American Institute of Mathematical Sciences

Advanced
January  2009, 24(1): 59-81. doi: 10.3934/dcds.2009.24.59

Over-populated tails for conservative-in-the-mean inelastic Maxwell models

José A. Carrillo 1, , Stéphane Cordier 2, and  Giuseppe Toscani 3,

1. 

ICREA-Departament de Matemàtiques, Universitat Autònoma de Barcelona, E-08193 Bellaterra, Spain
2. 

Fédération Denis Poisson (FR 2964), Department of Mathematics (MAPMO UMR 6628), University of Orléans and CNRS, F-45067 Orléans, France
3. 

Dipartimento di Matematica, Università di Pavia, via Ferrata 1, I-27100 Pavia, Italy

Received  May 2007 Revised  January 2008 Published  January 2009

We introduce and discuss spatially homogeneous Maxwell-type models of the nonlinear Boltzmann equation undergoing binary collisions with a random component. The random contribution to collisions is such that the usual collisional invariants of mass, momentum and energy do not hold pointwise, even if they all hold in the mean. Under this assumption it is shown that, while the Boltzmann equation has the usual conserved quantities, it possesses a steady state with power-like tails for certain random variables. A similar situation occurs in kinetic models of economy recently considered by two of the authors [24], which are conservative in the mean but possess a steady distribution with Pareto tails. The convolution-like gain operator is subsequently shown to have good contraction/expansion properties with respect to different metrics in the set of probability measures. Existence and regularity of isotropic stationary states is shown directly by constructing converging iteration sequences as done in [8]. Uniqueness, asymptotic stability and estimates of overpopulated high energy tails of the steady profile are derived from the basic property of contraction/expansion of metrics. For general initial conditions the solutions of the Boltzmann equation are then proved to converge with computable rate as $t\to\infty$ to the steady solution in these distances, which metricizes the weak convergence of measures. These results show that power-like tails in Maxwell models are obtained when the point-wise conservation of momentum and/or energy holds only globally.
Keywords: inelastic collisions, thick-tails., conservative in mean, asymptotic behavior, stochastic restitution coefficient.
Mathematics Subject Classification: 82C40, 35B4.
Citation: José A. Carrillo, Stéphane Cordier, Giuseppe Toscani. Over-populated tails for conservative-in-the-mean inelastic Maxwell models. Discrete & Continuous Dynamical Systems, 2009, 24 (1) : 59-81. doi: 10.3934/dcds.2009.24.59
[1]

Yadong Shang, Jianjun Paul Tian, Bixiang Wang. Asymptotic behavior of the stochastic Keller-Segel equations. Discrete & Continuous Dynamical Systems - B, 2019, 24 (3) : 1367-1391. doi: 10.3934/dcdsb.2019020
[2]

Giuseppe Da Prato, Arnaud Debussche. Asymptotic behavior of stochastic PDEs with random coefficients. Discrete & Continuous Dynamical Systems, 2010, 27 (4) : 1553-1570. doi: 10.3934/dcds.2010.27.1553
[3]

Daehwan Kim, Juncheol Pyo. Existence and asymptotic behavior of helicoidal translating solitons of the mean curvature flow. Discrete & Continuous Dynamical Systems, 2018, 38 (11) : 5897-5919. doi: 10.3934/dcds.2018256
[4]

Xiaobin Yao. Asymptotic behavior for stochastic plate equations with memory and additive noise on unbounded domains. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021050
[5]

Sanling Yuan, Xuehui Ji, Huaiping Zhu. Asymptotic behavior of a delayed stochastic logistic model with impulsive perturbations. Mathematical Biosciences & Engineering, 2017, 14 (5&6) : 1477-1498. doi: 10.3934/mbe.2017077
[6]

Tian Zhang, Huabin Chen, Chenggui Yuan, Tomás Caraballo. On the asymptotic behavior of highly nonlinear hybrid stochastic delay differential equations. Discrete & Continuous Dynamical Systems - B, 2019, 24 (10) : 5355-5375. doi: 10.3934/dcdsb.2019062
[7]

