American Institute of Mathematical Sciences

Advanced
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 and 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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[1]

Zhigang Wu, Wenjun Wang. Uniform stability of the Boltzmann equation with an external force near vacuum. Communications on Pure and Applied Analysis, 2015, 14 (3) : 811-823. doi: 10.3934/cpaa.2015.14.811
[2]

Seung-Yeal Ha, Eunhee Jeong, Robert M. Strain. Uniform $L^1$-stability of the relativistic Boltzmann equation near vacuum. Communications on Pure and Applied Analysis, 2013, 12 (2) : 1141-1161. doi: 10.3934/cpaa.2013.12.1141
[3]

Seung-Yeal Ha, Ho Lee, Seok Bae Yun. Uniform $L^p$-stability theory for the space-inhomogeneous Boltzmann equation with external forces. Discrete and Continuous Dynamical Systems, 2009, 24 (1) : 115-143. doi: 10.3934/dcds.2009.24.115
[4]

Raffaele Esposito, Yan Guo, Rossana Marra. Validity of the Boltzmann equation with an external force. Kinetic and Related Models, 2011, 4 (2) : 499-515. doi: 10.3934/krm.2011.4.499
[5]

Hongjun Yu. Global classical solutions to the Boltzmann equation with external force. Communications on Pure and Applied Analysis, 2009, 8 (5) : 1647-1668. doi: 10.3934/cpaa.2009.8.1647
[6]

Renjun Duan, Tong Yang, Changjiang Zhu. Boltzmann equation with external force and Vlasov-Poisson-Boltzmann system in infinite vacuum. Discrete and Continuous Dynamical Systems, 2006, 16 (1) : 253-277. doi: 10.3934/dcds.2006.16.253
[7]

Fei Meng, Fang Liu. On the inelastic Boltzmann equation for soft potentials with diffusion. Communications on Pure and Applied Analysis, 2020, 19 (11) : 5197-5217. doi: 10.3934/cpaa.2020233
[8]

Ralf Kirsch, Sergej Rjasanow. The uniformly heated inelastic Boltzmann equation in Fourier space. Kinetic and Related Models, 2010, 3 (3) : 445-456. doi: 10.3934/krm.2010.3.445
[9]

Rafael Sanabria. Inelastic Boltzmann equation driven by a particle thermal bath. Kinetic and Related Models, 2021, 14 (4) : 639-679. doi: 10.3934/krm.2021018
[10]

Anton Trushechkin. Microscopic and soliton-like solutions of the Boltzmann--Enskog and generalized Enskog equations for elastic and inelastic hard spheres. Kinetic and Related Models, 2014, 7 (4) : 755-778. doi: 10.3934/krm.2014.7.755
[11]

Daniel Han-Kwan. $L^1$ averaging lemma for transport equations with Lipschitz force fields. Kinetic and Related Models, 2010, 3 (4) : 669-683. doi: 10.3934/krm.2010.3.669
[12]

Marc Briant. Stability of global equilibrium for the multi-species Boltzmann equation in $L^\infty$ settings. Discrete and Continuous Dynamical Systems, 2016, 36 (12) : 6669-6688. doi: 10.3934/dcds.2016090
[13]

Yingzhe Fan, Yuanjie Lei. The Boltzmann equation with frictional force for very soft potentials in the whole space. Discrete and Continuous Dynamical Systems, 2019, 39 (7) : 4303-4329. doi: 10.3934/dcds.2019174
[14]

Jingwei Hu, Jie Shen, Yingwei Wang. A Petrov-Galerkin spectral method for the inelastic Boltzmann equation using mapped Chebyshev functions. Kinetic and Related Models, 2020, 13 (4) : 677-702. doi: 10.3934/krm.2020023
[15]

Kevin Zumbrun. L∞ resolvent bounds for steady Boltzmann's Equation. Kinetic and Related Models, 2017, 10 (4) : 1255-1257. doi: 10.3934/krm.2017048
[16]

Yong-Kum Cho. On the Boltzmann equation with the symmetric stable Lévy process. Kinetic and Related Models, 2015, 8 (1) : 53-77. doi: 10.3934/krm.2015.8.53
[17]

El Miloud Zaoui, Marc Laforest. Stability and modeling error for the Boltzmann equation. Kinetic and Related Models, 2014, 7 (2) : 401-414. doi: 10.3934/krm.2014.7.401
[18]

Hao Wang. Uniform stability estimate for the Vlasov-Poisson-Boltzmann system. Discrete and Continuous Dynamical Systems, 2021, 41 (2) : 657-680. doi: 10.3934/dcds.2020292
[19]

Nicolas Fournier. A recursive algorithm and a series expansion related to the homogeneous Boltzmann equation for hard potentials with angular cutoff. Kinetic and Related Models, 2019, 12 (3) : 483-505. doi: 10.3934/krm.2019020
[20]

Lvqiao Liu, Hao Wang. Global existence and decay of solutions for hard potentials to the fokker-planck-boltzmann equation without cut-off. Communications on Pure and Applied Analysis, 2020, 19 (6) : 3113-3136. doi: 10.3934/cpaa.2020135

2020 Impact Factor: 2.425

Tools

Metrics

  • PDF downloads (250)
  • HTML views (717)
  • Cited by (8)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy