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Global existence and asymptotic behavior of measure valued solutions to the kinetic Kuramoto--Daido model with inertia
Convergence of vanishing capillarity approximations for scalar conservation laws with discontinuous fluxes
1. | Department of Mathematics, University of Bari, Via E. Orabona 4, I--70125 Bari |
2. | Department of Mathematics, University of Bari, via E. Orabona 4, 70125 Bari, Italy |
3. | Seminar for Applied Mathematics (SAM), ETH Zürich, HG G 57.2, Rämistrasse 101, 8092 Zürich, Switzerland, Switzerland |
References:
[1] |
Adimurthi, S. Mishra and G. D. Veerappa Gowda, Optimal entropy solutions for conservation laws with discontinuous flux-functions, J. Hyp. Diff. Eqns., 2 (2005), 783-837.
doi: 10.1142/S0219891605000622. |
[2] |
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. |
[3] |
E. Audusse and B. Perthame, Uniqueness for scalar conservation laws with discontinuous flux via adapted entropies, Proc. Roy. Soc. Edinburgh Sect. A, 135 (2005), 253-265.
doi: 10.1017/S0308210500003863. |
[4] |
K. Aziz and A. Settari, Fundamentals of Petroleum Reservoir Simulation, Applied Science Publishers, London, 1979. |
[5] |
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. |
[6] |
G. M. Coclite and K. H. Karlsen, A singular limit problem for conservation laws related to the Camassa-Holm shallow water equation, Comm. Partial Differential Equations, 31 (2006), 1253-1272.
doi: 10.1080/03605300600781600. |
[7] |
G. M. Coclite, K. H. Karlsen, S. Mishra and N. H. Risebro, Convergence of vanishing viscosity approximations of $2\times2$ triangular systems of multi-dimensional conservation laws, Boll. Unione Mat. Ital. (9), 2 (2009), 275-284. |
[8] |
C. Dafermos, Hyperbolic Conservation laws in Continuum Physics, $3^{rd}$ edition, Springer-Verlag, New York, 2005. |
[9] |
E. vanDuijn, L. A. Peletier and S. Pop, A new class of entropy solutions of the Buckley-Leverett equation, SIAM J. Math. Anal., 39 (2007), 507-536.
doi: 10.1137/05064518X. |
[10] |
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. |
[11] |
R. Helmig, A. Weiss and B. I. Wohlmuth, Dynamic capillary effects in heterogeneous porous media, Comp. Geosci., 11 (2007), 261-274.
doi: 10.1007/s10596-007-9050-1. |
[12] |
S. Hassanizadeh and W. G. Gray, Mechanics and thermodynamics of multiphase flow in porous media including interphase boundaries, Adv. Wat. Res., 13 (1990), 169-186.
doi: 10.1016/0309-1708(90)90040-B. |
[13] |
H. Holden, K. H. Karlsen and D. Mitrovic, Zero diffusion-dispersion-smoothing limits for scalar conservation law with discontinuous flux function, International Journal of Differential Equations, 2009 (2009), Art. ID 279818, 1-33. |
[14] |
H. Holden, K. H. Karlsen, D. Mitrovic and E. Y. Panov, Strong compactness of approximate solutions to degenerate elliptic-hyperbolic equations with discontinuous flux function, Acta Mathematica Scientia, 29B (2009), 573-612.
doi: 10.1016/S0252-9602(10)60004-5. |
[15] |
K. H. Karlsen and F. Kissling, On the singular limit of a two-phase flow equation with heterogeneities and dynamic capillary pressure,, Z. Angew. Math. Mech., ().
|
[16] |
K. H. Karlsen, N. H. Risebro and J. Towers, $L^1$ stability for entropy solutions of nonlinear degenerate parabolic convection-diffusion equations with discontinuous coefficients, Skr. K. Nor. Vidensk. Selsk., 3 (2003), 1-49. |
[17] |
F. Kissling and C. Rohde, The computation of nonclassical shock waves with a heterogeneous multiscale method, Netw. Heterog. Media, 5 (2010), 661-674.
doi: 10.3934/nhm.2010.5.661. |
[18] |
P. LeFloch, Hyperbolic Systems of Conservation Laws: The Theory Of Classical and Non-Classical Shock Waves, Lecture notes in Mathematics., ETH Zürich, Birkhauser, 2002.
doi: 10.1007/978-3-0348-8150-0. |
[19] |
S. Mishra and J. Jaffré, On the upstream mobility scheme for two-phase flow in porous media, Comp. GeoSci., 14 (2010), 105-124.
doi: 10.1007/s10596-009-9135-0. |
[20] |
F. Murat, L'injection du cône positif de $H^{-1}$ dans $W^{-1,q}$ est compacte pour tout $q<2$, J. Math. Pures Appl. (9), 60 (1981), 309-322. |
[21] |
S. Mochon, An analysis of the traffic on highways with changing surface conditions, Math. Model., 9 (1987), 1-11.
doi: 10.1016/0270-0255(87)90068-6. |
[22] |
E. Yu. Panov, Existence and strong pre-compactness properties for entropy solutions of a first-order quasilinear equation with discontinuous flux, Arch. Ration. Mech. Anal., 195 (2010), 643-673.
