Transactions of the American Mathematical Society

Analysis of a coupled system of kinetic equations and conservation laws: Rigorous derivation and existence theory via defect measures

Author(s): M. Tidriri
Journal: Trans. Amer. Math. Soc. 357 (2005), 2133-2164.
MSC (2000): Primary 35L65, 82B40
Posted: January 21, 2005
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Abstract: In this paper we introduce a coupled system of kinetic equations of B.G.K. type and then study its hydrodynamic limit. We obtain as a consequence the rigorous derivation and existence theory for a coupled system of kinetic equations and their hydrodynamic (conservation laws) limit. The latter is a particular case of the coupled system of Boltzmann and Euler equations. A fundamental element in this study is the rigorous derivation and justification of the interface conditions between the kinetic model and its hydrodynamic conservation laws limit, which is obtained using a new regularity theory introduced herein.

1.: Bardos C., Golse F., Perthame B., and Sentis R., The nonaccretive radiative transfer equations: existence of solutions and Rosseland approximations, J. Funct. Anal. 77, 434-460 (1988). MR 0933978 (89f:35174)
2.: Bardos C., Leroux A.-Y., and Nedelec J.-C., First order quasilinear equations with boundary conditions, Comm. PDE 4 (9), 1017-1034 (1979). MR 0542510 (81b:35052)
3.: Brezis, H., Analyse functionnelle Théorie et applications, second edition, Collection Mathématiques appliquées pour la maîtrise sous la direction de P. G. Ciarlet et J. L. Lions, Masson, Paris, 1987. MR 0697382 (85a:46001)
4.: Chen G.-Q. and Frid H., On the theory of divergence-measure fields and its applications, Boletin da Sociedade Brasileira de Matematica, to appear.
5.: Chen G.-Q. and Perthame B., Well-posedness for anisotropic degenerate parabolic-hyperbolic equations. Annales de l'institut Henri Poincaré: Analyse Nonlinéaire, 2002, to appear.
6.: Dautray, R and Lions, J.-L., Analyse mathématique et calcul numérique pour les sciences et les techniques, Volume 9, Evolution: numérique, transport. Masson, Paris, 1988. MR 1016606 (90m:00006c)
7.: Diperna, R. J., Lions, P.-L. and Meyer, Y., regularity of velocity averages, Ann. Inst. Henri Poincaré, 8, Nos 3-4, pp. 271-287, 1991. MR 1127927 (92g:35036)
8.: Evans, L. C. and Gariepy, R. F., Measure Theory and Fine Properties of Functions, Studies in advanced mathematics, CRC Press, 1992. MR 1158660 (93f:28001)
9.: Golse F., Lions P.-L., Perthame B., and Sentis R., Regularity of the moments of the solution of a transport equation, J. Funct. Anal. 16 (1988). MR 0923047 (89a:35179)
10.: Krushkov S. N., First order quasilinear equations with several independent variables, Math. Sbornik. 81 (123), 228-255; Math USSR Sbornik 10, 217-243 (1970). MR 0267257 (42:2159)
11.: Lions, P.-L., Perthame B. and Souganidis P.E., Existence of entropy solutions for the hyperbolic systems of isentropic gas dynamics in Eulerian and Lagrangian coordinates, C.P.A.M. Vol. 49 (1996) 599-638. MR 1383202 (97e:35107)
12.: Lions, P.-L., Perthame B. and Tadmor E., A kinetic formulation of multidimensional scalar conservation laws and related equations, J. AMS, 7, No. 1, 169-190, 1994. MR 1201239 (94d:35100)
13.: Lions, P.-L., Perthame B. and Tadmor E., Kinetic formulation of the isentropic gas dynamics and -systems, Comm. Math. Phys. 163 (1994) 415-431. MR 1284790 (95c:76093)
14.: Perthame B., Uniqueness and error estimates in first order quasilinear conservation laws via the kinetic entropy defect measure, J. Math. Pures Appl., 77, 1998, pp. 1055-1064. MR 1661021 (2000e:35141)
15.: Perthame B., Kinetic formulations of conservation laws, Oxford Lecture Series in Mathematics and its Applications, vol. 21, Oxford University Press, Oxford, 2002. MR 2064166
16.: Perthame B. and Tadmor E., A kinetic equation with kinetic entropy functions for scalar conservation laws, Comm. Math. Phys. 136, 501-517 (1991). MR 1099693 (92d:82095)
17.: Schwartz L., Théorie des distributions, Hermann, Paris, 1978. MR 0209834 (35:730)
18.: Tartar L., Une nouvelle méthode de résolution d'équations aux dérivées partielles nonlinéaires, Lecture Notes in Math., vol. 665, Springer, Berlin, 1977, pp. 228-241. MR 0519433 (80b:35007)
19.: 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: Pitman Press, 1975. MR 0584398 (81m:35014)
20.: Tidriri M., Asymptotic analysis of a coupled system of kinetic equations. C. R. Acad. Sci. Paris, t. 328, Série I Math., 637-642, 1999. MR 1680029 (99m:76099)
21.: Tidriri M. D., Numerical analysis of coupling for a kinetic equation. Mathematical Modelling and Numerical Analysis M2AN, Vol. 33, No 6, 1999, p. 1121-1134. MR 1736892 (2000j:76128)
22.: Tidriri M. D., A novel class of multiscale models in mathematical physics. J. Nonlinear Analysis, Vol. 47 (2001) 4995-5008. MR 1975891 (2004d:35039)
23.: Tidriri M. D., Rigorous derivation and analysis of coupling of kinetic equations and their hydrodynamics limits for a simplified Boltzmann model. J. Stat. Phys. Vol. 104. Nos. 1/2 (2001), pp. 257-292. MR 1851389 (2002h:82075)
24.: Tidriri M. D., New models for the solution of intermediate regimes in transport theory and radiative transfer: existence theory, positivity, asymptotic analysis, and approximations. J. Stat. Phys., Vol. 104. Nos. 1/2 (2001), pp. 293-328. MR 1851390 (2002j:82104)
25.: Tidriri M. D., Analysis of a coupled system of kinetic equations and their hydrodynamic (conservation laws) limit: rigorous derivation and existence theory via BV theory. Accepted.
26.: Tidriri M., Hydrodynamic limit of a B.G.K. like model on domains with boundaries and analysis of kinetic boundary conditions for scalar multidimensional conservation laws. J. Stat. Phys., Vol. 115. Nos. 5/6 (2004), pp. 1715-1754. MR 2066297
27.: Tidriri M., Hydrodynamic limit of a B.G.K. kinetic model and kinetic formulation of conservation laws on domains with boundaries. To appear in Archive for Rational Mechanics and Analysis.

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 35L65, 82B40

M. Tidriri
Affiliation: Department of Mathematics, Iowa State University, Ames, Iowa 50011-2064
Email: tidriri@iastate.edu

DOI: 10.1090/S0002-9947-05-03830-4
PII: S 0002-9947(05)03830-4
Received by editor(s): January 17, 2003
Posted: January 21, 2005
Additional Notes: The author was partially supported by the Air Force Office of Scientific Research under Grant F49620-99-1-0197.
Copyright of article: Copyright 2005, American Mathematical Society

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