Hypocoercivity of the linearized BGK equation with stochastic coefficients

We consider an approximation of the Boltzmann equation, the Bathnagar-Gross-Krook (BGK) equation. This equation is used in many applications because it is very efficient in numerical simulations. In this paper we study the effect of randomness on a BGK-model. We prove exponential decay rate to a global equilibrium. In addition we prove the decay rate of the $n$-th derivative with respect to the stochastic variable of the solutions. The novelties are i) for the first time hypocoercivity is shown for a linearized BGK model that conserves mass, momentum and energy with randomness in the collision frequency, ii) new estimates for the decay of the derivatives of the solution with respect to the stochastic variable, which is very useful in applications.

Keywords

linear BGK-equation with uncertainties, hypocoercivity, decay estimate, Lyapunov’s direct method

2010 Mathematics Subject Classification

35A24, 35B30, 35Q20, 82B40

Full Text (PDF format)

Received 24 March 2022

Accepted 13 October 2023

Published 7 August 2024