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，傅里叶-伽辽金谱方法一直是对确定性玻尔兹曼方程进行数值逼近的主流方法，其谱精度已得到严格证明。本文我们将证明，对于碰撞核和初始条件均含不确定性的玻尔兹曼方程，傅里叶-伽辽金谱方法的谱收敛性同样成立。该证明基于新建立的函数空间与范数，这些空间与范数经精心设计，综合考虑了速度变量与随机变量及其高阶正则性。该理论结果将为后续研究提供坚实基础，进一步证明速度变量与随机变量同时离散的全离散系统的收敛性。