Abstract
We construct a nonlinear kinetic equation and prove that it is welladapted to describe general multidimensional scalar conservation laws. In particular we prove that it is well-posed uniformly in ε — the microscopic scale. We also show that the proposed kinetic equation is equipped with a family of kinetic entropy functions — analogous to Boltzmann's microscopicH-function, such that they recover Krushkov-type entropy inequality on the macroscopic scale. Finally, we prove by both — BV compactness arguments in the multidimensional case and by compensated compactness arguments in the one-dimensional case, that the local density of kinetic particles admits a “continuum” limit, as it converges strongly with ε↓0 to the unique entropy solution of the corresponding conservation law.
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Boltzmann, L.: Vorlesungen uber Gas theorie, Liepzig 1886
Y. Brenier, Y.: Averaged multivaried solutions for scalar conservation laws. SIAM J. Numer. Anal.21, 1013–1037 (1986)
Caflisch, R.: The fluid dynamic limit of the nonlinear Boltzmann equation. Commun. Pure Appl. Math.33, 651–666 (1980)
Crandall, M., Majda, A.: Monotone difference approximations for scalar conservation laws. Math. Comp.34, 1–21 (1980)
Crandall, M., Tartar, L.: Some relations between non-expansive and order preserving mappings. Proc. Am. Math. Soc.78 (3) 385–390 (1980)
DiPerna, R., Lions, P. L. On the Cauchy problem for Boltzmann equations: Global existence and weak stability. Ann. Math. (1989)
Giga, Y., Miyakawa, T.: A kinetic construction of global solutions of first order quasilinear equations. Duke Math. J.50, 505–515 (1983)
Krushkov, S. N.: Frist order quasilinear equations in several independent variables. Math. USSR Sb.10, 217–243 (1970)
Lax, P. D.: Hyperbolic systems of conservation laws and the mathematical theory of shock waves. SIAM Regional Conference Series in Applied Mathematics, vol. 11
Murat, F.: Compacité per compensation. Ann. Scuola Norm. Sup. Disa Sci. Math.5, 489–507 (1978) and8, 69–102 (1981)
Perthame, B.: Global existence of solutions to the BGK model of Boltzmann equations. J. Diff. Eq.81, 191–205 (1989)
Tadmor, E.: Semi-discrete approximations to nonlinear systems of conservation laws; consistency andL ∞-stability imply convergence. ICASE Report No. 88-41.
Tartar, L.: Compensated compactness and applications to partial differential equations. In: Research Notes in Mathematics, vol. 39, Nonlinear Analysis and Mechanics, Heriot-Watt Sympos., vol. 4. Knopps, R. J. (ed.) pp. 136–211. Boston, London: Pittman Press 1975
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Communicated by J. L. Lebowitz
Research was supported in part by the National Aeronautics and Space Administration under NASA Contract No. NAS1-18605 while the authors were in residence at the Institute for Computer Applications in Science and Engineering (ICASE), NASA Langley Research Center, Hampton, VA 23665. Additional support for the second author was provided by U.S.-Israel BSF Grant No. 85-00346. Part of this research was carried out while the first author was visiting Tel-Aviv University
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Perthame, B., Tadmor, E. A kinetic equation with kinetic entropy functions for scalar conservation laws. Commun.Math. Phys. 136, 501–517 (1991). https://doi.org/10.1007/BF02099071
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DOI: https://doi.org/10.1007/BF02099071