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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]., Comment: Revised version. To appear in SIAM/ASA Journal on Uncertainty Quantification

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., Comment: 30 Pages. Revised Version. To appear in Communications in Mathematical Sciences

Numerical approximation of the Boltzmann equation is a challenging problem due to its high-dimensional, nonlocal, and nonlinear collision integral. Over the past decade, the Fourier-Galerkin spectral method has become a popular deterministic method for solving the Boltzmann equation, manifested by its high accuracy and potential of being further accelerated by the fast Fourier transform. Albeit its practical success, the stability of the method is only recently proved by Filbet, F. & Mouhot, C. in [$ Trans. Amer. Math. Soc.$ 363, no. 4 (2011): 1947-1980.] by utilizing the "spreading" property of the collision operator. In this work, we provide a new proof based on a careful $L^2$ estimate of the negative part of the solution. We also discuss the applicability of the result to various initial data, including both continuous and discontinuous functions.

We introduce a fast Fourier spectral method for the spatially homogeneous Boltzmann equation with non-cutoff collision kernels. Such kernels contain non-integrable singularity in the deviation angle which arise in a wide range of interaction potentials (e.g., the inverse power law potentials). Albeit more physical, the non-cutoff kernels bring a lot of difficulties in both analysis and numerics, hence are often cut off in most studies (the well-known Grad's angular cutoff assumption). We demonstrate that the general framework of the fast Fourier spectral method can be extended to handle the non-cutoff kernels, achieving the accuracy/efficiency comparable to the cutoff case. We also show through several numerical examples that the solution to the non-cutoff Boltzmann equation enjoys the smoothing effect, a striking property absent in the cutoff case.