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]

Eduard Feireisl, Elisabetta Rocca, Giulio Schimperna, Arghir Zarnescu. Weak sequential stability for a nonlinear model of nematic electrolytes. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 219-241. doi: 10.3934/dcdss.2020366
[2]

Mohammad Ghani, Jingyu Li, Kaijun Zhang. Asymptotic stability of traveling fronts to a chemotaxis model with nonlinear diffusion. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021017
[3]

Skyler Simmons. Stability of broucke's isosceles orbit. Discrete & Continuous Dynamical Systems - A, 2021  doi: 10.3934/dcds.2021015
[4]

Marion Darbas, Jérémy Heleine, Stephanie Lohrengel. Numerical resolution by the quasi-reversibility method of a data completion problem for Maxwell's equations. Inverse Problems & Imaging, 2020, 14 (6) : 1107-1133. doi: 10.3934/ipi.2020056
[5]

Scipio Cuccagna, Masaya Maeda. A survey on asymptotic stability of ground states of nonlinear Schrödinger equations II. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020450
[6]

Patrick Martinez, Judith Vancostenoble. Lipschitz stability for the growth rate coefficients in a nonlinear Fisher-KPP equation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (2) : 695-721. doi: 10.3934/dcdss.2020362
[7]

Andrew Comech, Scipio Cuccagna. On asymptotic stability of ground states of some systems of nonlinear Schrödinger equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (3) : 1225-1270. doi: 10.3934/dcds.2020316
[8]

Jian-Xin Guo, Xing-Long Qu. Robust control in green production management. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2021011
[9]

Yi-Ming Tai, Zhengyang Zhang. Relaxation oscillations in a spruce-budworm interaction model with Holling's type II functional response. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021027
[10]

Jonathan J. Wylie, Robert M. Miura, Huaxiong Huang. Systems of coupled diffusion equations with degenerate nonlinear source terms: Linear stability and traveling waves. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 561-569. doi: 10.3934/dcds.2009.23.561
[11]

A. M. Elaiw, N. H. AlShamrani, A. Abdel-Aty, H. Dutta. Stability analysis of a general HIV dynamics model with multi-stages of infected cells and two routes of infection. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020441
[12]

Chao Xing, Jiaojiao Pan, Hong Luo. Stability and dynamic transition of a toxin-producing phytoplankton-zooplankton model with additional food. Communications on Pure & Applied Analysis, 2021, 20 (1) : 427-448. doi: 10.3934/cpaa.2020275
[13]

Imam Wijaya, Hirofumi Notsu. Stability estimates and a Lagrange-Galerkin scheme for a Navier-Stokes type model of flow in non-homogeneous porous media. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1197-1212. doi: 10.3934/dcdss.2020234
[14]

Omid Nikan, Seyedeh Mahboubeh Molavi-Arabshai, Hossein Jafari. Numerical simulation of the nonlinear fractional regularized long-wave model arising in ion acoustic plasma waves. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020466
[15]

Xin Zhao, Tao Feng, Liang Wang, Zhipeng Qiu. Threshold dynamics and sensitivity analysis of a stochastic semi-Markov switched SIRS epidemic model with nonlinear incidence and vaccination. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2021010
[16]

Xing-Bin Pan. Variational and operator methods for Maxwell-Stokes system. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3909-3955. doi: 10.3934/dcds.2020036
[17]

Yifan Chen, Thomas Y. Hou. Function approximation via the subsampled Poincaré inequality. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 169-199. doi: 10.3934/dcds.2020296
[18]

Shuai Huang, Zhi-Ping Fan, Xiaohuan Wang. Optimal financing and operational decisions of capital-constrained manufacturer under green credit and subsidy. Journal of Industrial & Management Optimization, 2021, 17 (1) : 261-277. doi: 10.3934/jimo.2019110
[19]

Pierre-Etienne Druet. A theory of generalised solutions for ideal gas mixtures with Maxwell-Stefan diffusion. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020458
[20]

Roland Schnaubelt, Martin Spitz. Local wellposedness of quasilinear Maxwell equations with absorbing boundary conditions. Evolution Equations & Control Theory, 2021, 10 (1) : 155-198. doi: 10.3934/eect.2020061

2019 Impact Factor: 1.311

Tools

Metrics

  • PDF downloads (45)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences