# American Institute of Mathematical Sciences

November  2011, 30(4): 1191-1210. doi: 10.3934/dcds.2011.30.1191

## New entropy conditions for scalar conservation laws with discontinuous flux

 1 Faculty of Mathematics, University of Montenegro, Cetinjski put bb, 81000 Podgorica

Received  April 2010 Revised  July 2010 Published  May 2011

We propose new Kruzhkov type entropy conditions for one dimensional scalar conservation law with a discontinuous flux. We prove existence and uniqueness of the entropy admissible weak solution to the corresponding Cauchy problem merely under assumptions on the flux which provide the maximum principle. In particular, we allow multiple flux crossings and we do not need any kind of genuine nonlinearity conditions.
Citation: Darko Mitrovic. New entropy conditions for scalar conservation laws with discontinuous flux. Discrete & Continuous Dynamical Systems - A, 2011, 30 (4) : 1191-1210. doi: 10.3934/dcds.2011.30.1191
##### References:
 [1] Adimurthi, G. D. Veerappa Gowda, Conservation laws with discontinuous flux,, J. Math. (Kyoto University), 43 (2003), 27. Google Scholar [2] Adimurthi, S. Mishra and G. D. Veerappa Gowda, Optimal entropy solutions for conservation laws with discontinuous flux functions,, J. of Hyperbolic Differ. Equ., 2 (2005), 783. doi: 10.1142/S0219891605000622. Google Scholar [3] Adimurthi, S. Mishra and G. D. Veerappa Gowda, Existence and stability of entropy solutions for a conservation law with discontinuous non-convex fluxes,, Netw. Heterog. Media, 2 (2007), 127. doi: 10.3934/nhm.2007.2.127. Google Scholar [4] J. Aleksic and D. Mitrovic, On the compactness for two dimensional scalar conservation law with discontinuous flux,, Comm. Math. Sciences, 4 (2009), 963. Google Scholar [5] B. Andreianov, K. H. Karlsena and N. H. Risebro, On vanishing viscosity approximation of conservation laws with discontinuous flux,, preprint. Available from: , (). Google Scholar [6] E. Audusse and B. Perthame, Uniqueness for scalar conservation law via adapted entropies,, Proc. Roy. Soc. Edinburgh Sect. A, 135 (2005), 253. doi: 10.1017/S0308210500003863. Google Scholar [7] F. Bachmann and J. Vovelle, Existence and uniqueness of entropy solution of scalar conservation law with a flux function involving discontinuous coefficients,, Comm. Partial Differential Equations, 31 (2006), 371. doi: 10.1080/03605300500358095. Google Scholar [8] R. Burger, K. H. Karlsen and J. Towers, On Enquist-Osher-type scheme for conservation laws with discontinuous flux adapted to flux connections,, SIAM J. Numer. Anal., 3 (2009), 1684. doi: 10.1137/07069314X. Google Scholar [9] R. Burger, A. Garcia, K. H. Karlsen and J. Towers, A family of schemes for kinematic flows with discontinuous flux,, J. Engrg. Math., 60 (2008), 387. doi: 10.1007/s10665-007-9148-4. Google Scholar [10] S. Diehl, On scalar conservation law with point source and discontinuous flux function modelling continuous sedimentation,, SIAM J. Math. Anal., 6 (1995), 1425. doi: 10.1137/S0036141093242533. Google Scholar [11] S. Diehl, A conservation law with point source and discontinuous flux function modelling continuous sedimentation,, SIAM J. Appl. Anal., 2 (1996), 388. Google Scholar [12] S. Diehl, A uniqueness condition for non-linear convection-diffusion equations with discontinuous coefficients,, J. Hyperbolic Diff. Eq., 6 (2009), 127. doi: 10.1142/S0219891609001794. Google Scholar [13] R. J. DiPerna, Measure-valued solutions to conservation laws,, Arch. Ration. Mech. Anal., 88 (1985), 223. doi: 10.1007/BF00752112. Google Scholar [14] L. C. Evans, "Weak Convergence Methods in Nonlinear Partial Differential Equations,", AMS, 74 (1990). Google Scholar [15] H. Holden, K. Karlsen and D. Mitrovic, Zero diffusion