Communications in Mathematical Sciences

Volume 20 (2022)

Number 3

Stability of measure solutions to a generalized Boltzmann equation with collisions of a random number of particles

Pages: 877 – 896



Henryk Gacki (Faculty of Science and Technology, University of Silesia, Katowice, Poland)

Łukasz Stettner (Institute of Mathematics, Polish Academy of Sciences, Warsaw, Poland)


In the paper we study a measure version of the evolutionary nonlinear Boltzmann-type equation in which we admit a random number of collisions of particles. We consider first a stationary model and use two methods to find its fixed points: the first based on Zolotarev seminorm and the second on Kantorovich–Rubinstein maximum principle. Then a dynamic version of Boltzmann-type equation is considered and its asymptotic stability is shown.


generalized Boltzmann equation, stability, collisions of particles, Zolotariev seminorm, Kantorovich–Rubinstein maximum principle

2010 Mathematics Subject Classification

35Q20, 82B21, 82B31

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Received 6 April 2020

Received revised 16 April 2021

Accepted 24 October 2021

Published 21 March 2022