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June  2016, 11(2): 331-348. doi: 10.3934/nhm.2016.11.331

## BV regularity near the interface for nonuniform convex discontinuous flux

 1 Gran Sasso Science Institute, Viale Francesco Crispi, 7, 67100 LAquila, Italy

Received  April 2015 Revised  September 2015 Published  March 2016

In this paper, we discuss the total variation bound for the solution of scalar conservation laws with discontinuous flux. We prove the smoothing effect of the equation forcing the $BV_{loc}$ solution near the interface for $L^\infty$ initial data without the assumption on the uniform convexity of the fluxes made as in [1,21]. The proof relies on the method of characteristics and the explicit formulas.
Citation: Shyam Sundar Ghoshal. BV regularity near the interface for nonuniform convex discontinuous flux. Networks & Heterogeneous Media, 2016, 11 (2) : 331-348. doi: 10.3934/nhm.2016.11.331
##### References:
 [1] Adimurthi, R. Dutta, S. S. Ghoshal and G. D. Veerappa Gowda, Existence and nonexistence of TV bounds for scalar conservation laws with discontinuous flux, Comm. Pure Appl. Math., 64 (2011), 84-115. doi: 10.1002/cpa.20346.  Google Scholar [2] Adimurthi, S. S. Ghoshal and G. D. Veerappa Gowda, Finer regularity of an entropy solution for $1$-$d$ scalar conservation laws with non uniform convex flux, Rend. Sem. Mat. Univ. Padova, 132 (2014), 1-24. doi: 10.4171/RSMUP/132-1.  Google Scholar [3] Adimurthi, S. S. Ghoshal and G. D. Veerappa Gowda, Structure of an entropy solution of a scalar conservation law with strict convex flux, J. Hyperbolic Differ. Equ., 9 (2012), 571-611. doi: 10.1142/S0219891612500191.  Google Scholar [4] Adimurthi and G. D. Veerappa Gowda, Conservation laws with discontinuous flux, J. Math. Kyoto Univ., 43 (2003), 27-70.  Google Scholar [5] Adimurthi, J. Jaffre and G. D. Veerappa Gowda, Godunov type methods for scalar conservation laws with flux function discontinuous in the space variable, SIAM J. Numer. Anal., 42 (2004), 179-208. doi: 10.1137/S003614290139562X.  Google Scholar [6] Adimurthi, S. Mishra and G. D. Veerappa Gowda, Optimal entropy solutions for conservation laws with discontinuous flux-functions, J. Hyperbolic Differ. Equ., 2 (2005), 783-837. doi: 10.1142/S0219891605000622.  Google Scholar [7] Adimurthi, S. Mishra and G. D. Veerappa Gowda, Explicit Hopf-Lax type formulas for Hamilton-Jacobi equations and conservation laws with discontinuous coefficients， J. Differential Equations, 241 (2007), 1-31. doi: 10.1016/j.jde.2007.05.039.  Google Scholar [8] Adimurthi, S. Mishra and G. D. Veerappa Gowda, Convergence of Godunov type methods for a conservation law with a spatially varying discontinuous flux function, Math. Comp., 76 (2007), 1219-1242. doi: 10.1090/S0025-5718-07-01960-6.  Google Scholar [9] B. Andreianov, K. H. Karlsen and N. H. Risebro, A theory of $L^1$ - dissipative solvers for scalar conservation laws with discontinuous flux, Arch. Ration. Mech. Anal., 201 (2011), 27-86. doi: 10.1007/s00205-010-0389-4.  Google Scholar [10] B. Andreianov and C. Cances, The Godunov scheme for scalar conservation laws with discontinuous bell-shaped flux functions, Appl. Math. Lett., 25 (2012), 1844-1848. doi: 10.1016/j.aml.2012.02.044.  Google Scholar [11] R. Bürger, A. García, K. H. Karlsen and J. D. Towers, A family of numerical schemes for kinematic flows with discontinuous flux, J. Engrg. Math., 60 (2008), 387-425. doi: 10.1007/s10665-007-9148-4.  Google Scholar [12] R. Bürger, K. H. Karlsen, N. H. Risebro and J. D. Towers, Well-posedness in $BV_t$ and convergence of a difference scheme for continuous sedimentation in ideal clarifier thickener units, Numer. Math., 97 (2004), 25-65. doi: 10.1007/s00211-003-0503-8.  