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Time-asymptotic behavior of the Boltzmann equation with random inputs in whole space and its stochastic Galerkin approximation

Shi Jin, Qi Shao, Haitao Wang

TL;DR

The paper analyzes the Boltzmann equation with random inputs in the whole space and its gPC-SG approximation. It develops a Green's-function framework with macro-micro decomposition to obtain time-decay estimates for the linearized problem and extends them to nonlinear dynamics, including higher-order derivatives in the random input with controlled growth. It then establishes weighted, spectrally accurate gPC-SG bounds, proving that SG errors decay in time with explicit rates independent of the truncation level. These results extend prior torus-domain analyses to the whole space and provide rigorous justification for SG methods in uncertainty quantification for kinetic equations.

Abstract

We consider the Boltzmann equation with random uncertainties arising from the initial data and collision kernel in the {\it whole space}, along with their stochastic Galerkin (SG) approximations. By employing Green's function method, we show that, the higher-order derivatives of the solution with respect to the random variable exhibit polynomial decay over time. These results are then applied to analyze the SG method for the SG system and to demonstrate the polynomial decay of the numerical error over time.

Time-asymptotic behavior of the Boltzmann equation with random inputs in whole space and its stochastic Galerkin approximation

TL;DR

The paper analyzes the Boltzmann equation with random inputs in the whole space and its gPC-SG approximation. It develops a Green's-function framework with macro-micro decomposition to obtain time-decay estimates for the linearized problem and extends them to nonlinear dynamics, including higher-order derivatives in the random input with controlled growth. It then establishes weighted, spectrally accurate gPC-SG bounds, proving that SG errors decay in time with explicit rates independent of the truncation level. These results extend prior torus-domain analyses to the whole space and provide rigorous justification for SG methods in uncertainty quantification for kinetic equations.

Abstract

We consider the Boltzmann equation with random uncertainties arising from the initial data and collision kernel in the {\it whole space}, along with their stochastic Galerkin (SG) approximations. By employing Green's function method, we show that, the higher-order derivatives of the solution with respect to the random variable exhibit polynomial decay over time. These results are then applied to analyze the SG method for the SG system and to demonstrate the polynomial decay of the numerical error over time.

Paper Structure

This paper contains 21 sections, 18 theorems, 257 equations.

Table of Contents

  1. Introduction
  2. The Boltzmann equation
  3. Notations
  4. Review of previous works and main objectives
  5. The main results
  6. Organization of the paper
  7. Preliminaries
  8. Linearized collision operator and macro-micro projections
  9. The Green's function
  10. The linearized problem
  11. The solution operator for the linearized problem
  12. Time decay of derivatives of the solutions
  13. Time decay of higher derivatives
  14. The nonlinear problem
  15. Solutions to the nonlinear problems
  16. ...and 6 more sections

Key Result

Theorem 1

Assuming that the collision kernel satisfies (b1) and (b2), then for given $\alpha\in \mathbb{N}$, $C_{b_1},C_{b_2}$ and $C_{b_*}$ in (b1) and (b2), there exist positive constants $\delta$ and $C_\alpha$ such that if then there exists a unique global solution $f=f(x,t,\xi,z)$ to equation (UC), satisfying and Moreover, for $1\le k\le \alpha$, it holds that and In this theorem, all constants ar

Theorems & Definitions (26)

  • Theorem 1
  • Remark 2
  • Theorem 3
  • Remark 4
  • Lemma 5
  • Lemma 6
  • Lemma 7
  • Theorem 8
  • Remark 9
  • Theorem 10: shu[LiuYu1]liu2011solving
  • ...and 16 more