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New entropy conditions for scalar conservation laws with discontinuous flux
1. | Faculty of Mathematics, University of Montenegro, Cetinjski put bb, 81000 Podgorica |
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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