Google Scholar [13] R. Bürger, K. H. Karlsen, C. Klingenberg and N. H. Risebro, A front tracking approach to a model of continuous sedimentation in ideal clarifier-thickener units, Nonlin. Anal. Real World Appl., 4 (2003), 457-481. doi: 10.1016/S1468-1218(02)00071-8.  Google Scholar [14] R. Bürger, K. H. Karlsen and N. H. Risebro, A relaxation scheme for continuous sedimentation in ideal clarifier-thickner units, Comput. Math. Applic., 50 (2005), 993-1009. doi: 10.1016/j.camwa.2005.08.019.  Google Scholar [15] R. Bürger, K. H. Karlsen, N. H. Risebro and J. D. Towers, Monotone difference approximations for the simulation of clarifier-thickener units, Comput. Visual. Sci., 6 (2004), 83-91. doi: 10.1007/s00791-003-0112-1.  Google Scholar [16] R. Bürger, K. H. Karlsen and J. D. Towers, A mathematical model of continuous sedimentation of flocculated suspensions in clarifier-thickener units, SIAM J. Appl. Math., 65 (2005), 882-940. doi: 10.1137/04060620X.  Google Scholar [17] S. Diehl, Dynamic and steady-state behaviour of continuous sedimentation, SIAM J. Appl. Math., 57 (1997), 991-1018. doi: 10.1137/S0036139995290101.  Google Scholar [18] S. Diehl, Operating charts for continuous sedimentation II: Step responses, J. Engrg. Math., 53 (2005), 139-185. doi: 10.1007/s10665-005-6430-1.  Google Scholar [19] T. Gimse and N. H. Risebro, Solution of the Cauchy problem for a conservation law with a discontinuous flux function, SIAM J. Math. Anal., 23 (1992), 635-648. doi: 10.1137/0523032.  Google Scholar [20] T. Gimse and N. H. Risebro, Riemann problems with discontinuous flux function, Proc. 3rd Internat. Conf. Hyperbolic Problems, Studentlitteratur, Uppsala, (1991), 488-502.  Google Scholar [21] S. S. Ghoshal, Optimal results on TV bounds for scalar conservation laws with discontinuous flux, J. Differential Equations, 258 (2015), 980-1014. doi: 10.1016/j.jde.2014.10.014.  Google Scholar [22] S. S. Ghoshal, Finer Analysis of Characteristic Curve and its Application to Exact, Optimal Controllability, Structure of the Entropy Solution of a Scalar Conservation Law with Convex Flux, Ph.D thesis, TIFR, Centre for Applicable Mathematics, 2012. Google Scholar [23] K. T. Joseph and G. D. Veerappa Gowda, Explicit formula for solution of convex conservation laws with boundary condition, Duke Math. J., 62 (1991), 401-416. doi: 10.1215/S0012-7094-91-06216-2.  Google Scholar [24] E. Kaasschieter, Solving the Buckley-Leverret equation with gravity in a heterogeneous porous media, Comput. Geosci., 3 (1999), 23-48. doi: 10.1023/A:1011574824970.  Google Scholar [25] K. H. Karlsen, N. H. Risebro and J. D. Towers, On a Nonlinear Degenerate Parabolic Transport Diffusion Equation with Discontinuous Coefficient, Electron. J. Differential Equations, 2002.  Google Scholar [26] C. Klingenberg and N. H. Risebro, Convex conservation laws with discontinuous coefficients, existence, uniqueness and asymptotic behavior, Comm. Partial Differential Equations, 20 (1995), 1959-1990. doi: 10.1080/03605309508821159.  Google Scholar [27] S. N. Kružkov, First order quasilinear equations with several independent variables. (Russian), Mat. Sb., 81 (1970), 228-255. English transl. in Math. USSR Sb., 10 (1970), 217-243.  Google Scholar [28] P. D. Lax, Hyperbolic systems of conservation laws II, Comm. Pure Appl. Math., 10 (1957), 537-566. doi: 10.1002/cpa.3160100406.  Google Scholar [29] S. Mochon, An analysis for the traffic on highways with changing surface conditions, Math. Model., 9 (1987), 1-11. doi: 10.1016/0270-0255(87)90068-6.  Google Scholar [30] D. N. Ostrov, Solutions of Hamilton-Jacobi equations and conservation laws with discontinuous space-time dependence, J. Differential Equations, 182 (2002), 51-77. doi: 10.1006/jdeq.2001.4088.  Google Scholar [31] D. Serre, Systémes De Lois De Conservation. I., Hyperbolicité, Entropies, Ondes de Choc, Diderot Editeur, Paris, 1996.  Google Scholar [32] J. D. Towers, Convergence of a difference scheme for conservation laws with a discontinuous flux, SIAM J. Numer. Anal., 38 (2000), 681-698. doi: 10.1137/S0036142999363668.  Google Scholar

