Convergence of the Fourier-Galerkin spectral method for the Boltzmann equation with uncertainties

TL;DR

This work extends the spectral convergence guarantees of the Fourier-Galerkin method from the deterministic Boltzmann equation to the Boltzmann equation with random uncertainties in the initial data and collision kernel. A novel analytical framework is developed to handle coupled velocity and random variables via new Sobolev-type norms that incorporate $z$-derivatives up to order $r$, enabling rigorous well-posedness and stability results for the semi-discretized system. The authors prove propagation of high-regularity in $v$ and $z$, control the negative part, local and global well-posedness, and finally establish spectral convergence in the velocity and random spaces with a rate $\|e_N\|_{L^2_v,r} \le C_r(T,f^0) N^{-k}$. This provides a solid foundation for fully discretized schemes and motivates future work on multi-species kinetic models with uncertainties.

Abstract

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.

Theorems & Definitions (23)

• remark 1
• remark 2
• lemma 1
• remark 3
• proposition 1
• proof
• corollary 1
• proof
• proposition 2
• proof