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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

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 & 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.  doi: 10.3934/nhm.2007.2.383.  Google Scholar

[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.  doi: 10.3934/nhm.2006.1.167.  Google Scholar

[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.  doi: 10.1142/S0219891608001489.  Google Scholar

[4]

T.-P. Liu, Pointwise convergence to shock waves for viscous conservarion laws,, Commun. Pure Appl. Math., 50 (1997), 1113.  doi: 10.1002/(SICI)1097-0312(199711)50:11<1113::AID-CPA3>3.0.CO;2-D.  Google Scholar

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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.  doi: 10.1002/cpa.20172.  Google Scholar

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T.-P. Liu and S.-H. Yu, On boundary relation for some dissipative systems,, Bull. Inst. Math. Acad. Sin. (N.S.), 6 (2011), 245.   Google Scholar

[7]

Y. Sone, Kinetic Theory and Fluid Dynamics,, Modeling and Simulation in Science, (2002).  doi: 10.1007/978-1-4612-0061-1.  Google Scholar

show all references

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.  doi: 10.3934/nhm.2007.2.383.  Google Scholar

[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.  doi: 10.3934/nhm.2006.1.167.  Google Scholar

[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.  doi: 10.1142/S0219891608001489.  Google Scholar

[4]

T.-P. Liu, Pointwise convergence to shock waves for viscous conservarion laws,, Commun. Pure Appl. Math., 50 (1997), 1113.  doi: 10.1002/(SICI)1097-0312(199711)50:11<1113::AID-CPA3>3.0.CO;2-D.  Google Scholar

[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.  doi: 10.1002/cpa.20172.  Google Scholar

[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.   Google Scholar

[7]

Y. Sone, Kinetic Theory and Fluid Dynamics,, Modeling and Simulation in Science, (2002).  doi: 10.1007/978-1-4612-0061-1.  Google Scholar

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