The Maxwell problem and the Chapman projection

• V. V Palin ^[1] ; E. V Radkevich ^[1]
  1. [1] Moscow State University

     Moscow State University

     Rusia

     
• Localización: Cubo: A Mathematical Journal, ISSN 0716-7776, ISSN-e 0719-0646, Vol. 12, Nº. 2, 2010, págs. 275-298
• Idioma: inglés
• DOI: 10.4067/S0719-06462010000200017
• Enlaces
  • Texto completo
• Resumen
  • español

    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 pequeños 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.

  • English

    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.

• Referencias bibliográficas
  • Chen, G. Q,Levermore, C. D,Lui, T.-P. (1994). Hyperbolic conservation laws with stiff relaxation terms and entropy. Commun. Pure Appl. Math....
  • Boltzmann, L. (1894). Rep. Brit. Assoc. 579.
  • Zhura, N. A. Hyperbolic first order systems and quantum mechanics [in Russian.
  • Bardos, C,Levermore, C. D. (1993). Fluid dynamic of kinetic equation II: convergence proofs for the Boltzmann equation. Comm. Pure Appl. Math....
  • Bardos, C,Golse, F,Levermore, C. D. (1991). Fluid dynamics limits of discrete velocity kinetic equations: In: Advances in Kinetic Theory and...
  • Caffish, R. E,Papanicolaou, G. C. (1979). The fluid dynamical limit of nonlinear model Boltzmann equations. Comm. Pure Appl. Math. 32. 103-130
  • Chen, G. Q,Frid, H. (1999). Divergence-measure fields and hyperbolic conservation laws. Arch. Ration. Mech. Anal. 147. 89-118
  • Chapman, S. (1922). On Certain Integrals Occurring in the Kinetic Theory of Gases: Manchester Mem. 66.
  • Chapman, S. C.,Cowling, T. C. (1970). The Mathematical Theory of Non-Uniform Gases. Cambridge Univ. Press. Cambridge.
  • Grad, H. (1949). On the kinetic theory of rarefied gases. Commun. Pure Appl. Math. 2. 331-406
  • Radkevich, E. V. (2007). In: Instability in Models Connected with Fluid Flows. Springer. New York.
  • Radkevich, E. V.. (2005). Kinetic equations and the Chapman projection problem [in Russian: Tr. Mat. Inst. Steklova 250, 219-225 (2005); English...
  • Radkevich, E. V. (2007). Mathematical Aspects of Nonequilibrium Processes [in Russian. Tamara Rozhkovskaya Publisher. Novosibirsk.
  • Palin, V. V. (2008). On the solvability of quadratic matrix equations [in Russian] Vestn. MGU, Ser. 1. 36-42
  • Palin, V. V. (2008). On the solvability of the Riccati matrix equations [in Russian. Tr. Semin. I. G. Petrovskogo. 27. 281-298
  • Palin, V. V. (2008). Dynamics separation in conservation laws with relaxation [in Russian. Vestn. SamGU. 6. 407-427
  • Palin, V. V,Radkevich, E. V. (2008). Hyperbolic regularizations of conservation laws. Russian J. Math. Phys. 15. 343-363
  • Palin, V. V.,Radkevich, E. V. (2009). On the Maxwell problem. Springer. New York.
  • Radkevich, E. V,Fursikov, A,Galdi, G.P,Pukhnachov, V. (2009). Problems with insufficient informationabout initial-boundary data, Advances...
  • Hsiao, L,Liu, T.-P. (1992). Convergence to nonlinear diffusion waves for solutions of a system of hyperbolic conservation laws with damping....
  • Dreyer, W,Struchtrup, H. (1993). Heat pulse experiments revisted. Continuum Mech. Thermodyn. 5. 3-50
  • Palin, V. V. (2009). Dynamics separation in conservation laws with relaxation [in Russian. Vestn. MGU, Ser. 1.