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

New entropy conditions for scalar conservation laws with discontinuous flux

Darko Mitrovic 1,

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.
Keywords: discontinuous flux, Scalar conservation law, entropy conditions., existence and uniqueness.
Mathematics Subject Classification: Primary: 35L6.
Citation: Darko Mitrovic. New entropy conditions for scalar conservation laws with discontinuous flux. Discrete and Continuous Dynamical Systems, 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), 27-70.

[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), 783-837. doi: 10.1142/S0219891605000622.

[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), 127-157. doi: 10.3934/nhm.2007.2.127.

[4]

J. Aleksic and D. Mitrovic, On the compactness for two dimensional scalar conservation law with discontinuous flux, Comm. Math. Sciences, 4 (2009), 963-971.

[5]

B. Andreianov, K. H. Karlsena and N. H. Risebro, On vanishing viscosity approximation of conservation laws with discontinuous flux,, preprint. Available from: , (). 

[6]

E. Audusse and B. Perthame, Uniqueness for scalar conservation law via adapted entropies, Proc. Roy. Soc. Edinburgh Sect. A, 135 (2005), 253-265. doi: 10.1017/S0308210500003863.

[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), 371-395. doi: 10.1080/03605300500358095.

[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), 1684-1712. doi: 10.1137/07069314X.

[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), 387-425. doi: 10.1007/s10665-007-9148-4.

[10]

S. Diehl, On scalar conservation law with point source and discontinuous flux function modelling continuous sedimentation, SIAM J. Math. Anal., 6 (1995), 1425-1451. doi: 10.1137/S0036141093242533.

[11]

S. Diehl, A conservation law with point source and discontinuous flux function modelling continuous sedimentation, SIAM J. Appl. Anal., 2 (1996), 388-419.

[12]

S. Diehl, A uniqueness condition for non-linear convection-diffusion equations with discontinuous coefficients, J. Hyperbolic Diff. Eq., 6 (2009), 127-159. doi: 10.1142/S0219891609001794.

[13]

R. J. DiPerna, Measure-valued solutions to conservation laws, Arch. Ration. Mech. Anal., 88 (1985), 223-270. doi: 10.1007/BF00752112.

[14]

L. C. Evans, "Weak Convergence Methods in Nonlinear Partial Differential Equations," AMS, 74, Providence, Rhode Island, 1990.

[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), 33 pp. doi: 10.1155/2009/279818.

[16]

P. Gerard, Microlocal defect measures, Comm. Partial Differential Equations, 11 (1991), 1761-1794. doi: 10.1080/03605309108820822.

[17]

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

[18]

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.

[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), 1-49.

[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 (). 

[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), 253-265.

[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), 287-318. doi: 10.1142/S0252959904000299.

[23]

S. N. Kruzhkov, First order quasilinear equations in several independent variables, Mat. Sb., 81 (1970), 217-243. doi: 10.1070/SM1970v010n02ABEH002156.

[24]

Y. S. Kwon and A. Vasseur, Strong traces for scalar conservation laws with general flux, Arch. Rat. Mech. Anal., 3 (2007), 495-513. doi: 10.1007/s00205-007-0055-7.

[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), 169-191. doi: 10.1090/S0894-0347-1994-1201239-3.

[26]

D. Mitrovic, Estence amd stability of a multidimensional scalar conservation law with discontinuous flux, Netw. Het. Media, 5 (2010), 163-188. doi: 10.3934/nhm.2010.5.163.

[27]

E. Yu. Panov, Existence of Strong Traces for Quasi-Solutions of Multidimensional Conservation Laws, J. of Hyperbolic Differential Equations, 4 (2007), 729-770. doi: 10.1142/S0219891607001343.

[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), 525-548. doi: 10.1142/S0219891609001915.

[29]

E. Yu. Panov, On weak completeness of the set of entropy solutions to a scalar conservation law, SIAM J. Math. Anal., 1 (2009), 26-36. doi: 10.1137/080724587.

[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), 643-673. doi: 10.1007/s00205-009-0217-x.

[31]

P. Pedregal, "Parametrized Measures and Variational Principles," Progress in Nonlinear Partial Differential Equations and Their Applications, 30, Birkhäuser, Basel, 1997.

[32]

B. Perthame, Kinetic approach to systems of conservation laws, Journées équations aux derivées partielles, (1992), 13.

[33]

L. Tartar, Comp. compactness and application to PDEs, Nonlin. Anal. and Mech.: Heriot-Watt symposium, IV, Pitman, Boston, Mass., 1979.

[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), 193-230.

[35]

B. Temple, Global solution of the Cauchy problem for a class of 2x2 nonstrictly hyperbolic conservation laws, Adv. in Appl. Math., 3 (1982), 335-375. doi: 10.1016/S0196-8858(82)80010-9.

