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

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

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.
Citation: José A. Carrillo, Stéphane Cordier, Giuseppe Toscani. Over-populated tails for conservative-in-the-mean inelastic Maxwell models. Discrete & Continuous Dynamical Systems - A, 2009, 24 (1) : 59-81. doi: 10.3934/dcds.2009.24.59
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