International Journal of Differential Equations

Zero Diffusion-Dispersion-Smoothing Limits for a Scalar Conservation Law with Discontinuous Flux Function

H. Holden, K. H. Karlsen, and D. Mitrovic

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We consider multidimensional conservation laws with discontinuous flux, which are regularized with vanishing diffusion and dispersion terms and with smoothing of the flux discontinuities. We use the approach of H -measures to investigate the zero diffusion-dispersion-smoothing limit.

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Int. J. Differ. Equ., Volume 2009 (2009), Article ID 279818, 33 pages.

Received: 2 April 2009
Revised: 24 August 2009
Accepted: 24 September 2009
First available in Project Euclid: 26 January 2017

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Holden, H.; Karlsen, K. H.; Mitrovic, D. Zero Diffusion-Dispersion-Smoothing Limits for a Scalar Conservation Law with Discontinuous Flux Function. Int. J. Differ. Equ. 2009 (2009), Article ID 279818, 33 pages. doi:10.1155/2009/279818.

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  • S. N. Kružhkov, “First order quasilinear quations in several independent variables,” Mathematics of the USSR-Sbornik, vol. 10, pp. 217–243, 1970.
  • R. J. DiPerna, “Measure-valued solutions to conservation laws,” Archive for Rational Mechanics and Analysis, vol. 88, no. 3, pp. 223–270, 1985.MR775191
  • K. H. Karlsen, N. H. Risebro, and J. D. Towers, “${L}_{1}$ stability for entropy solutions of nonlinear degenerate parabolic convection-diffusion equations with discontinuous coefficients,” Skrifter. Det Kongelige Norske Videnskabers Selskab, no. 3, pp. 1–49, 2003.MR2024741
  • Adimurthi, S. Mishra, and G. D. Veerappa Gowda, “Optimal entropy solutions for conservation laws with discontinuous flux-functions,” Journal of Hyperbolic Differential Equations, vol. 2, no. 4, pp. 783–837, 2005.MR2195983
  • L. Tartar, “Compensated compactness and applications to partial differential equations,” in Nonlinear Analysis & Mechanics: Heriot-Watt Symposium, Vol. IV, vol. 39 of Research Notes in Mathematics, pp. 136–212, Pitman, Boston, Mass, USA, 1979.MR584398
  • K. H. Karlsen, N. H. Risebro, and J. D. Towers, “On a nonlinear degenerate parabolic transport-diffusion equation with a discontinuous coefficient,” Electronic Journal of Differential Equations, vol. 93, pp. 1–23, 2002.MR1938389
  • P.-L. Lions, B. Perthame, and E. Tadmor, “A kinetic formulation of multidimensional scalar conservation laws and related equations,” Journal of the American Mathematical Society, vol. 7, no. 1, pp. 169–191, 1994.MR1201239
  • F. Bachmann and J. Vovelle, “Existence and uniqueness of entropy solution of scalar conservation laws with a flux function involving discontinuous coefficients,” Communications in Partial Differential Equations, vol. 31, no. 1–3, pp. 371–395, 2006.MR2209759
  • K. H. Karlsen, M. Rascle, and E. Tadmor, “On the existence and compactness of a two-dimensional resonant system of conservation laws,” Communications in Mathematical Sciences, vol. 5, no. 2, pp. 253–265, 2007.MR2334842
  • E. Yu. Panov, “Existence and strong precompactness properties for entropy solutions of čommentComment on ref. [18?]: Please update the information of this reference, if possible. a first-order quasilinear equation with discontinuous flux,” Archive for Rational Mechanics and Analysis. In press.
  • L. Tartar, “$H$-measures, a new approach for studying homogenisation, oscillations and concentration effects in partial differential equations,” Proceedings of the Royal Society of Edinburgh: Section A, vol. 115, no. 3-4, pp. 193–230, 1990.MR1069518
  • P. Gérard, “Microlocal defect measures,” Communications in Partial Differential Equations, vol. 16, no. 11, pp. 1761–1794, 1991.MR1135919
  • M. E. Schonbek, “Convergence of solutions to nonlinear dispersive equations,” Communications in Partial Differential Equations, vol. 7, no. 8, pp. 959–1000, 1982.MR668586
  • J. M. C. Correia and P. G. Lefloch, “Nonlinear diffusive-dispersive limits for multidimensional conservation laws,” in Advances in Nonlinear Partial Differential Equations and Related Areas (Beijing, 1997), pp. 103–123, World Scientific, Edge, NJ, USA, 1998.MR1690825
  • P. G. LeFloch, Hyperbolic Systems of Conservation Laws, Birkhäuser, Basel, Switzerland, 2002.MR1927887
  • P. G. LeFloch and R. Natalini, “Conservation laws with vanishing nonlinear diffusion and dispersion,” Nonlinear Analysis: Theory, Methods & Applications, vol. 36, no. 2, pp. 213–230, 1999.MR1668856
  • P. G. LeFloch and C. I. Kondo, “Zero diffusion-dispersion limits for scalar conservation laws,” SIAM Journal on Mathematical Analysis, vol. 33, no. 6, pp. 1320–1329, 2002.MR1920633
  • A. Szepessy, “An existence result for scalar conservation laws using measure valued solutions,” Communications in Partial Differential Equations, vol. 14, no. 10, pp. 1329–1350, 1989.MR1022989
  • S. Hwang, “Nonlinear diffusive-dispersive limits for scalar multidimensional conservation laws,” Journal of Differential Equations, vol. 225, no. 1, pp. 90–102, 2006.MR2228693
  • B. Perthame and P. E. Souganidis, “A limiting case for velocity averaging,” Annales Scientifiques de l'École Normale Supérieure, vol. 31, no. 4, pp. 591–598, 1998.MR1634024
  • E. Yu. Panov, “A condition for the strong precompactness of bounded sets of measure-valued solutions of a first-order quasilinear equation,” Sbornik: Mathematics, vol. 190, pp. 427–446, 1999.MR1700996
  • L. C. Evans, Weak Convergence Methods for Nonlinear Partial Differential Equations, vol. 74, American Mathematical Society, Providence, RI, USA, 1990.MR1034481
  • G. Dolzmann, N. Hungerbühler, and S. Müller, “Uniqueness and maximal regularity for nonlinear elliptic systems of $n$-Laplace type with measure valued right hand side,” Journal für die Reine und Angewandte Mathematik, vol. 520, pp. 1–35, 2000.MR1748270
  • P. Pedregal, Parametrized Measures and Variational Principles, vol. 30 of Progress in Nonlinear Differential Equations and Their Applications, Birkhäuser, Basel, Switzerland, 1997.MR1452107