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##### References:
 [1] Adimurthi, R. Dutta, S. S. Ghoshal and G. D. Veerappa Gowda, Existence and nonexistence of TV bounds for scalar conservation laws with discontinuous flux, Comm. Pure Appl. Math., 64 (2011), 84-115. doi: 10.1002/cpa.20346.  Google Scholar [2] Adimurthi, S. S. Ghoshal and G. D. Veerappa Gowda, Finer regularity of an entropy solution for $1$-$d$ scalar conservation laws with non uniform convex flux, Rend. Sem. Mat. Univ. Padova, 132 (2014), 1-24. doi: 10.4171/RSMUP/132-1.  Google Scholar [3] Adimurthi, S. S. Ghoshal and G. D. Veerappa Gowda, Structure of an entropy solution of a scalar conservation law with strict convex flux, J. Hyperbolic Differ. Equ., 9 (2012), 571-611. doi: 10.1142/S0219891612500191.  Google Scholar [4] Adimurthi and G. D. Veerappa Gowda, Conservation laws with discontinuous flux, J. Math. Kyoto Univ., 43 (2003), 27-70.  Google Scholar [5] Adimurthi, J. Jaffre and G. D. Veerappa Gowda, Godunov type methods for scalar conservation laws with flux function discontinuous in the space variable, SIAM J. Numer. Anal., 42 (2004), 179-208. doi: 10.1137/S003614290139562X.  Google Scholar [6] Adimurthi, S. Mishra and G. D. Veerappa Gowda, Optimal entropy solutions for conservation laws with discontinuous flux-functions, J. Hyperbolic Differ. Equ., 2 (2005), 783-837. doi: 10.1142/S0219891605000622.  Google Scholar [7] Adimurthi, S. Mishra and G. D. Veerappa Gowda, Explicit Hopf-Lax type formulas for Hamilton-Jacobi equations and conservation laws with discontinuous coefficients， J. Differential Equations, 241 (2007), 1-31. doi: 10.1016/j.jde.2007.05.039.  Google Scholar [8] Adimurthi, S. Mishra and G. D. Veerappa Gowda, Convergence of Godunov type methods for a conservation law with a spatially varying discontinuous flux function, Math. Comp., 76 (2007), 1219-1242. doi: 10.1090/S0025-5718-07-01960-6.  Google Scholar [9] B. Andreianov, K. H. Karlsen and N. H. Risebro, A theory of $L^1$ - dissipative solvers for scalar conservation laws with discontinuous flux, Arch. Ration. Mech. Anal., 201 (2011), 27-86. doi: 10.1007/s00205-010-0389-4.  Google Scholar [10] B. Andreianov and C. Cances, The Godunov scheme for scalar conservation laws with discontinuous bell-shaped flux functions, Appl. Math. Lett., 25 (2012), 1844-1848. doi: 10.1016/j.aml.2012.02.044.  Google Scholar [11] R. Bürger, A. García, K. H. Karlsen and J. D. Towers, A family of numerical schemes for kinematic flows with discontinuous flux, J. Engrg. Math., 60 (2008), 387-425. doi: 10.1007/s10665-007-9148-4.  Google Scholar [12] R. Bürger, K. H. Karlsen, N. H. Risebro and J. D. Towers, Well-posedness in $BV_t$ and convergence of a difference scheme for continuous sedimentation in ideal clarifier thickener units, Numer. Math., 97 (2004), 25-65. doi: 10.1007/s00211-003-0503-8.  Google Scholar [13] R. Bürger, K. H. Karlsen, C. Klingenberg and N. H. Risebro, A front tracking approach to a model of continuous sedimentation in ideal clarifier-thickener units, Nonlin. Anal. Real World Appl., 4 (2003), 457-481. doi: 10.1016/S1468-1218(02)00071-8.  Google Scholar [14] R. Bürger, K. H. Karlsen and N. H. Risebro, A relaxation scheme for continuous sedimentation in ideal clarifier-thickner units, Comput. Math. Applic., 50 (2005), 993-1009. doi: 10.1016/j.camwa.2005.08.019.  