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Abstract
We introduce and discuss spatially homogeneous Maxwell-type models of thenonlinear Boltzmann equation undergoing binary collisions with a randomcomponent. The random contribution to collisions is such that the usualcollisional invariants of mass, momentum and energy do not hold pointwise, evenif they all hold in the mean. Under this assumption it is shown that, while theBoltzmann equation has the usual conserved quantities, it possesses a steadystate with power-like tails for certain random variables. A similar situationoccurs in kinetic models of economy recently considered by two of the authors[24], which are conservative in the mean but possess a steadydistribution with Pareto tails. The convolution-like gain operator issubsequently shown to have good contraction/expansion properties with respectto different metrics in the set of probability measures. Existence andregularity of isotropic stationary states is shown directly by constructingconverging iteration sequences as done in [8]. Uniqueness, asymptotic stability and estimates of overpopulated high energytails of the steady profile are derived from the basic property ofcontraction/expansion of metrics. For general initial conditions the solutionsof the Boltzmann equation are then proved to converge with computable rate as$t\to\infty$ to the steady solution in these distances, which metricizes theweak convergence of measures. These results show that power-like tails inMaxwell models are obtained when the point-wise conservation of momentumand/or energy holds only globally.
Highlights
We introduce and discuss the possibility to obtain steady solutions with power-like tails starting from conservative molecular systems described by the Boltzmann equation with Maxwell-type collision kernels
The starting point of our model is to consider binary collisions that result in a linear combination of an inelastic collision and a random contribution
Other results concerned with self-similar solutions in the theory of the classical Boltzmann equation for Maxwell molecules were recently published in [7, 8]
Summary
We introduce and discuss the possibility to obtain steady solutions with power-like tails starting from conservative molecular systems described by the Boltzmann equation with Maxwell-type collision kernels. Estimates of the evolution of the Wasserstein distance [32, 29, 30] between solutions will be presented for the economic and the inelastic model since they give complementary information with respect to the results in [24] Concerning this second aspect, we will take advantage of the recent analysis of Bolley and Carrillo [14, 17] of the inelastic Boltzmann equation for Maxwell molecules. Let us summarize the two main results of this paper : first, we give some insight into conditions for a collision operator to lead to power-law tails (conservatism in mean being some kind of necessary condition); second, we propose a new form for the thermal bath with a physically relevant origin (the restitution coefficient taking into account the randomness of granular media)
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