American Institute of Mathematical Sciences

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

Nonlinear stability of Broadwell model with Maxwell diffuse boundary condition

Shijin Deng 1, , Linglong Du 2, and  Shih-Hsien Yu 2,

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.
Keywords: Broadwell model, Maxwell diffuse, nonlinear stability., Green's function.
Mathematics Subject Classification: Primary: 82C4.
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.  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.  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.  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.  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.  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.  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.  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.  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.  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.  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.   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.   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.  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.  doi: 10.1007/BF00376025.  Google Scholar

[16]

Y. Sone, Kinetic Theory and Fluid Dynamics,, Modeling and Simulation in Science, (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, (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.  doi: 10.1002/cpa.3160440604.  Google Scholar

show all references

References:
[1]

R.-E. Caflisch and T.-P. Liu, Stability of shock waves for the Broadwell equations,, Comm. Math. Phys., 114 (1988), 103.  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.  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.  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.  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.  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.  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.  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.  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.  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.  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.   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.   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.  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.  doi: 10.1007/BF00376025.  Google Scholar

[16]

Y. Sone, Kinetic Theory and Fluid Dynamics,, Modeling and Simulation in Science, (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, (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.  doi: 10.1002/cpa.3160440604.  Google Scholar

[1]

Chiu-Ya Lan, Huey-Er Lin, Shih-Hsien Yu. The Green's functions for the Broadwell Model in a half space problem. Networks & Heterogeneous Media, 2006, 1 (1) : 167-183. doi: 10.3934/nhm.2006.1.167
[2]

Peter Bella, Arianna Giunti. Green's function for elliptic systems: Moment bounds. Networks & Heterogeneous Media, 2018, 13 (1) : 155-176. doi: 10.3934/nhm.2018007
[3]

Virginia Agostiniani, Rolando Magnanini. Symmetries in an overdetermined problem for the Green's function. Discrete & Continuous Dynamical Systems - S, 2011, 4 (4) : 791-800. doi: 10.3934/dcdss.2011.4.791
[4]

Sungwon Cho. Alternative proof for the existence of Green's function. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1307-1314. doi: 10.3934/cpaa.2011.10.1307
[5]

Hasib Khan, Cemil Tunc, Aziz Khan. Green function's properties and existence theorems for nonlinear singular-delay-fractional differential equations. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 0-0. doi: 10.3934/dcdss.2020139
[6]

W. Wei, H. M. Yin. Global solvability for a singular nonlinear Maxwell's equations. Communications on Pure & Applied Analysis, 2005, 4 (2) : 431-444. doi: 10.3934/cpaa.2005.4.431
[7]

Jeremiah Birrell. A posteriori error bounds for two point boundary value problems: A green's function approach. Journal of Computational Dynamics, 2015, 2 (2) : 143-164. doi: 10.3934/jcd.2015001
[8]

Wen-ming He, Jun-zhi Cui. The estimate of the multi-scale homogenization method for Green's function on Sobolev space $W^{1,q}(\Omega)$. Communications on Pure & Applied Analysis, 2012, 11 (2) : 501-516. doi: 10.3934/cpaa.2012.11.501
[9]

Björn Birnir, Niklas Wellander. Homogenized Maxwell's equations; A model for ceramic varistors. Discrete & Continuous Dynamical Systems - B, 2006, 6 (2) : 257-272. doi: 10.3934/dcdsb.2006.6.257
[10]

Khalid Latrach, Hatem Megdiche. Time asymptotic behaviour for Rotenberg's model with Maxwell boundary conditions. Discrete & Continuous Dynamical Systems - A, 2011, 29 (1) : 305-321. doi: 10.3934/dcds.2011.29.305
[11]

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
[12]

Shijin Deng, Weike Wang, Shih-Hsien Yu. Pointwise convergence to a Maxwellian for a Broadwell model with a supersonic boundary. Networks & Heterogeneous Media, 2007, 2 (3) : 383-395. doi: 10.3934/nhm.2007.2.383
[13]

Kyoungsun Kim, Gen Nakamura, Mourad Sini. The Green function of the interior transmission problem and its applications. Inverse Problems & Imaging, 2012, 6 (3) : 487-521. doi: 10.3934/ipi.2012.6.487
[14]

Jongkeun Choi, Ki-Ahm Lee. The Green function for the Stokes system with measurable coefficients. Communications on Pure & Applied Analysis, 2017, 16 (6) : 1989-2022. doi: 10.3934/cpaa.2017098
[15]

Dirk Pauly. On Maxwell's and Poincaré's constants. Discrete & Continuous Dynamical Systems - S, 2015, 8 (3) : 607-618. doi: 10.3934/dcdss.2015.8.607
[16]

J. J. Morgan, Hong-Ming Yin. On Maxwell's system with a thermal effect. Discrete & Continuous Dynamical Systems - B, 2001, 1 (4) : 485-494. doi: 10.3934/dcdsb.2001.1.485
[17]

Zhi-Min Chen. Straightforward approximation of the translating and pulsating free surface Green function. Discrete & Continuous Dynamical Systems - B, 2014, 19 (9) : 2767-2783. doi: 10.3934/dcdsb.2014.19.2767
[18]

Claudia Bucur. Some observations on the Green function for the ball in the fractional Laplace framework. Communications on Pure & Applied Analysis, 2016, 15 (2) : 657-699. doi: 10.3934/cpaa.2016.15.657
[19]

Eduard Feireisl. On weak solutions to a diffuse interface model of a binary mixture of compressible fluids. Discrete & Continuous Dynamical Systems - S, 2016, 9 (1) : 173-183. doi: 10.3934/dcdss.2016.9.173
[20]

Jiann-Sheng Jiang, Chi-Kun Lin, Chi-Hua Liu. Homogenization of the Maxwell's system for conducting media. Discrete & Continuous Dynamical Systems - B, 2008, 10 (1) : 91-107. doi: 10.3934/dcdsb.2008.10.91

2018 Impact Factor: 1.38

Tools

Metrics

  • PDF downloads (11)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences