In this paper, we study the Boltzmann equation with uncertainties and prove that the spectral convergence of the semi-discretized numerical system holds in a combined velocity and random space, where the Fourier-spectral method is applied for approximation in the velocity space whereas the generalized polynomial chaos (gPC)-based stochastic Galerkin (SG) method is employed to discretize the random variable. Our proof is based on a delicate energy estimate for showing the well-posedness of the numerical solution as well as a rigorous control of its negative part in our well-designed functional space that involves high-order derivatives of both the velocity and random variables. This paper rigorously justifies the statement proposed in [Remark 4.4, J. Hu and S. Jin, J. Comput. Phys., 315 (2016), pp. 150-168].

翻译：本文研究含不确定性的Boltzmann方程，证明了在速度与随机变量联合空间中半离散数值系统的谱收敛性，其中速度空间采用Fourier谱方法近似，而随机变量则采用基于广义多项式混沌（gPC）的随机Galerkin（SG）方法离散。我们的证明基于精细的能量估计，以展示数值解的良好适定性，并在我们精心设计的包含速度和随机变量高阶导数的函数空间中，对其负部进行严格控制。本文严格论证了文献[J. Hu and S. Jin, J. Comput. Phys., 315 (2016), pp. 150-168]中[注记4.4]所提出的论断。