Google Scholar [15] R. Bürger, K. H. Karlsen, N. H. Risebro and J. D. Towers, Monotone difference approximations for the simulation of clarifier-thickener units, Comput. Visual. Sci., 6 (2004), 83-91. doi: 10.1007/s00791-003-0112-1.  Google Scholar [16] R. Bürger, K. H. Karlsen and J. D. Towers, A mathematical model of continuous sedimentation of flocculated suspensions in clarifier-thickener units, SIAM J. Appl. Math., 65 (2005), 882-940. doi: 10.1137/04060620X.  Google Scholar [17] S. Diehl, Dynamic and steady-state behaviour of continuous sedimentation, SIAM J. Appl. Math., 57 (1997), 991-1018. doi: 10.1137/S0036139995290101.  Google Scholar [18] S. Diehl, Operating charts for continuous sedimentation II: Step responses, J. Engrg. Math., 53 (2005), 139-185. doi: 10.1007/s10665-005-6430-1.  Google Scholar [19] T. Gimse and N. H. Risebro, Solution of the Cauchy problem for a conservation law with a discontinuous flux function, SIAM J. Math. Anal., 23 (1992), 635-648. doi: 10.1137/0523032.  Google Scholar [20] T. Gimse and N. H. Risebro, Riemann problems with discontinuous flux function, Proc. 3rd Internat. Conf. Hyperbolic Problems, Studentlitteratur, Uppsala, (1991), 488-502.  Google Scholar [21] S. S. Ghoshal, Optimal results on TV bounds for scalar conservation laws with discontinuous flux, J. Differential Equations, 258 (2015), 980-1014. doi: 10.1016/j.jde.2014.10.014.  Google Scholar [22] S. S. Ghoshal, Finer Analysis of Characteristic Curve and its Application to Exact, Optimal Controllability, Structure of the Entropy Solution of a Scalar Conservation Law with Convex Flux, Ph.D thesis, TIFR, Centre for Applicable Mathematics, 2012. Google Scholar [23] K. T. Joseph and G. D. Veerappa Gowda, Explicit formula for solution of convex conservation laws with boundary condition, Duke Math. J., 62 (1991), 401-416. doi: 10.1215/S0012-7094-91-06216-2.  Google Scholar [24] E. Kaasschieter, Solving the Buckley-Leverret equation with gravity in a heterogeneous porous media, Comput. Geosci., 3 (1999), 23-48. doi: 10.1023/A:1011574824970.  Google Scholar [25] K. H. Karlsen, N. H. Risebro and J. D. Towers, On a Nonlinear Degenerate Parabolic Transport Diffusion Equation with Discontinuous Coefficient, Electron. J. Differential Equations, 2002.  Google Scholar [26] C. Klingenberg and N. H. Risebro, Convex conservation laws with discontinuous coefficients, existence, uniqueness and asymptotic behavior, Comm. Partial Differential Equations, 20 (1995), 1959-1990. doi: 10.1080/03605309508821159.  Google Scholar [27] S. N. Kružkov, First order quasilinear equations with several independent variables. (Russian), Mat. Sb., 81 (1970), 228-255. English transl. in Math. USSR Sb., 10 (1970), 217-243.  Google Scholar [28] P. D. Lax, Hyperbolic systems of conservation laws II, Comm. Pure Appl. Math., 10 (1957), 537-566. doi: 10.1002/cpa.3160100406.  Google Scholar [29] S. Mochon, An analysis for the traffic on highways with changing surface conditions, Math. Model., 9 (1987), 1-11. doi: 10.1016/0270-0255(87)90068-6.  Google Scholar [30] D. N. Ostrov, Solutions of Hamilton-Jacobi equations and conservation laws with discontinuous space-time dependence, J. Differential Equations, 182 (2002), 51-77. doi: 10.1006/jdeq.2001.4088.  Google Scholar [31] D. Serre, Systémes De Lois De Conservation. I., Hyperbolicité, Entropies, Ondes de Choc, Diderot Editeur, Paris, 1996.  Google Scholar [32] J. D. Towers, Convergence of a difference scheme for conservation laws with a discontinuous flux, SIAM J. Numer. Anal., 38 (2000), 681-698. doi: 10.1137/S0036142999363668.  Google Scholar
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