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: evrad07@gmail.com

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ABSTRACT

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

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RESUMEN

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

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Notas

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

²Corresponding author

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Received: July 2009.

Revised: August 2009.