doi: 10.1007/s00205-009-0217-x. |
[23] |
E. Yu. Panov, Erratum to: Existence and strong pre-compactness properties for entropy solutions of a first-order quasilinear equation with discontinuous flux, Arch. Ration. Mech. Anal., 196 (2010), 1077-1078.
doi: 10.1007/s00205-010-0303-0. |
show all references
References:
[1] |
Adimurthi, S. Mishra and G. D. Veerappa Gowda, Optimal entropy solutions for conservation laws with discontinuous flux-functions, J. Hyp. Diff. Eqns., 2 (2005), 783-837.
doi: 10.1142/S0219891605000622. |
[2] |
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. |
[3] |
E. Audusse and B. Perthame, Uniqueness for scalar conservation laws with discontinuous flux via adapted entropies, Proc. Roy. Soc. Edinburgh Sect. A, 135 (2005), 253-265.
doi: 10.1017/S0308210500003863. |
[4] |
K. Aziz and A. Settari, Fundamentals of Petroleum Reservoir Simulation, Applied Science Publishers, London, 1979. |
[5] |
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. |
[6] |
G. M. Coclite and K. H. Karlsen, A singular limit problem for conservation laws related to the Camassa-Holm shallow water equation, Comm. Partial Differential Equations, 31 (2006), 1253-1272.
doi: 10.1080/03605300600781600. |
[7] |
G. M. Coclite, K. H. Karlsen, S. Mishra and N. H. Risebro, Convergence of vanishing viscosity approximations of $2\times2$ triangular systems of multi-dimensional conservation laws, Boll. Unione Mat. Ital. (9), 2 (2009), 275-284. |
[8] |
C. Dafermos, Hyperbolic Conservation laws in Continuum Physics, $3^{rd}$ edition, Springer-Verlag, New York, 2005. |
[9] |
E. vanDuijn, L. A. Peletier and S. Pop, A new class of entropy solutions of the Buckley-Leverett equation, SIAM J. Math. Anal., 39 (2007), 507-536.
doi: 10.1137/05064518X. |
[10] |
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. |
[11] |
R. Helmig, A. Weiss and B. I. Wohlmuth, Dynamic capillary effects in heterogeneous porous media, Comp. Geosci., 11 (2007), 261-274.
doi: 10.1007/s10596-007-9050-1. |
[12] |
S. Hassanizadeh and W. G. Gray, Mechanics and thermodynamics of multiphase flow in porous media including interphase boundaries, Adv. Wat. Res., 13 (1990), 169-186.
doi: 10.1016/0309-1708(90)90040-B. |
[13] |
H. Holden, K. H. Karlsen and D. Mitrovic, Zero diffusion-dispersion-smoothing limits for scalar conservation law with discontinuous flux function, International Journal of Differential Equations, 2009 (2009), Art. ID 279818, 1-33. |
[14] |
H. Holden, K. H. Karlsen, D. Mitrovic and E. Y. Panov, Strong compactness of approximate solutions to degenerate elliptic-hyperbolic equations with discontinuous flux function, Acta Mathematica Scientia, 29B (2009), 573-612.
doi: 10.1016/S0252-9602(10)60004-5. |
[15] |
K. H. Karlsen and F. Kissling, On the singular limit of a two-phase flow equation with heterogeneities and dynamic capillary pressure,, Z. Angew. Math. Mech., ().
|
[16] |
K. H. Karlsen, N. H. Risebro and J. Towers, $L^1$ stability for entropy solutions of nonlinear degenerate parabolic convection-diffusion equations with discontinuous coefficients, Skr. K. Nor. Vidensk. Selsk., 3 (2003), 1-49. |
[17] |
F. Kissling and C. Rohde, The computation of nonclassical shock waves with a heterogeneous multiscale method, Netw. Heterog. Media, 5 (2010), 661-674.
doi: 10.3934/nhm.2010.5.661. |
[18] |
P. LeFloch, Hyperbolic Systems of Conservation Laws: The Theory Of Classical and Non-Classical Shock Waves, Lecture notes in Mathematics., ETH Zürich, Birkhauser, 2002.
doi: 10.1007/978-3-0348-8150-0. |
[19] |
S. Mishra and J. Jaffré, On the upstream mobility scheme for two-phase flow in porous media, Comp. GeoSci., 14 (2010), 105-124.
doi: 10.1007/s10596-009-9135-0. |
[20] |
F. Murat, L'injection du cône positif de $H^{-1}$ dans $W^{-1,q}$ est compacte pour tout $q<2$, J. Math. Pures Appl. (9), 60 (1981), 309-322. |
[21] |
S. Mochon, An analysis of the traffic on highways with changing surface conditions, Math. Model., 9 (1987), 1-11.
doi: 10.1016/0270-0255(87)90068-6. |
[22] |
E. Yu. Panov, Existence and strong pre-compactness properties for entropy solutions of a first-order quasilinear equation with discontinuous flux, Arch. Ration. Mech. Anal., 195 (2010), 643-673.
doi: 10.1007/s00205-009-0217-x. |
[23] |
E. Yu. Panov, Erratum to: Existence and strong pre-compactness properties for entropy solutions of a first-order quasilinear equation with discontinuous flux, Arch. Ration. Mech. Anal., 196 (2010), 1077-1078.
doi: 10.1007/s00205-010-0303-0. |
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