It is well-known that the Fourier-Galerkin spectral method has been a popular approach for the numerical approximation of the deterministic Boltzmann equation with spectral accuracy rigorously proved. In this paper, we will show that such a spectral convergence of the Fourier-Galerkin spectral method also holds for the Boltzmann equation with uncertainties arising from both collision kernel and initial condition. Our proof is based on newly-established spaces and norms that are carefully designed and take the velocity variable and random variables with their high regularities into account altogether. For future studies, this theoretical result will provide a solid foundation for further showing the convergence of the full-discretized system where both the velocity and random variables are discretized simultaneously.

翻译：众所周知，Fourier-Galerkin谱方法已成为确定性Boltzmann方程数值逼近的常用方法，且已严格证明了其谱精度。本文证明，对于碰撞核和初始条件均存在不确定性的Boltzmann方程，Fourier-Galerkin谱方法同样具有谱收敛性。我们的证明基于新建立的空间与范数，这些空间与范数经过精心设计，能够同时考虑速度变量和随机变量及其高正则性。该理论结果将为未来研究中同时离散速度变量和随机变量的全离散系统的收敛性证明提供坚实基础。