Electronic Journal of Probability

Infinite energy solutions to inelastic homogeneous Boltzmann equations

Federico Bassetti, Lucia Ladelli, and Daniel Matthes

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Abstract

This paper is concerned with the existence, shape and dynamical stability of infinite-energy equilibria for a class of spatially homogeneous kinetic equations in space dimensions $d\ge2$. Our results cover in particular Bobylev's model for inelastic Maxwell molecules.  First, we show under certain conditions on the collision kernel, that there exists an index $\alpha\in(0,2)$  such that the equation possesses a nontrivial stationary solution, which is a scale mixture of radially symmetric $\alpha$-stable laws. We also characterize the mixing distribution as the fixed point of a smoothing transformation.  Second, we prove that any transient solution that emerges from the NDA of some (not necessarily radial symmetric) $\alpha$-stable distribution  converges to an equilibrium. The key element of the convergence proof is an application of the central limit theorem to a representation of the transient solution as a weighted sum of projections  of randomly rotated  i.i.d. random vectors.

Article information

Source
Electron. J. Probab. Volume 20 (2015), paper no. 89, 34 pp.

Dates
Accepted: 8 September 2015
First available in Project Euclid: 4 June 2016

Permanent link to this document
https://projecteuclid.org/euclid.ejp/1465067195

Digital Object Identifier
doi:10.1214/EJP.v20-3531

Mathematical Reviews number (MathSciNet)
MR3399825

Zentralblatt MATH identifier
1345.60060

Subjects
Primary: 60F05: Central limit and other weak theorems
Secondary: 60K40: Other physical applications of random processes 82C40: Kinetic theory of gases

Keywords
Central Limit Theorems Inelastic Boltzmann Equation Infinite Energy Solutions Normal Domain of Attraction Multidimensional Stable Laws

Rights
This work is licensed under a Creative Commons Attribution 3.0 License.

Citation

Bassetti, Federico; Ladelli, Lucia; Matthes, Daniel. Infinite energy solutions to inelastic homogeneous Boltzmann equations. Electron. J. Probab. 20 (2015), paper no. 89, 34 pp. doi:10.1214/EJP.v20-3531. https://projecteuclid.org/euclid.ejp/1465067195


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