Article Contents

2019, Volume 14, Issue 4: 677-707. Doi: 10.3934/nhm.2019027

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A local sensitivity and regularity analysis for the Vlasov-Poisson-Fokker-Planck system with multi-dimensional uncertainty and the spectral convergence of the stochastic Galerkin method

Yuhua Zhu^,

Department of Mathematics, University of Wisconsin-Madison, Madison, Wisconsin, 53706, USA

^* Corresponding author: Yuhua Zhu
^* Corresponding author: Yuhua Zhu

Received: August 2018

Revised: April 2019

Published: December 2019

The author was supported in part by Prof. Shi Jin's NSF grants DMS-1522184 and DMS1107291: RNMS KI-Net

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Abstract

We study the Vlasov-Poisson-Fokker-Planck (VPFP) system with uncertainty and multiple scales. Here the uncertainty, modeled by multi-dimensional random variables, enters the system through the initial data, while the multiple scales lead the system to its high-field or parabolic regimes. We obtain a sharp decay rate of the solution to the global Maxwellian, which reveals that the VPFP system is decreasingly sensitive to the initial perturbation as the Knudsen number goes to zero. The sharp regularity estimates in terms of the Knudsen number lead to the stability of the generalized Polynomial Chaos stochastic Galerkin (gPC-SG) method. Based on the smoothness of the solution in the random space and the stability of the numerical method, we conclude the gPC-SG method has spectral accuracy uniform in the Knudsen number.

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Mathematics Subject Classification: Primary: 65M12, 35Q83, 35Q84; Secondary: 82C40.

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