dispersion limits for a scalar conservation law with discontinuous flux function,, International Journal of Differential Equations, (2009) (2009). doi: 10.1155/2009/279818. Google Scholar [16] P. Gerard, Microlocal defect measures,, Comm. Partial Differential Equations, 11 (1991), 1761. doi: 10.1080/03605309108820822. Google Scholar [17] T. Gimse and N. H. Risebro, Riemann problems with discontinuous flux function,, in Proc. 3rd Int. Conf. Hyperbolic Problems Studentlitteratur, (1991), 488. Google Scholar [18] E. Kaasschieter, Solving the Buckley-Leverret equation with gravity in a heterogeneous porous media,, Comput. Geosci., 3 (1999), 23. doi: 10.1023/A:1011574824970. Google Scholar [19] K. H. Karslen, N. H. Risebro and J. Towers, $L$1-stability for entropy solutions of nonlinear degenerate parabolic connection-diffusion equations with disc. coeff.,, Skr. K. Nor. Vid. Selsk, 3 (2003), 1. Google Scholar [20] K. Karlsen, N. H. Risebro and J. Towers, On a nonlin. degenerate parabolic transport-diff. eq. with a disc. coeff.,, Electronic J. of Differential Equations, 2002 (). Google Scholar [21] K. Karlsen, M. Rascle and E. Tadmor, On the existence and compactness of a two-dimensional resonant system of conservation laws,, Communications in Mathematical Sciences 2 (2007), 2 (2007), 253. Google Scholar [22] K. Karlsen and J. Towers, Convergence of the Lax-Friedrichs scheme and stability for conservation laws with a discontinous space-time dependent flux,, Chinese Ann. Math. Ser. B, 3 (2004), 287. doi: 10.1142/S0252959904000299. Google Scholar [23] S. N. Kruzhkov, First order quasilinear equations in several independent variables,, Mat. Sb., 81 (1970), 217. doi: 10.1070/SM1970v010n02ABEH002156. Google Scholar [24] Y. S. Kwon and A. Vasseur, Strong traces for scalar conservation laws with general flux,, Arch. Rat. Mech. Anal., 3 (2007), 495. doi: 10.1007/s00205-007-0055-7. Google Scholar [25] P. L. Lions, B. Perthame and E. Tadmor, A kinetic formulation of multidim. scalar cons. law and related equations,, J. Amer. Math. Soc., 1 (1994), 169. doi: 10.1090/S0894-0347-1994-1201239-3. Google Scholar [26] D. Mitrovic, Estence amd stability of a multidimensional scalar conservation law with discontinuous flux,, Netw. Het. Media, 5 (2010), 163. doi: 10.3934/nhm.2010.5.163. Google Scholar [27] E. Yu. Panov, Existence of Strong Traces for Quasi-Solutions of Multidimensional Conservation Laws,, J. of Hyperbolic Differential Equations, 4 (2007), 729. doi: 10.1142/S0219891607001343. Google Scholar [28] E. Yu. Panov, On existence and uniqueness of entropy solutions to the Cauchy problem for a conservation law with discontinuous flux,, J. of Hyperbolic Differential Equations, 3 (2009), 525. doi: 10.1142/S0219891609001915. Google Scholar [29] E. Yu. Panov, On weak completeness of the set of entropy solutions to a scalar conservation law,, SIAM J. Math. Anal., 1 (2009), 26. doi: 10.1137/080724587. Google Scholar [30] E. Yu. Panov, Existence and strong pre-compactness properties for entropy solutions of a first-order quasilinear equation with discontinuous flux,, Arch. Rational Mech. Anal., 195 (2010), 643. doi: 10.1007/s00205-009-0217-x. Google Scholar [31] P. Pedregal, "Parametrized Measures and Variational Principles,", Progress in Nonlinear Partial Differential Equations and Their Applications, 30 (1997). Google Scholar [32] B. Perthame, Kinetic approach to systems of conservation laws,, Journées équations aux derivées partielles, (1992). Google Scholar [33] L. Tartar, Comp. compactness and application to PDEs,, Nonlin. Anal. and Mech.: Heriot-Watt symposium, IV (1979). Google Scholar [34] L. Tartar, H-measures, a new approach for studying homogenisation, oscillation and concentration effects in PDEs,, Proc. Roy. Soc. Edinburgh. Sect. A, 3-4 (1990), 3. Google Scholar [35] B. Temple, Global solution of the Cauchy problem for a class of 2x2 nonstrictly hyperbolic conservation laws,, Adv. in Appl. Math., 3 (1982), 335. doi: 10.1016/S0196-8858(82)80010-9. Google Scholar [36] A. Vasseur, Strong traces for solutions of multidimensional conservation laws,, Arch. Rat. Mech. Anal., 160 (2001), 181. doi: 10.1007/s002050100157. Google Scholar

