Volume 55, Number 4, July-August 2021
|Page(s)||1323 - 1345|
|Published online||07 July 2021|
On the multi-species Boltzmann equation with uncertainty and its stochastic Galerkin approximation*
Institute for Analysis and Scientific Computing, Vienna University of Technology, Wiedner Hauptstraße 8—10, 1040 Wien, Austria
2 School of Mathematical Sciences, Institute of Natural Sciences, MOE-LSC and SHL-MAC, Shanghai Jiao Tong University, Shanghai 200240, China
3 Department of Mathematics, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong SAR
* Corresponding author: firstname.lastname@example.org
Accepted: 6 May 2021
In this paper the nonlinear multi-species Boltzmann equation with random uncertainty coming from the initial data and collision kernel is studied. Well-posedness and long-time behavior – exponential decay to the global equilibrium – of the analytical solution, and spectral gap estimate for the corresponding linearized gPC-based stochastic Galerkin system are obtained, by using and extending the analytical tools provided in [M. Briant and E.S. Daus, Arch. Ration. Mech. Anal. 3 (2016) 1367–1443] for the deterministic problem in the perturbative regime, and in [E.S. Daus, S. Jin and L. Liu, Kinet. Relat. Models 12 (2019) 909–922] for the single-species problem with uncertainty. The well-posedness result of the sensitivity system presented here has not been obtained so far neither in the single species case nor in the multi-species case.
Mathematics Subject Classification: 35Q20 / 65M70
Key words: Multi-species Boltzmann equation / uncertainty quantification / hypocoercivity / stochastic Galerkin
© The authors. Published by EDP Sciences, SMAI 2021
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