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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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##### References:
 [1] Adimurthi; Dutta, R.; Ghoshal, S. S.; Gowda, G. D.V., Existence and nonexistence of TV bounds for scalar conservation laws with discontinuous flux, Commun. Pure Appl. Math., 64, 84-115 (2011) · Zbl 1223.35222 [2] Adimurthi; Mishra, S.; Gowda, G. D.V., Optimal entropy solutions for conservation laws with discontinuous flux-functions, J. Hyperbolic Differ. Equ., 2, 783-837 (2005) · Zbl 1093.35045 [3] Andreianov, B.; Cancès, C., Vanishing capillarity solutions of Buckley-Leverett equation with gravity in two-rocks’ medium, Comput. Geosci., 17, 551-572 (2013) · Zbl 1392.76033 [4] Andreianov, B.; Karlsen, K. H.; Risebro, N. H., A theory of $$L^1$$-dissipative solvers for scalar conservation laws with discontinuous flux, Arch. Ration. Mech. Anal., 201, 27-86 (2011) · Zbl 1261.35088 [5] Audusse, E.; Perthame, B., Uniqueness for scalar conservation laws with discontinuous flux via adapted entropies, Proc. R. Soc. Edinb., Sect. A, 135, 253-265 (2005) · Zbl 1071.35079 [6] Baiti, P.; Jenssen, H. K., Well-posedness for a class of $$2 \times 2$$ conservation laws with $$L^\infty$$ data, J. Differ. Equ., 140, 161-185 (1997) · Zbl 0892.35097 [7] Bressan, A.; Guerra, G.; Shen, W., Vanishing viscosity solutions for conservation laws with regulated flux, J. Differ. Equ., 266, 312-351 (2019) · Zbl 1421.35212 [8] Bürger, R.; Karlsen, K. H.; Risebro, N. H.; Towers, J. D., Well-posedness in $$B V_t$$ and convergence of a difference scheme for continuous sedimentation in ideal clarifier-thickener units, Numer. Math., 97, 25-65 (2004) · Zbl 1053.76047 [9] Bürger, R.; Karlsen, K. H.; Towers, J. D., An Engquist-Osher-type scheme for conservation laws with discontinuous flux adapted to flux connections, SIAM J. Numer. Anal., 47, 1684-1712 (2009) · Zbl 1201.35022 [10] Crandall, M. G.; Majda, A., Monotone difference approximations for scalar conservation laws, Math. Comput., 34, 1-21 (1980) · Zbl 0423.65052 [11] Crasta, G.; DeCicco, V.; DePhilippis, G., Structure of solutions of multidimensional conservation laws with discontinuous flux and applications to uniqueness, Arch. Ration. Mech. Anal., 221, 961-985 (2016) · Zbl 1338.35288 [12] Diehl, S., A conservation law with point source and discontinuous flux function modelling continuous sedimentation, SIAM J. Appl. Math., 56, 388-419 (1996) · Zbl 0849.35142 [13] Garavello, M.; Natalini, R.; Piccoli, B.; Terracina, A., Conservation laws with discontinuous flux, Netw. Heterog. Media, 2, 159-179 (2007) · Zbl 1142.35511 [14] Ghoshal, S.; Jana, A.; Towers, J., Convergence of a Godunov scheme to an Audusse-Perthame adapted entropy solution for conservation laws with BV spatial flux, preprint [15] Klingenberg, C.; Risebro, N. H., Convex conservation laws with discontinuous coefficients, existence, uniqueness and asymptotic behavior, Commun. Partial Differ. Equ., 20, 1959-1990 (1995) · Zbl 0836.35090 [16] Karlsen, K. H.; Risebro, N. H.; Towers, J. D., $$L^1$$ stability for entropy solutions of nonlinear degenerate parabolic convection-diffusion equations with discontinuous coefficients, Skr. K. Nor. Vidensk. Selsk. (2003), 49 pp. · Zbl 1036.35104 [17] Karlsen, K. H.; Towers, J. D., Convergence of the Lax-Friedrichs scheme and stability for conservation laws with a discontinuous space-time dependent flux, Chin. Ann. Math., 25B, 287-318 (2004) · Zbl 1112.65085 [18] Karlsen, K. H.; Towers, J. D., Convergence of a Godunov scheme for conservation laws with a discontinuous flux lacking the crossing condition, J. Hyperbolic Differ. Equ., 14, 671-702 (2017) · Zbl 1380.65158 [19] May, L.; Shearer, M.; Davis, K., Scalar conservation laws with nonconstant coefficients with application to particle size segregation in granular flow, J. Nonlinear Sci., 20, 689-707 (2010) · Zbl 1209.35080 [20] Mishra, S., Numerical Methods for Conservation Laws with Discontinuous Coefficients, Handbook of Numerical Analysis, vol. 18, 479-506 (2017) · Zbl 1368.65156 [21] Panov, E. Y., On existence and uniqueness of entropy solutions to the Cauchy problem for a conservation law with discontinuous flux, J. Hyperbolic Differ. Equ., 06, 525-548 (2009) · Zbl 1181.35145 [22] Piccoli, B.; Tournus, M., A general BV existence result for conservation laws with spatial heterogeneities, SIAM J. Math. Anal., 50, 2901-2927 (2018) · Zbl 1402.35171 [23] Risebro, N. H., An introduction to the theory of scalar conservation laws with spatially discontinuous flux functions, (Applied Wave Mathematics (2009), Springer: Springer Berlin), 395-464 · Zbl 1191.74032 [24] Seguin, N.; Vovelle, J., Analysis and approximation of a scalar conservation law with a flux function with discontinuous coefficients, Math. Models Methods Appl. Sci., 13, 221-257 (2003) · Zbl 1078.35011 [25] Shen, W., On the uniqueness of vanishing viscosity solutions for Riemann problems for polymer flooding, Nonlinear Differ. Equ. Appl., 24, Article 37 pp. (2017) · Zbl 1379.35003 [26] Towers, J. D., Convergence of a difference scheme for conservation laws with a discontinuous flux, SIAM J. Numer. Anal., 38, 681-698 (2000) · Zbl 0972.65060
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