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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 and 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.

[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.

[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.

[4]

Adimurthi and G. D. Veerappa Gowda, Conservation laws with discontinuous flux, J. Math. Kyoto Univ., 43 (2003), 27-70.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[17]

S. Diehl, Dynamic and steady-state behaviour of continuous sedimentation, SIAM J. Appl. Math., 57 (1997), 991-1018. doi: 10.1137/S0036139995290101.

[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.

[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.

[20]

T. Gimse and N. H. Risebro, Riemann problems with discontinuous flux function, Proc. 3rd Internat. Conf. Hyperbolic Problems, Studentlitteratur, Uppsala, (1991), 488-502.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[28]

P. D. Lax, Hyperbolic systems of conservation laws II, Comm. Pure Appl. Math., 10 (1957), 537-566. doi: 10.1002/cpa.3160100406.

[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.

[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.

[31]

D. Serre, Systémes De Lois De Conservation. I., Hyperbolicité, Entropies, Ondes de Choc, Diderot Editeur, Paris, 1996.

[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.

show all references

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.

[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.

[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.

[4]

Adimurthi and G. D. Veerappa Gowda, Conservation laws with discontinuous flux, J. Math. Kyoto Univ., 43 (2003), 27-70.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[17]

S. Diehl, Dynamic and steady-state behaviour of continuous sedimentation, SIAM J. Appl. Math., 57 (1997), 991-1018. doi: 10.1137/S0036139995290101.

[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.

[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.

[20]

T. Gimse and N. H. Risebro, Riemann problems with discontinuous flux function, Proc. 3rd Internat. Conf. Hyperbolic Problems, Studentlitteratur, Uppsala, (1991), 488-502.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[28]

P. D. Lax, Hyperbolic systems of conservation laws II, Comm. Pure Appl. Math., 10 (1957), 537-566. doi: 10.1002/cpa.3160100406.

[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.

[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.

[31]

D. Serre, Systémes De Lois De Conservation. I., Hyperbolicité, Entropies, Ondes de Choc, Diderot Editeur, Paris, 1996.

[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.

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