# American Institute of Mathematical Sciences

December  2013, 6(4): 865-882. doi: 10.3934/krm.2013.6.865

## Nonlinear stability of Broadwell model with Maxwell diffuse boundary condition

 1 Department of Mathematics, Shanghai Jiao Tong University, Shanghai 200240, China 2 Department of Mathematics, National University of Singapore, 10 Lower Kent Ridge Road, 119076, Singapore, Singapore

Received  August 2013 Revised  September 2013 Published  November 2013

This is a continuation of the paper [5] on the Broadwell model with conservative boundary condition. In this paper, based on the full boundary data obtained by LY algorithm and recombination lemma, we construct the Green's function for the initial boundary value problem. We also establish the pointwise convergence estimate of the solution and nonlinear stability of the global Maxwellian under sufficiently small initial perturbation.
Citation: Shijin Deng, Linglong Du, Shih-Hsien Yu. Nonlinear stability of Broadwell model with Maxwell diffuse boundary condition. Kinetic & Related Models, 2013, 6 (4) : 865-882. doi: 10.3934/krm.2013.6.865
##### References:
 [1] R.-E. Caflisch and T.-P. Liu, Stability of shock waves for the Broadwell equations, Comm. Math. Phys., 114 (1988), 103-130. doi: 10.1007/BF01218291.  Google Scholar [2] R.-E. Caflisch, Navier-Stokes and Boltzmann shock profiles for a model of gas dynamics, Comm. Pure. Appl. Math, 32 (1979), 521-554. doi: 10.1002/cpa.3160320404.  Google Scholar [3] S.-J. Deng, W.-K. Wang and S.-H. Yu, Pointwise convergence to a Maxwellian for a Broadwell model with a supersonic boundary, Netw. Heterog. Media, 2 (2007), 383-395. doi: 10.3934/nhm.2007.2.383.  Google Scholar [4] S.-J. Deng, W.-K. Wang and S.-H. Yu, Broadwell model and conservative supersonic boundary, Arch. Ration. Mech. Anal., 200 (2011), 203-223. doi: 10.1007/s00205-010-0344-4.  Google Scholar [5] S.-J. Deng, W.-K. Wang and S.-H. Yu, Bifurcation on Boundary Data for Linear Broadwell Model with Conservative Boundary Condition,, to appear., ().   Google Scholar [6] 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.  Google Scholar [7] H.-E. Lin, Nonlinear stability of the initial-boundary value problem for the Broadwell model around a Maxwellian, J. Hyperbolic Differ. Equ., 8 (2011), 131-157. doi: 10.1142/S0219891611002354.  Google Scholar [8] J.-G. Liu and Z.-P. Xin, Boundary-layer behavior in the fluid-dynamic limit for a nonlinear model Boltzmann equation, Arch. Rational Mech. Anal., 135 (1996), 61-105. doi: 10.1007/BF02198435.  Google Scholar [9] 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.  Google Scholar [10] T.-P. Liu and S.-H. Yu, Boltzmann equation: micro-macro decompostions and positivity of shock profiles, Comm. Math. Phys., 246 (2004), 133-179. doi: 10.1007/s00220-003-1030-2.  Google Scholar [11] 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.  Google Scholar [12] 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.  Google Scholar [13] T.-P. Liu and S.-H. Yu, Dirichlet-Neumann kernel for dissipative system in half-space, Bull. Inst. Math. Acad. Sin. (N.S.), 7 (2012), 477-543.  Google Scholar [14] M. Slemrod and A.-E. Tzavaras, Self-similiar fluid-dynamic limits for the Broadwell system, Arch. Ration. Mech. Anal., 122 (1993), 353-392. doi: 10.1007/BF00375140.  Google Scholar [15] M. Slemrod, Large time behavior of the Broadwell model of a discrete velocity gas with specularly reflective boundary conditions, Arch. Ration. Mech. Anal., 111 (1990), 323-342. doi: 10.1007/BF00376025.  Google Scholar [16] Y. Sone, Kinetic Theory and Fluid Dynamics, Modeling and Simulation in Science, Engineering and Technology. Birkhäuster Boston, Inc., Boston, MA, 2002. doi: 10.1007/978-1-4612-0061-1.  Google Scholar [17] Y. Sone, Molecular Gas Dynamics: Theory, Techniques, and Applications, Modeling and Simulation in Science, Engineering and Technology. Birkhäuster Boston, Inc., Boston, MA, 2007. doi: 10.1007/978-0-8176-4573-1.  Google Scholar [18] Z.-P. Xin, The fluid-dynamic limit of the Broadwell model of the nonlinear Boltzmann equation in the presence of shocks, Comm. Pure Appl. Math, 44 (1991), 679-713. doi: 10.1002/cpa.3160440604.  Google Scholar

