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玻尔兹曼方程，并证明其在速度与随机变量联合空间中的半离散数值系统具有谱收敛性。其中，速度空间采用傅里叶谱方法进行近似，而随机变量则采用基于广义多项式混沌(gPC)的随机伽辽金(SG)方法进行离散化。我们的证明基于精细的能量估计，以论证数值解的良好适定性，并在精心设计的包含速度与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[注释4.4]所提出的论断。