Transmission boundary conditions in a model-kinetic decomposition

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February 2002, 2(1): 69-94. doi: 10.3934/dcdsb.2002.2.69

Transmission boundary conditions in a model-kinetic decomposition

C. Bourdarias ^1, , M. Gisclon ^1, and A. Omrane ^2,

1.	Laboratoire de Mathématiques, Université de Savoie, 73376 Le Bourget-du-Lac Cedex, France, France
2.	Laboratoire de Mathématiques et informatique, Université des Antilles et de la Guyane, Campus de Fouillole, 97159 Pointe à Pitre, Guadeloupe (French)

Received May 2001 Revised July 2001 Published November 2001

Abstract
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This paper deals with the fluid limit using the Perthame-Tadmor model with initial and boundary conditions of transmission type within two positive parameters $\varepsilon_1$ and $\varepsilon_2$ for the kinetic dynamical problem. We show that the kinetic problem is well posed in $L^\infty \bigcap L^1(0,T;L^1(\mathbb{R}^n \times \mathbb{R}_v))$. We also prove a BV estimate which allows us to pass to the limit in each kinetic region or, under restrictive conditions, in a single region. This result can be applied to scalar conservation laws with decomposition domain.

Keywords: kinetic approach, conditions of transmission, Cauchy problem, decomposition domain., boundary conditions, Multidimensional scalar conservation laws.

Mathematics Subject Classification: 35F30, 35L60, 35L6.

Citation: C. Bourdarias, M. Gisclon, A. Omrane. Transmission boundary conditions in a model-kinetic decomposition. Discrete & Continuous Dynamical Systems - B, 2002, 2 (1) : 69-94. doi: 10.3934/dcdsb.2002.2.69

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