Communications in Mathematical Sciences

Volume 19 (2021)

Number 3

Hypocoercivity of stochastic Galerkin formulations for stabilization of kinetic equations

Pages: 787 – 806



Stephan Gerster (Institut für Geometrie und Praktische Mathematik, RWTH Aachen University, Aachen, Germany)

Michael Herty (Institut für Geometrie und Praktische Mathematik, RWTH Aachen University, Aachen, Germany)

Hui Yu (Yau Mathematical Sciences Center, Tsinghua University, Beijing, China)


We consider the stabilization of linear kinetic equations with a random relaxation term. The well-known framework of hypocoercivity by J. Dolbeault, C. Mouhot and C. Schmeiser (2015) ensures the stability in the deterministic case. This framework, however, cannot be applied directly for arbitrarily small random relaxation parameters. Therefore, we introduce a Galerkin formulation, which reformulates the stochastic system as a sequence of deterministic ones. We prove for the $\gamma$-distribution that the hypocoercivity framework ensures the stability of this series and hence the stochastic stability of the underlying random kinetic equation. The presented approach also yields a convergent numerical approximation.


systems of kinetic and hyperbolic balance laws, exponential stability, asymptotic stability, stochastic Galerkin

2010 Mathematics Subject Classification

35B30, 35B35, 35R60, 37L45, 93D20

This work is supported by DFG HE5386/18,19, BMBF ENet 05M18PAA, DFG 320021702/GRK2326, NSFC 11901339 and NSFC 11971258. The authors would like to offer special thanks to Giuseppe Visconti, Tabea Tscherpel and to the support from RWTH Aachen-Tsinghua Senior Research Fellowships.

Received 27 June 2020

Accepted 1 November 2020

Published 5 May 2021