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##### References:
 [1] R.-E. Caflisch and T.-P. Liu, Stability of shock waves for the Broadwell equations, Comm. Math. Phys., 114 (1988), 103-130. doi: 10.1007/BF01218291.  Google Scholar [2] R.-E. Caflisch, Navier-Stokes and Boltzmann shock profiles for a model of gas dynamics, Comm. Pure. Appl. Math, 32 (1979), 521-554. doi: 10.1002/cpa.3160320404.  Google Scholar [3] S.-J. Deng, W.-K. Wang and S.-H. Yu, Pointwise convergence to a Maxwellian for a Broadwell model with a supersonic boundary, Netw. Heterog. Media, 2 (2007), 383-395. doi: 10.3934/nhm.2007.2.383.  Google Scholar [4] S.-J. Deng, W.-K. Wang and S.-H. Yu, Broadwell model and conservative supersonic boundary, Arch. Ration. Mech. Anal., 200 (2011), 203-223. doi: 10.1007/s00205-010-0344-4.  Google Scholar [5] S.-J. Deng, W.-K. Wang and S.-H. Yu, Bifurcation on Boundary Data for Linear Broadwell Model with Conservative Boundary Condition,, to appear., ().   Google Scholar [6] 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.  Google Scholar [7] H.-E. Lin, Nonlinear stability of the initial-boundary value problem for the Broadwell model around a Maxwellian, J. Hyperbolic Differ. Equ., 8 (2011), 131-157. doi: 10.1142/S0219891611002354.  Google Scholar [8] J.-G. Liu and Z.-P. Xin, Boundary-layer behavior in the fluid-dynamic limit for a nonlinear model Boltzmann equation, Arch. Rational Mech. Anal., 135 (1996), 61-105. doi: 10.1007/BF02198435.  Google Scholar [9] 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.  Google Scholar [10] T.-P. Liu and S.-H. Yu, Boltzmann equation: micro-macro decompostions and positivity of shock profiles, Comm. Math. Phys., 246 (2004), 133-179. doi: 10.1007/s00220-003-1030-2.  Google Scholar [11] 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.  Google Scholar [12] 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.  Google Scholar [13] T.-P. Liu and S.-H. Yu, Dirichlet-Neumann kernel for dissipative system in half-space, Bull. Inst. Math. Acad. Sin. (N.S.), 7 (2012), 477-543.  Google Scholar [14] M. Slemrod and A.-E. Tzavaras, Self-similiar fluid-dynamic limits for the Broadwell system, Arch. Ration. Mech. Anal., 122 (1993), 353-392. doi: 10.1007/BF00375140.  Google Scholar [15] M. Slemrod, Large time behavior of the Broadwell model of a discrete velocity gas with specularly reflective boundary conditions, Arch. Ration. Mech. Anal., 111 (1990), 323-342. doi: 10.1007/BF00376025.  Google Scholar [16] Y. Sone, Kinetic Theory and Fluid Dynamics, Modeling and Simulation in Science, Engineering and Technology. Birkhäuster Boston, Inc., Boston, MA, 2002. doi: 10.1007/978-1-4612-0061-1.  Google Scholar [17] Y. Sone, Molecular Gas Dynamics: Theory, Techniques, and Applications, Modeling and Simulation in Science, Engineering and Technology. Birkhäuster Boston, Inc., Boston, MA, 2007. doi: 10.1007/978-0-8176-4573-1.  Google Scholar [18] Z.-P. Xin, The fluid-dynamic limit of the Broadwell model of the nonlinear Boltzmann equation in the presence of shocks, Comm. Pure Appl. Math, 44 (1991), 679-713. doi: 10.1002/cpa.3160440604.  Google Scholar
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