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##### References:
 [1] Adimurthi, G. D. Veerappa Gowda, Conservation laws with discontinuous flux,, J. Math. (Kyoto University), 43 (2003), 27. Google Scholar [2] Adimurthi, S. Mishra and G. D. Veerappa Gowda, Optimal entropy solutions for conservation laws with discontinuous flux functions,, J. of Hyperbolic Differ. Equ., 2 (2005), 783. doi: 10.1142/S0219891605000622. Google Scholar [3] Adimurthi, S. Mishra and G. D. Veerappa Gowda, Existence and stability of entropy solutions for a conservation law with discontinuous non-convex fluxes,, Netw. Heterog. Media, 2 (2007), 127. doi: 10.3934/nhm.2007.2.127. Google Scholar [4] J. Aleksic and D. Mitrovic, On the compactness for two dimensional scalar conservation law with discontinuous flux,, Comm. Math. Sciences, 4 (2009), 963. Google Scholar [5] B. Andreianov, K. H. Karlsena and N. H. Risebro, On vanishing viscosity approximation of conservation laws with discontinuous flux,, preprint. Available from: , (). Google Scholar [6] E. Audusse and B. Perthame, Uniqueness for scalar conservation law via adapted entropies,, Proc. Roy. Soc. Edinburgh Sect. A, 135 (2005), 253. doi: 10.1017/S0308210500003863. Google Scholar [7] F. Bachmann and J. Vovelle, Existence and uniqueness of entropy solution of scalar conservation law with a flux function involving discontinuous coefficients,, Comm. Partial Differential Equations, 31 (2006), 371. doi: 10.1080/03605300500358095. Google Scholar [8] R. Burger, K. H. Karlsen and J. Towers, On Enquist-Osher-type scheme for conservation laws with discontinuous flux adapted to flux connections,, SIAM J. Numer. Anal., 3 (2009), 1684. doi: 10.1137/07069314X. Google Scholar [9] R. Burger, A. Garcia, K. H. Karlsen and J. Towers, A family of schemes for kinematic flows with discontinuous flux,, J. Engrg. Math., 60 (2008), 387. doi: 10.1007/s10665-007-9148-4. Google Scholar [10] S. Diehl, On scalar conservation law with point source and discontinuous flux function modelling continuous sedimentation,, SIAM J. Math. Anal., 6 (1995), 1425. doi: 10.1137/S0036141093242533. Google Scholar [11] S. Diehl, A conservation law with point source and discontinuous flux function modelling continuous sedimentation,, SIAM J. Appl. Anal., 2 (1996), 388. Google Scholar [12] S. Diehl, A uniqueness condition for non-linear convection-diffusion equations with discontinuous coefficients,, J. Hyperbolic Diff. Eq., 6 (2009), 127. doi: 10.1142/S0219891609001794. Google Scholar [13] R. J. DiPerna, Measure-valued solutions to conservation laws,, Arch. Ration. Mech. Anal., 88 (1985), 223. doi: 10.1007/BF00752112. Google Scholar [14] L. C. Evans, "Weak Convergence Methods in Nonlinear Partial Differential Equations,", AMS, 74 (1990). Google Scholar [15] H. Holden, K. Karlsen and D. Mitrovic, Zero diffusion dispersion limits for a scalar conservation law with discontinuous flux function,, International Journal of Differential Equations, (2009) (2009). doi: 10.1155/2009/279818. Google Scholar [16] P. Gerard, Microlocal defect measures,, Comm. Partial Differential Equations, 11 (1991), 1761. doi: 10.1080/03605309108820822. Google Scholar [17] T. Gimse and N. H. Risebro, Riemann problems with discontinuous flux function,, in Proc. 3rd Int. Conf. Hyperbolic Problems Studentlitteratur, (1991), 