Abstract
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. Models12(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.
Highlights
We consider the multi-species Boltzmann equation describing the evolution of a multi-species mono-atomic nonreactive gaseous mixture with additional uncertainty coming from the initial data and collision kernel, which was studied analytically in the deterministic setting in [1, 2, 4,5,6, 8, 12]
We deal with the multi-species Boltzmann equation with an additional random parameter described by the random variable z, which lies in the random space Iz with a probability measure π(z)dz
We remark that our work relies on several existing literature on uncertainty quantification (UQ) for general kinetic models [24], sensitivity analysis [27], spectral convergence of the gPC-Galerkin method [11] and multi-species Boltzmann equations [8]
Summary
We consider the multi-species Boltzmann equation describing the evolution of a multi-species mono-atomic nonreactive gaseous mixture with additional uncertainty coming from the initial data and collision kernel, which was studied analytically in the deterministic setting in [1, 2, 4,5,6, 8, 12]. The main goal of this paper is to study the well-posedness and long-time behavior of the nonlinear multispecies Boltzmann equation under the impact of random uncertainty and its stochastic Galerkin approximation in the perturbative regime. Compared to [11] on the single-species gPC-SG Boltzmann system, the generalization to the multispecies case here can be done by adapting techniques from the proof for the multi-species H-theorem, see for instance [12,14] Establishing this spectral estimate is essential in order to understanding the long-time behavior of the gPC-SG approximation. We remark that our work relies on several existing literature on UQ for general kinetic models [24], sensitivity analysis [27], spectral convergence of the gPC-Galerkin method [11] and multi-species Boltzmann equations [8].
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