Glasnik Matematicki, Vol. 55, No. 1 (2020), 101-111.

SCALAR CONSERVATION LAWS WITH CHARATHEODORY FLUX REVISITED

Nikola Konatar

Department of Natural Sciences and Mathematics, Faculty of Mathematics and Natural Sciences, University of Montenegro, Džordža Vašingtona bb, 81000 Podgorica, Crna Gora
e-mail: nikola.k@ucg.ac.me

Abstract.   We introduce a new approach for dealing with scalar conservation laws with the flux discontinuous with respect to the space variable and merely continuous with respect to the state variable which employs a variant of the kinetic formulation. We use it to improve results about the existence of solutions for non-degenerate scalar conservation laws with Caratheodory flux under a variant of non-degeneracy conditions.

2010 Mathematics Subject Classification.   35L65, 65M25

Key words and phrases.   Multidimensional scalar conservation law, discontinuous flux, existence, kinetic formulation, Caratheodory flux

Full text (PDF) (free access)

https://doi.org/10.3336/gm.55.1.09

References:

  1. Adimurthi, S. Mishra and G.D.V. Gowda, Optimal entropy solutions for conservation laws with discontinuous flux, J.Hyperbolic Differ. Equ. 2 (2005), 783-837.
    MathSciNet     CrossRef

  2. B. Andreianov and D. Mitrovic, Entropy conditions for scalar conservation laws with discontinuous flux revisited, Ann. Inst. H. Poincaré Anal. Non Linéaire 32 (2015), 1307-1335.
    MathSciNet     CrossRef

  3. B. Andreianov, K. H. Karlsen and N. H. Risebro, A theory of L1-dissipative solvers for scalar conservation laws with discontinuous flux, Arch. Ration. Mech. Anal. 201 (2011), 27-86.
    MathSciNet     CrossRef

  4. N. Antonić, M. Erceg and M. Lazar, Localisation principle for one-scale H-measures, J. Funct. Anal. 272 (2017), 3410-3454.
    MathSciNet     CrossRef

  5. N. Antonić and M. Lazar, Parabolic H-measures, J. Funct. Anal. 265 (2013), 1190-1239.
    MathSciNet     CrossRef

  6. N. Antonić and D. Mitrović, H-distributions, an extension of H-measures to an lp-lq setting, Abstr. Appl. Anal. 2011, Article ID 901084, 12 pp.
    MathSciNet     CrossRef

  7. R. Bürger, K. H. Karlsen, H. Torres and J. Towers, Second-order schemes for conservation laws with discontinuous flux modelling clarifier-thickener units, Numer. Math. 116 (2010), 579-617.
    MathSciNet     CrossRef

  8. R. Colombo, K. H. Karlsen, F. Lagoutiere and A. Marson, Special issue on contemporary topics in conservation laws, Netw. Heterog. Media 11 (2016), i-ii.
    MathSciNet     CrossRef

  9. P. Gérard, Microlocal defect measures, Comm. Partial Differential Equations 16 (1991), 1761-1794.
    MathSciNet     CrossRef

  10. M. Garavello, R. Natalini, B. Piccoli and A. Terracina, Conservation laws with discontinuous flux, Netw. Heter. Media 2 (2007), 159-179.
    MathSciNet     CrossRef

  11. L. Grafakos, Classical Fourier analysis, Springer, New York, 2008.
    MathSciNet    

  12. K. H. Karlsen, N. H. Risebro and J. Towers, L1-stability for entropy solutions of nonlinear degenerate parabolic convection-diffusion equations with discontinuous coefficients, Skr. K. Nor. Vidensk. Selsk. 3 (2003), 1-49.
    MathSciNet    

  13. S. N. Kruzhkov, First order quasilinear equations with several independent variables, Mat. Sb. (N.S.) 81(123) (1970), 228-255.
    MathSciNet    

  14. M. Lazar and D. Mitrović, Velocity averaging - a general framework, Dynamics of PDEs 9 (2012) 239-260.
    MathSciNet     CrossRef

  15. M. Lazar and D. Mitrović, On an extension of a bilinear functional on Lp(ℝn)× E to Bôchner spaces with an application to velocity averaging, C. R. Math. Acad. Sci. Paris 351 (2013), 261-264.
    MathSciNet    

  16. M. Lazar and D. Mitrović, Existence of solutions for a scalar conservation law with a flux of low regularity, Electron. J. Differential Equations 2016, No. 325, 18 pp.
    MathSciNet    

  17. M. Lazar and D. Mitrović, On a new class of functional spaces with application to the velocity averaging, Glas. Mat. Ser. III 52(72) (2017), 115-130.
    MathSciNet     CrossRef

  18. P.L. Lions, B. Perthame and E. Tadmor, A kinetic formulation of multidimensional scalar conservation law and related equations, J. Amer. Math. Soc. 7 (1994), 169-191.
    MathSciNet     CrossRef

  19. M. Mišur and D. Mitrović, On a generalization of compensated compactness in the Lp-Lq setting, J. Funct. Anal. 268 (2015), 1904-1927.
    MathSciNet     CrossRef

  20. W. Neves, E.Yu. Panov and J. Silva, Strong traces for conservation laws with general nonautonomous flux, SIAM J. Math. Anal. 50 (2018), 6049-6081.
    MathSciNet     CrossRef

  21. E. Yu. Panov, Generalized solutions of the Cauchy problem for quasilinear conservation laws, Dissertation Can. Phys.-Math. Sci., Mosk. Gos. Univ., Moscow, 1991 (Russian).

  22. E.Yu. Panov, On existence and uniqueness of entropy solutions to the Cauchy problem for a conservation law with discontinuous flux, J. Hyperbolic Differ. Equ. 6 (2009), 525-548.
    MathSciNet     CrossRef

  23. 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.
    MathSciNet     CrossRef

  24. E.Yu. Panov, Ultra-parabolic H-measures and compensated compactness, Ann. Inst. H. Poincaré Anal. Non Linéaire 28 (2011), 47-62.
    MathSciNet     CrossRef

  25. E.Yu. Panov, Ultra-parabolic H-measures and compensated compactness, Ann. Inst. H.Poincaré Anal. Non Linéare 28 (2011), 47-62.

  26. F. Rindler, Directional oscillations, concentrations, and compensated compactness via microlocal compactness forms, Arch. Ration. Mech. Anal. 215 (2015), 1-63.
    MathSciNet     CrossRef

  27. L. Tartar, H-measures, a new approach for studying homogenisation, oscillations and concentration effects in partial differential equations, Proc. Roy. Soc. Edinburgh Sect. A 115 (1990), 193-230.
    MathSciNet     CrossRef

  28. L. Tartar, Multi-scales H-measures, Discrete Contin. Dyn. Syst. Ser. S 8 (2015) 77-90.
    MathSciNet     CrossRef

  29. B. Temple, Global solution of the Cauchy problem for a class of 2× 2 nonstrictly hyperbolic conservation laws, Adv. in Appl. Math. 3 (1982), 335-375.
    MathSciNet     CrossRef
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