[36]

A. Vasseur, Strong traces for solutions of multidimensional conservation laws, Arch. Rat. Mech. Anal., 160 (2001), 181-193. doi: 10.1007/s002050100157.

show all references

References:
[1]

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

[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), 783-837. doi: 10.1142/S0219891605000622.

[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), 127-157. doi: 10.3934/nhm.2007.2.127.

[4]

J. Aleksic and D. Mitrovic, On the compactness for two dimensional scalar conservation law with discontinuous flux, Comm. Math. Sciences, 4 (2009), 963-971.

[5]

B. Andreianov, K. H. Karlsena and N. H. Risebro, On vanishing viscosity approximation of conservation laws with discontinuous flux,, preprint. Available from: , (). 

[6]

E. Audusse and B. Perthame, Uniqueness for scalar conservation law via adapted entropies, Proc. Roy. Soc. Edinburgh Sect. A, 135 (2005), 253-265. doi: 10.1017/S0308210500003863.

[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), 371-395. doi: 10.1080/03605300500358095.

[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), 1684-1712. doi: 10.1137/07069314X.

[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), 387-425. doi: 10.1007/s10665-007-9148-4.

[10]

S. Diehl, On scalar conservation law with point source and discontinuous flux function modelling continuous sedimentation, SIAM J. Math. Anal., 6 (1995), 1425-1451. doi: 10.1137/S0036141093242533.

[11]

S. Diehl, A conservation law with point source and discontinuous flux function modelling continuous sedimentation, SIAM J. Appl. Anal., 2 (1996), 388-419.

[12]

S. Diehl, A uniqueness condition for non-linear convection-diffusion equations with discontinuous coefficients, J. Hyperbolic Diff. Eq., 6 (2009), 127-159. doi: 10.1142/S0219891609001794.

[13]

R. J. DiPerna, Measure-valued solutions to conservation laws, Arch. Ration. Mech. Anal., 88 (1985), 223-270. doi: 10.1007/BF00752112.

[14]

L. C. Evans, "Weak Convergence Methods in Nonlinear Partial Differential Equations," AMS, 74, Providence, Rhode Island, 1990.

[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), 33 pp. doi: 10.1155/2009/279818.

[16]

P. Gerard, Microlocal defect measures, Comm. Partial Differential Equations, 11 (1991), 1761-1794. doi: 10.1080/03605309108820822.

[17]

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

[18]

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.

[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), 1-49.

[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 (). 

[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), 253-265.

[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), 287-318. doi: 10.1142/S0252959904000299.

[23]

S. N. Kruzhkov, First order quasilinear equations in several independent variables, Mat. Sb., 81 (1970), 217-243. doi: 10.1070/SM1970v010n02ABEH002156.

[24]

Y. S. Kwon and A. Vasseur, Strong traces for scalar conservation laws with general flux, Arch. Rat. Mech. Anal., 3 (2007), 495-513. doi: 10.1007/s00205-007-0055-7.

[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), 169-191. doi: 10.1090/S0894-0347-1994-1201239-3.

[26]

D. Mitrovic, Estence amd stability of a multidimensional scalar conservation law with discontinuous flux, Netw. Het. Media, 5 (2010), 163-188. doi: 10.3934/nhm.2010.5.163.

[27]

E. Yu. Panov, Existence of Strong Traces for Quasi-Solutions of Multidimensional Conservation Laws, J. of Hyperbolic Differential Equations, 4 (2007), 729-770. doi: 10.1142/S0219891607001343.

[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), 525-548. doi: 10.1142/S0219891609001915.

[29]

E. Yu. Panov, On weak completeness of the set of entropy solutions to a scalar conservation law, SIAM J. Math. Anal., 1 (2009), 26-36. doi: 10.1137/080724587.

[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), 643-673. doi: 10.1007/s00205-009-0217-x.

[31]

P. Pedregal, "Parametrized Measures and Variational Principles," Progress in Nonlinear Partial Differential Equations and Their Applications, 30, Birkhäuser, Basel, 1997.

[32]

B. Perthame, Kinetic approach to systems of conservation laws, Journées équations aux derivées partielles, (1992), 13.

[33]

L. Tartar, Comp. compactness and application to PDEs, Nonlin. Anal. and Mech.: Heriot-Watt symposium, IV, Pitman, Boston, Mass., 1979.

[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), 193-230.

[35]

B. Temple, Global solution of the Cauchy problem for a class of 2x2 nonstrictly hyperbolic conservation laws, Adv. in Appl. Math., 3 (1982), 335-375. doi: 10.1016/S0196-8858(82)80010-9.

[36]

A. Vasseur, Strong traces for solutions of multidimensional conservation laws, Arch. Rat. Mech. Anal., 160 (2001), 181-193. doi: 10.1007/s002050100157.

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