Hypocoercivity and Uniform Regularity for the Vlasov-Poisson-Fokker-Planck System with Uncertainty and Multiple Scales

TL;DR

This paper establishes uniform regularity and exponential decay for the Vlasov-Poisson-Fokker-Planck system with uncertainty and multiple scales, advancing the understanding of its sensitivity and numerical analysis.

Contribution

It provides the first hypocoercivity results for a nonlinear kinetic system with random input, crucial for uncertainty quantification and spectral convergence analysis.

Findings

Proves uniform regularity in random space under Sobolev norms.

Establishes exponential decay to the global Maxwellian.

First hypocoercivity results for nonlinear kinetic systems with randomness.

Abstract

We study the Vlasov-Poisson-Fokker-Planck system with uncertainty and multiple scales. Here the uncertainty, modeled by random variables, enters the solution through initial data, while the multiple scales lead the system to its high-field or parabolic regimes. With the help of proper Lyapunov-type inequalities, under some mild conditions on the initial data, the regularity of the solution in the random space, as well as exponential decay of the solution to the global Maxwellian, are established under Sobolev norms, which are ${\it uniform}$ in terms of the scaling parameters. These are the first hypocoercivity results for a nonlinear kinetic system with random input, which are important for the understanding of the sensitivity of the system under random perturbations, and for the establishment of spectral convergence of popular numerical methods for uncertainty quantification based on (spectrally accurate) polynomial chaos expansions.