) [29] Kružkov, First order quasilinear equations with several independent variables, Mat. Sb. (N.S.) (2) pp 228– (1970) [30] Kružhkov, First-order conservative quasilinear laws with an infinite domain of dependence on the initial data, Dokl. Akad. Nauk SSSR 314 (1) pp 79– (1990) [31] Lin, A comparison of convergence rates for Godunov’s method and Glimm’s method in resonant nonlinear systems of conservation laws, SIAM J. Numer. Anal. 32 (3) pp 824– (1995) · Zbl 0830.35079 [32] Mochon, An analysis for the traffic on highways with changing surface conditions, Math. Modelling 9 (1) pp 1– (1987) [33] Ostrov, Solutions of Hamilton-Jacobi equations and conservation laws with discontinuous space-time dependence, J. Differential Equations 182 (2) pp 51– (2002) · Zbl 1009.35015 [34] Ross, Two new moving boundary problems for scalar conservation laws, Comm. Pure Appl. Math. 41 (5) pp 725– (1988) · Zbl 0632.35078 [35] Towers, Convergence of a difference scheme for conservation laws with a discontinuous flux, SIAM J. Numer. Anal. 38 (2) pp 681– (2000) · Zbl 0972.65060
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.