SciELO - Scientific Electronic Library Online

vol.12 número2Real and stable ranks for certain crossed products of Toeplitz algebrasOperator Homology and Cohomology in Clifford Algebras índice de autoresíndice de assuntospesquisa de artigos
Home Pagelista alfabética de periódicos  

Serviços Personalizados




Links relacionados


Cubo (Temuco)

versão On-line ISSN 0719-0646

Cubo vol.12 no.2 Temuco  2010 

CUBO A Mathematical Journal Vol.12, N° 02, (275–298). June 2010


The Maxwell problem and the Chapman projection1


V. V. Palin, E. V. Radkevich 2

Department of Mech.-Math., Moscow State University, Moscow 119899, Vorobievy Gory, Russia. email:


We study the large-time behavior of global smooth solutions to the Cauchy problem for hyperbolic regularization of conservation laws. An attracting manifold of special smooth global solutions is determined by the Chapman projection onto the phase space of consolidated variables. For small initial data we construct the Chapman projection and describe its properties in the case of the Cauchy problem for moment approximations of kinetic equations. The existence conditions for the Chapman projection are expressed in terms of the solvability of the Riccati matrix equations with parameter.

Key words and phrases: closure, the state equation, the Chapman projection, matrix equation, dynamic separation, inertional manifold


Nosotros estudiamos el comportamiento temporal de soluciones globales suaves del problema de Cauchy para regularización hiperbólica de leyes de conservación. Una variedad atractora de soluciones globales suaves es determinada por la proyección de Chapman sobre el espacio de fase de las variables consolidadas. Para datos iniciales peque˜nos nosotros construimos la proyección de Chapman y descubrimos sus propiedades en el caso del problema de Cauchy para aproximación de momentos en ecuaciones kineticas. Las condiciones de existencia para la proyección de Chapman son expresadas en términos de la solubilidad de las ecuaciones matriciales de Riccati con parámetros.

AMS (MOS) Subj. Class.: UDC 517.9


1This work was supported by the Russian Foundation of Basic Researches (grant no. 09-01-00288)

2Corresponding author


[1] G. Q. Chen, C. D. Levermore, and T.-P. Lui, Hyperbolic conservation laws with stiff relaxation terms and entropy, Commun. Pure Appl. Math. 47, No. 6, 787-830 (1994).         [ Links ]

[2] L. Boltzmann, Rep. Brit. Assoc. 579 (1894).         [ Links ]

[3] N. A. Zhura, Hyperbolic first order systems and quantum mechanics [in Russian], Mat. Zametki [Submitted]        [ Links ]

[4] C. Bardos and C. D. Levermore, Fluid dynamic of kinetic equation II: convergence proofs for the Boltzmann equation, Comm. Pure Appl. Math. 46, 667-753 (1993).         [ Links ]

[5] C. Bardos, F. Golse, and C. D. Levermore, Fluid dynamics limits of discrete velocity kinetic equations, In: Advances in Kinetic Theory and Continuum Mechanics Springer- Verlag, Berlin-New-York (1991), pp. 57-71.         [ Links ]

[6] R. E. Caffish and G. C. Papanicolaou, The fluid dynamical limit of nonlinear model Boltzmann equations, Comm. Pure Appl. Math. 32, 103-130 (1979).         [ Links ]

[7] G. Q. Chen and H. Frid, Divergence-measure fields and hyperbolic conservation laws, Arch. Ration. Mech. Anal. 147, 89-118 (1999).         [ Links ]

[8] S. Chapman, On Certain Integrals Occurring in the Kinetic Theory of Gases, Manchester Mem. 66 (1922).         [ Links ]

[9] S. C. Chapman and T. C. Cowling, The Mathematical Theory of Non-Uniform Gases, Cambridge Univ. Press, Cambridge (1970).         [ Links ]

[10] Grad H., On the kinetic theory of rarefied gases, Commun. Pure Appl. Math. v. 2 (1949) no. 4, 331-406.         [ Links ]

[11] E. V. Radkevich, Irreducible Chapman projections and Navier-Stokes approximations, In: Instability in Models Connected with Fluid Flows. II Springer, New York (2007), pp. 85-151.         [ Links ]

[12] E. V. Radkevich, Kinetic equations and the Chapman projection problem [in Russian], Tr. Mat. Inst. Steklova 250, 219-225 (2005); English transl.: Proc. Steklov Inst. Math. 250, 204-210 (2005).         [ Links ]

[13] E. V. Radkevich, Mathematical Aspects of Nonequilibrium Processes [in Russian], Tamara Rozhkovskaya Publisher, Novosibirsk (2007).         [ Links ]

[14] V. V. Palin, On the solvability of quadratic matrix equations [in Russian] Vestn. MGU, Ser. 1, No. 6, 36-42 (2008).         [ Links ]

[15] V. V. Palin, On the solvability of the Riccati matrix equations [in Russian], Tr. Semin. I. G. Petrovskogo 27, 281-298 (2008).         [ Links ]

[16] V. V. Palin, Dynamics separation in conservation laws with relaxation [in Russian], Vestn. SamGU, No. 6 (65), 407-427 (2008).         [ Links ]

[17] V. V. Palin and E. V. Radkevich, Hyperbolic regularizations of conservation laws, Russian J. Math. Phys. 15, No. 3, 343-363 (2008). Submitted date: January 9, 2009 857        [ Links ]

[18] V. V. Palin and E. V. Radkevich, On the Maxwell problem, Journal of Mathematical Sciences, Springer New York Vol. 157, No. 6, 2009. The date for the Table of Contents is March 28, 2009.         [ Links ]

[19] E. V. Radkevich, Problems with insufficient informationabout initial-boundary data, Advances in Mathematical Fluid Mechanics (AMFM), Special AMFM Volume in Honour of Professor Kazhikhov Volume editor(s): A. Fursikov, G.P.Galdi, V. Pukhnachov, Birkhauser Verlag(2009), to appear        [ Links ]

[20] L. Hsiao and T.-P. Liu, Convergence to nonlinear diffusion waves for solutions of a system of hyperbolic conservation laws with damping, Commun. Math. Phys. 143(1992), pp. 599-605        [ Links ]

[21] W. Dreyer and H. Struchtrup, Heat pulse experiments revisted, Continuum Mech. Thermodyn. 5 (1993), pp. 3-50        [ Links ]

[22] V. V. Palin, Dynamics separation in conservation laws with relaxation [in Russian], Vestn. MGU, Ser. 1(2009), to appear.         [ Links ]

Received: July 2009.

Revised: August 2009.

Creative Commons License Todo o conteúdo deste periódico, exceto onde está identificado, está licenciado sob uma Licença Creative Commons