Guanggan Chen, Jian Zhang. Asymptotic behavior for a stochastic wave equation with dynamical boundary conditions. Discrete & Continuous Dynamical Systems - B, 2012, 17 (5) : 1441-1453. doi: 10.3934/dcdsb.2012.17.1441
[8]

Ciro D’Apice, Umberto De Maio, T. A. Mel'nyk. Asymptotic analysis of a perturbed parabolic problem in a thick junction of type 3:2:2. Networks & Heterogeneous Media, 2007, 2 (2) : 255-277. doi: 10.3934/nhm.2007.2.255
[9]

Evelyn Buckwar, Girolama Notarangelo. A note on the analysis of asymptotic mean-square stability properties for systems of linear stochastic delay differential equations. Discrete & Continuous Dynamical Systems - B, 2013, 18 (6) : 1521-1531. doi: 10.3934/dcdsb.2013.18.1521
[10]

Xiaobin Yao, Qiaozhen Ma, Tingting Liu. Asymptotic behavior for stochastic plate equations with rotational inertia and Kelvin-Voigt dissipative term on unbounded domains. Discrete & Continuous Dynamical Systems - B, 2019, 24 (4) : 1889-1917. doi: 10.3934/dcdsb.2018247
[11]

Dingshi Li, Xiaohu Wang. Asymptotic behavior of stochastic complex Ginzburg-Landau equations with deterministic non-autonomous forcing on thin domains. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 449-465. doi: 10.3934/dcdsb.2018181
[12]

Dandan Ma, Ji Shu, Ling Qin. Wong-Zakai approximations and asymptotic behavior of stochastic Ginzburg-Landau equations. Discrete & Continuous Dynamical Systems - B, 2020, 25 (11) : 4335-4359. doi: 10.3934/dcdsb.2020100
[13]

G. Deugoué, T. Tachim Medjo. The Stochastic 3D globally modified Navier-Stokes equations: Existence, uniqueness and asymptotic behavior. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2593-2621. doi: 10.3934/cpaa.2018123
[14]

Yiju Chen, Xiaohu Wang. Asymptotic behavior of non-autonomous fractional stochastic lattice systems with multiplicative noise. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021271
[15]

Veronica Felli, Alberto Ferrero, Susanna Terracini. On the behavior at collisions of solutions to Schrödinger equations with many-particle and cylindrical potentials. Discrete & Continuous Dynamical Systems, 2012, 32 (11) : 3895-3956. doi: 10.3934/dcds.2012.32.3895
[16]

Jingwei Hu, Shi Jin, Li Wang. An asymptotic-preserving scheme for the semiconductor Boltzmann equation with two-scale collisions: A splitting approach. Kinetic & Related Models, 2015, 8 (4) : 707-723. doi: 10.3934/krm.2015.8.707
[17]

Zhipeng Qiu, Jun Yu, Yun Zou. The asymptotic behavior of a chemostat model. Discrete & Continuous Dynamical Systems - B, 2004, 4 (3) : 721-727. doi: 10.3934/dcdsb.2004.4.721
[18]

Nguyen Huu Du, Nguyen Thanh Dieu. Long-time behavior of an SIR model with perturbed disease transmission coefficient. Discrete & Continuous Dynamical Systems - B, 2016, 21 (10) : 3429-3440. doi: 10.3934/dcdsb.2016105
[19]

Tomasz Komorowski, Łukasz Stȩpień. Kinetic limit for a harmonic chain with a conservative Ornstein-Uhlenbeck stochastic perturbation. Kinetic & Related Models, 2018, 11 (2) : 239-278. doi: 10.3934/krm.2018013
[20]

Mykhailo Potomkin. Asymptotic behavior of thermoviscoelastic Berger plate. Communications on Pure & Applied Analysis, 2010, 9 (1) : 161-192. doi: 10.3934/cpaa.2010.9.161

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (62)
  • HTML views (0)
  • Cited by (13)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences | Privacy Policy