American Institute of Mathematical Sciences

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August & September  2019, 12(4&5): 1005-1013. doi: 10.3934/dcdss.2019068

Uniform $L^1$ stability of the inelastic Boltzmann equation with large external force for hard potentials

Shaofei Wu 1,, , Mingqing Wang 2, , Maozhu Jin 3, , Yuntao Zou 4, and  Lijun Song 5,

1. 

Hubei Province Key Laboratory of Intelligent Robots, School of Computer Science and Engineering, Wuhan Institute of Technology, Wuhan, China
2. 

Wuhan inCarCloud Technologies Pte.Ltd. China
3. 

Business School, Sichuan University, Chengdu, China
4. 

Wuhan Winphone Technology Co., Ltd, China
5. 

Chengdu University of Information Technology, Chengdu 610225, China

* Corresponding author: Shaofei Wu

Received  June 2017 Revised  November 2017 Published  November 2018

In this paper, we will study the uniform $L^1$ stability of the inelastic Boltzmann equation. More precisely, according to the existence result on the inelastic Boltzmann equation with external force near vacuum, we obtain the uniform $L^1$ stability estimates of mild solution for the hard potentials under the assumptions on the characteristic generated by force term which can be arbitrarily large. The proof is based on the exponentially decay estimate and Lu's trick in [10].

Keywords: Inelastic Boltzmann equation, hard potential, external force, uniform $L^1$ stability.
Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.
Citation: Shaofei Wu, Mingqing Wang, Maozhu Jin, Yuntao Zou, Lijun Song. Uniform $L^1$ stability of the inelastic Boltzmann equation with large external force for hard potentials. Discrete & Continuous Dynamical Systems - S, 2019, 12 (4&5) : 1005-1013. doi: 10.3934/dcdss.2019068
References:
[1]

R. J. Alonso, Existence of global solutions to the Cauchy problem for the inelastic Boltzmann equation with near-vacuum data, Indiana Univ. Math. J., 58 (2009), 999-1022.  doi: 10.1512/iumj.2009.58.3506.  Google Scholar

[2]

L. Arkeryd, Stability in $L^1$ for the spatially homogeneous Boltzmann equation, Arch. Ration. Mech. Anal., 103 (1988), 151-167.  doi: 10.1007/BF00251506.  Google Scholar

[3]

C. H. Cheng, Uniform stability of solutions of Boltzmann equation for soft potential with external force, J. Math. Anal. Appl., 352 (2009), 724-732.  doi: 10.1016/j.jmaa.2008.11.027.  Google Scholar

[4]

R. J. DiPerna and P. L. Lions, On the Cauchy problem for Boltzmann equations, Ann. Math., 130 (1989), 321-366.  doi: 10.2307/1971423.  Google Scholar

[5]

R. J. Duan, T. Yang and C. J. Zhu, Global existence to Boltzmann equation with external force in infinite vacuum J. Math. Phys. 46 (2005), 053307, 13pp. doi: 10.1063/1.1899985.  Google Scholar

[6]

R. J. Duan, T. Yang and C. J. Zhu, $L^1$ and BV-type stability of the Boltzmann equation with external forces, J. Differential Equations, 227 (2006), 1-28.  doi: 10.1016/j.jde.2006.01.010.  Google Scholar

[7]

S. Y. Ha, $L^1$ stability of the Boltzmann equation for the hard sphere model, Arch. Ration. Mech. Anal., 171 (2004), 279-296.  doi: 10.1007/s00205-004-0321-x.  Google Scholar

[8]

S. Y. Ha, Nonlinear functionals of the Boltzmann equation and uniform stability estimates, J. Differential Equations, 215 (2005), 178-205.  doi: 10.1016/j.jde.2004.07.022.  Google Scholar

[9]

S. Y. Ha, $L^1$-stability of the Boltzmann equation for Maxwellian molecules, Nonlinearity, 18 (2005), 981-1001.  doi: 10.1088/0951-7715/18/3/003.  Google Scholar

[10]

X. Lu, Spatial decay solutions of the Boltzmann equation: Converse properties of long time limiting behavior, SIAM J. Math. Anal., 30 (1999), 1151-1174.  doi: 10.1137/S0036141098334985.  Google Scholar

[11]

J. B. Wei and X. W. Zhang, On the Cauchy problem for the inelastic Boltzmann equation with external force, J. Stat. Phys., 146 (2012), 592-609.  doi: 10.1007/s10955-011-0410-9.  Google Scholar

[12]

J. B. Wei and X. W. Zhang, Infinite energy solutions of the inelastic Boltzmann equation with external force, Acta Mathematica Scientia, 32 (2012), 2131-2140.  doi: 10.1016/S0252-9602(12)60165-9.  Google Scholar

[13]

J. B. Weiand X. W. Zhang, On the inelastic Enskog equation with external force J. Math. Phy. 53 (2012), 103505, 12pp. doi: 10.1063/1.4753988.  Google Scholar

[14]

B. Wennberg, Stability and exponential convergence in $L^p$ for the spatially homogeneous Boltzmann equation, Nonlinear Anal. Theory, Methods Appl., 20 (1993), 935-964.  doi: 10.1016/0362-546X(93)90086-8.  Google Scholar

[15]

Z. G. Wu, $L^1$ and BV-type stability of the inelastic Boltzmann equation near vacuum, Continuum Mech. Thermodyn, 22 (2010), 239-249.  doi: 10.1007/s00161-009-0127-z.  Google Scholar

[16]

S. B. Yun, $L^p$ stability estimate of the Boltzmann equation around a traveling local Maxwellian, J. Differential Equations, 251 (2011), 45-57.  doi: 10.1016/j.jde.2011.03.001.  Google Scholar

show all references

References:
[1]

R. J. Alonso, Existence of global solutions to the Cauchy problem for the inelastic Boltzmann equation with near-vacuum data, Indiana Univ. Math. J., 58 (2009), 999-1022.  doi: 10.1512/iumj.2009.58.3506.  Google Scholar

[2]

L. Arkeryd, Stability in $L^1$ for the spatially homogeneous Boltzmann equation, Arch. Ration. Mech. Anal., 103 (1988), 151-167.  doi: 10.1007/BF00251506.  Google Scholar

[3]

C. H. Cheng, Uniform stability of solutions of Boltzmann equation for soft potential with external force, J. Math. Anal. Appl., 352 (2009), 724-732.  doi: 10.1016/j.jmaa.2008.11.027.  Google Scholar

[4]

R. J. DiPerna and P. L. Lions, On the Cauchy problem for Boltzmann equations, Ann. Math., 130 (1989), 321-366.  doi: 10.2307/1971423.  Google Scholar

[5]

R. J. Duan, T. Yang and C. J. Zhu, Global existence to Boltzmann equation with external force in infinite vacuum J. Math. Phys. 46 (2005), 053307, 13pp. doi: 10.1063/1.1899985.  Google Scholar

[6]

R. J. Duan, T. Yang and C. J. Zhu, $L^1$ and BV-type stability of the Boltzmann equation with external forces, J. Differential Equations, 227 (2006), 1-28.  doi: 10.1016/j.jde.2006.01.010.  Google Scholar

[7]

S. Y. Ha, $L^1$ stability of the Boltzmann equation for the hard sphere model, Arch. Ration. Mech. Anal., 171 (2004), 279-296.  doi: 10.1007/s00205-004-0321-x.  Google Scholar

[8]

S. Y. Ha, Nonlinear functionals of the Boltzmann equation and uniform stability estimates, J. Differential Equations, 215 (2005), 178-205.  doi: 10.1016/j.jde.2004.07.022.  Google Scholar

[9]

S. Y. Ha, $L^1$-stability of the Boltzmann equation for Maxwellian molecules, Nonlinearity, 18 (2005), 981-1001.  doi: 10.1088/0951-7715/18/3/003.  Google Scholar

[10]

X. Lu, Spatial decay solutions of the Boltzmann equation: Converse properties of long time limiting behavior, SIAM J. Math. Anal., 30 (1999), 1151-1174.  doi: 10.1137/S0036141098334985.  Google Scholar

[11]

J. B. Wei and X. W. Zhang, On the Cauchy problem for the inelastic Boltzmann equation with external force, J. Stat. Phys., 146 (2012), 592-609.  doi: 10.1007/s10955-011-0410-9.  Google Scholar

[12]

J. B. Wei and X. W. Zhang, Infinite energy solutions of the inelastic Boltzmann equation with external force, Acta Mathematica Scientia, 32 (2012), 2131-2140.  doi: 10.1016/S0252-9602(12)60165-9.  Google Scholar

[13]

J. B. Weiand X. W. Zhang, On the inelastic Enskog equation with external force J. Math. Phy. 53 (2012), 103505, 12pp. doi: 10.1063/1.4753988.  Google Scholar

[14]

B. Wennberg, Stability and exponential convergence in $L^p$ for the spatially homogeneous Boltzmann equation, Nonlinear Anal. Theory, Methods Appl., 20 (1993), 935-964.  doi: 10.1016/0362-546X(93)90086-8.  Google Scholar

[15]

Z. G. Wu, $L^1$ and BV-type stability of the inelastic Boltzmann equation near vacuum, Continuum Mech. Thermodyn, 22 (2010), 239-249.  doi: 10.1007/s00161-009-0127-z.  Google Scholar

[16]

S. B. Yun, $L^p$ stability estimate of the Boltzmann equation around a traveling local Maxwellian, J. Differential Equations, 251 (2011), 45-57.  doi: 10.1016/j.jde.2011.03.001.  Google Scholar

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