488. Google Scholar [18] E. Kaasschieter, Solving the Buckley-Leverret equation with gravity in a heterogeneous porous media,, Comput. Geosci., 3 (1999), 23. doi: 10.1023/A:1011574824970. Google Scholar [19] K. H. Karslen, N. H. Risebro and J. Towers, $L$1-stability for entropy solutions of nonlinear degenerate parabolic connection-diffusion equations with disc. coeff.,, Skr. K. Nor. Vid. Selsk, 3 (2003), 1. Google Scholar [20] K. Karlsen, N. H. Risebro and J. Towers, On a nonlin. degenerate parabolic transport-diff. eq. with a disc. coeff.,, Electronic J. of Differential Equations, 2002 (). Google Scholar [21] K. Karlsen, M. Rascle and E. Tadmor, On the existence and compactness of a two-dimensional resonant system of conservation laws,, Communications in Mathematical Sciences 2 (2007), 2 (2007), 253. Google Scholar [22] K. Karlsen and J. Towers, Convergence of the Lax-Friedrichs scheme and stability for conservation laws with a discontinous space-time dependent flux,, Chinese Ann. Math. Ser. B, 3 (2004), 287. doi: 10.1142/S0252959904000299. Google Scholar [23] S. N. Kruzhkov, First order quasilinear equations in several independent variables,, Mat. Sb., 81 (1970), 217. doi: 10.1070/SM1970v010n02ABEH002156. Google Scholar [24] Y. S. Kwon and A. Vasseur, Strong traces for scalar conservation laws with general flux,, Arch. Rat. Mech. Anal., 3 (2007), 495. doi: 10.1007/s00205-007-0055-7. Google Scholar [25] P. L. Lions, B. Perthame and E. Tadmor, A kinetic formulation of multidim. scalar cons. law and related equations,, J. Amer. Math. Soc., 1 (1994), 169. doi: 10.1090/S0894-0347-1994-1201239-3. Google Scholar [26] D. Mitrovic, Estence amd stability of a multidimensional scalar conservation law with discontinuous flux,, Netw. Het. Media, 5 (2010), 163. doi: 10.3934/nhm.2010.5.163. Google Scholar [27] E. Yu. Panov, Existence of Strong Traces for Quasi-Solutions of Multidimensional Conservation Laws,, J. of Hyperbolic Differential Equations, 4 (2007), 729. doi: 10.1142/S0219891607001343. Google Scholar [28] E. Yu. Panov, On existence and uniqueness of entropy solutions to the Cauchy problem for a conservation law with discontinuous flux,, J. of Hyperbolic Differential Equations, 3 (2009), 525. doi: 10.1142/S0219891609001915. Google Scholar [29] E. Yu. Panov, On weak completeness of the set of entropy solutions to a scalar conservation law,, SIAM J. Math. Anal., 1 (2009), 26. doi: 10.1137/080724587. Google Scholar [30] E. Yu. Panov, Existence and strong pre-compactness properties for entropy solutions of a first-order quasilinear equation with discontinuous flux,, Arch. Rational Mech. Anal., 195 (2010), 643. doi: 10.1007/s00205-009-0217-x. Google Scholar [31] P. Pedregal, "Parametrized Measures and Variational Principles,", Progress in Nonlinear Partial Differential Equations and Their Applications, 30 (1997). Google Scholar [32] B. Perthame, Kinetic approach to systems of conservation laws,, Journées équations aux derivées partielles, (1992). Google Scholar [33] L. Tartar, Comp. compactness and application to PDEs,, Nonlin. Anal. and Mech.: Heriot-Watt symposium, IV (1979). Google Scholar [34] L. Tartar, H-measures, a new approach for studying homogenisation, oscillation and concentration effects in PDEs,, Proc. Roy. Soc. Edinburgh. Sect. A, 3-4 (1990), 3. Google Scholar [35] B. Temple, Global solution of the Cauchy problem for a class of 2x2 nonstrictly hyperbolic conservation laws,, Adv. in Appl. Math., 3 (1982), 335. doi: 10.1016/S0196-8858(82)80010-9. Google Scholar [36] A. Vasseur, Strong traces for solutions of multidimensional conservation laws,, Arch. Rat. Mech. Anal., 160 (2001), 181. doi: 10.1007/s002050100157. Google Scholar
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