Characteristic half space problem for the Broadwell model

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March 2014, 9(1): 97-110. doi: 10.3934/nhm.2014.9.97

Characteristic half space problem for the Broadwell model

Linglong Du ^1,

1.	Department of Mathematics, National University of Singapore, Block S17, 10 Lower Kent Ridge Road, 119076, Singapore

Received July 2013 Revised August 2013 Published April 2014

Abstract
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We study an initial boundary value problem for the Broadwell model in half space. The Green's function for the initial boundary value problem is decomposed into two parts: one is the Green's function for the initial value problem, we call it the fundamental solution for the whole space; the other is the convolution of this fundamental solution with full boundary data. A new approach to obtain the full boundary data is established here. Finally, a nonlinear time-asymptotic stability of an equilibrium state is proved.

Keywords: Broadwell model, Incoming-outgoing map., Green's function, fundamental pair, characteristic boundary.

Mathematics Subject Classification: Primary: 82C4.

Citation: Linglong Du. Characteristic half space problem for the Broadwell model. Networks and Heterogeneous Media, 2014, 9 (1) : 97-110. doi: 10.3934/nhm.2014.9.97

References:

[1]	S.-J. Deng, W.-K. Wang and S.-H. Yu, Pointwise convergence to a Maxwellian for a Broadwellw model with a supersonic boundary, Netw. Heterog. Media, 2 (2007), 383-395. doi: 10.3934/nhm.2007.2.383.
[2]	C.-Y. Lan, H.-E. Lin and S.-H. Yu, The Green's functions for the Broadwell model in a half space problem, Netw. Heterog. Media, 1 (2006), 167-183. doi: 10.3934/nhm.2006.1.167.
[3]	C.-Y. Lan, H.-E. Lin and S.-H. Yu, The Green's functions for the Broadwell model with a transonic boundary, J. Hyperbolic Differ. Equ., 5 (2008), 279-294. doi: 10.1142/S0219891608001489.
[4]	T.-P. Liu, Pointwise convergence to shock waves for viscous conservarion laws, Commun. Pure Appl. Math., 50 (1997), 1113-1182. doi: 10.1002/(SICI)1097-0312(199711)50:11<1113::AID-CPA3>3.0.CO;2-D.
[5]	T.-P. Liu and S.-H. Yu, Initial-boundary value problem for one-dimensional wave solutions of the Boltzmann equation, Commun. Pure Appl. Math., 60 (2007), 295-356. doi: 10.1002/cpa.20172.
[6]	T.-P. Liu and S.-H. Yu, On boundary relation for some dissipative systems, Bull. Inst. Math. Acad. Sin. (N.S.), 6 (2011), 245-267.
[7]	Y. Sone, Kinetic Theory and Fluid Dynamics, Modeling and Simulation in Science, Engineering and Technology, Birkhäuser Boston, Inc., Boston, MA, 2002. doi: 10.1007/978-1-4612-0061-1.

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References:

[1]	S.-J. Deng, W.-K. Wang and S.-H. Yu, Pointwise convergence to a Maxwellian for a Broadwellw model with a supersonic boundary, Netw. Heterog. Media, 2 (2007), 383-395. doi: 10.3934/nhm.2007.2.383.
[2]	C.-Y. Lan, H.-E. Lin and S.-H. Yu, The Green's functions for the Broadwell model in a half space problem, Netw. Heterog. Media, 1 (2006), 167-183. doi: 10.3934/nhm.2006.1.167.
[3]	C.-Y. Lan, H.-E. Lin and S.-H. Yu, The Green's functions for the Broadwell model with a transonic boundary, J. Hyperbolic Differ. Equ., 5 (2008), 279-294. doi: 10.1142/S0219891608001489.
[4]	T.-P. Liu, Pointwise convergence to shock waves for viscous conservarion laws, Commun. Pure Appl. Math., 50 (1997), 1113-1182. doi: 10.1002/(SICI)1097-0312(199711)50:11<1113::AID-CPA3>3.0.CO;2-D.
[5]	T.-P. Liu and S.-H. Yu, Initial-boundary value problem for one-dimensional wave solutions of the Boltzmann equation, Commun. Pure Appl. Math., 60 (2007), 295-356. doi: 10.1002/cpa.20172.
[6]	T.-P. Liu and S.-H. Yu, On boundary relation for some dissipative systems, Bull. Inst. Math. Acad. Sin. (N.S.), 6 (2011), 245-267.
[7]	Y. Sone, Kinetic Theory and Fluid Dynamics, Modeling and Simulation in Science, Engineering and Technology, Birkhäuser Boston, Inc., Boston, MA, 2002. doi: 10.1007/978-1-4612-0061-1.

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