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

Tidriri, M.

doi:10.1090/S0002-9947-05-03830-4

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

Trans. Amer. Math. Soc. 357 (2005), 2133-2164

DOI: https://doi.org/10.1090/S0002-9947-05-03830-4

Published electronically: 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.

References

C. Bardos, F. Golse, B. Perthame, and R. Sentis, The nonaccretive radiative transfer equations: existence of solutions and Rosseland approximation, J. Funct. Anal. 77 (1988), no. 2, 434–460. MR 933978, DOI 10.1016/0022-1236(88)90096-1
C. Bardos, A. Y. le Roux, and J.-C. Nédélec, First order quasilinear equations with boundary conditions, Comm. Partial Differential Equations 4 (1979), no. 9, 1017–1034. MR 542510, DOI 10.1080/03605307908820117
Haïm Brezis, Analyse fonctionnelle, Collection Mathématiques Appliquées pour la Maîtrise. [Collection of Applied Mathematics for the Master’s Degree], Masson, Paris, 1983 (French). Théorie et applications. [Theory and applications]. MR 697382
Chen G.-Q. and Frid H., On the theory of divergence-measure fields and its applications, Boletin da Sociedade Brasileira de Matematica, to appear.
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.
Robert Dautray and Jacques-Louis Lions, Analyse mathématique et calcul numérique pour les sciences et les techniques. Vol. 7, INSTN: Collection Enseignement. [INSTN: Teaching Collection], Masson, Paris, 1988 (French). Evolution: Fourier, Laplace; Reprint of the 1985 edition. MR 1016604
R. J. DiPerna, P.-L. Lions, and Y. Meyer, $L^p$ regularity of velocity averages, Ann. Inst. H. Poincaré C Anal. Non Linéaire 8 (1991), no. 3-4, 271–287 (English, with French summary). MR 1127927, DOI 10.1016/S0294-1449(16)30264-5
Lawrence C. Evans and Ronald F. Gariepy, Measure theory and fine properties of functions, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1992. MR 1158660
François Golse, Pierre-Louis Lions, Benoît Perthame, and Rémi Sentis, Regularity of the moments of the solution of a transport equation, J. Funct. Anal. 76 (1988), no. 1, 110–125. MR 923047, DOI 10.1016/0022-1236(88)90051-1
S. N. Kružkov, First order quasilinear equations with several independent variables. , Mat. Sb. (N.S.) 81 (123) (1970), 228–255 (Russian). MR 0267257
Pierre-Louis Lions, Benoît Perthame, and Panagiotis E. Souganidis, Existence and stability of entropy solutions for the hyperbolic systems of isentropic gas dynamics in Eulerian and Lagrangian coordinates, Comm. Pure Appl. Math. 49 (1996), no. 6, 599–638. MR 1383202, DOI 10.1002/(SICI)1097-0312(199606)49:6<599::AID-CPA2>3.0.CO;2-5
P.-L. Lions, B. Perthame, and E. Tadmor, A kinetic formulation of multidimensional scalar conservation laws and related equations, J. Amer. Math. Soc. 7 (1994), no. 1, 169–191. MR 1201239, DOI 10.1090/S0894-0347-1994-1201239-3
P.-L. Lions, B. Perthame, and E. Tadmor, Kinetic formulation of the isentropic gas dynamics and $p$-systems, Comm. Math. Phys. 163 (1994), no. 2, 415–431. MR 1284790, DOI 10.1007/BF02102014
B. Perthame, Uniqueness and error estimates in first order quasilinear conservation laws via the kinetic entropy defect measure, J. Math. Pures Appl. (9) 77 (1998), no. 10, 1055–1064 (English, with English and French summaries). MR 1661021, DOI 10.1016/S0021-7824(99)80003-8
Benoît Perthame, Kinetic formulation of conservation laws, Oxford Lecture Series in Mathematics and its Applications, vol. 21, Oxford University Press, Oxford, 2002. MR 2064166
Benoît Perthame and Eitan Tadmor, A kinetic equation with kinetic entropy functions for scalar conservation laws, Comm. Math. Phys. 136 (1991), no. 3, 501–517. MR 1099693, DOI 10.1007/BF02099071
Laurent Schwartz, Théorie des distributions, Publications de l’Institut de Mathématique de l’Université de Strasbourg, IX-X, Hermann, Paris, 1966 (French). Nouvelle édition, entiérement corrigée, refondue et augmentée. MR 0209834
L. Tartar, Une nouvelle méthode de résolution d’équations aux dérivées partielles non linéaires, Journées d’Analyse Non Linéaire (Proc. Conf., Besançon, 1977) Lecture Notes in Math., vol. 665, Springer, Berlin, 1978, pp. 228–241 (French). MR 519433
L. Tartar, Compensated compactness and applications to partial differential equations, Nonlinear analysis and mechanics: Heriot-Watt Symposium, Vol. IV, Res. Notes in Math., vol. 39, Pitman, Boston, Mass.-London, 1979, pp. 136–212. MR 584398
Moulay D. Tidriri, Asymptotic analysis of a coupled system of kinetic equations, C. R. Acad. Sci. Paris Sér. I Math. 328 (1999), no. 7, 637–642 (English, with English and French summaries). MR 1680029, DOI 10.1016/S0764-4442(99)80261-4
Moulay Tidriri, Numerical analysis of coupling for a kinetic equation, M2AN Math. Model. Numer. Anal. 33 (1999), no. 6, 1121–1134. MR 1736892, DOI 10.1051/m2an:1999137
M. D. Tidriri, A novel class of multiscale models in mathematical physics, Proceedings of the Third World Congress of Nonlinear Analysts, Part 7 (Catania, 2000), 2001, pp. 4995–5008. MR 1975891, DOI 10.1016/S0362-546X(01)00611-3
M. Tidriri, Rigorous derivation and analysis of coupling of kinetic equations and their hydrodynamic limits for a simplified Boltzmann model, J. Statist. Phys. 104 (2001), no. 1-2, 255–290. MR 1851389, DOI 10.1023/A:1010313828663
M. Tidriri, New models for the solution of intermediate regimes in transport theory and radiative transfer: existence theory, positivity, asymptotic analysis, and approximations, J. Statist. Phys. 104 (2001), no. 1-2, 291–325. MR 1851390, DOI 10.1023/A:1010365812733
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.
M. Tidriri, 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. Statist. Phys. 115 (2004), no. 5-6, 1715–1754. MR 2066297, DOI 10.1023/B:JOSS.0000028079.51072.06
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.