Finding the jump rate for fastest decay in the Goldstein-Taylor model

Helge Dietert (1, 2) , Josephine Evans (3)

1 IMJ-PRG (UMR_7586) - Institut de Mathématiques de Jussieu - Paris Rive Gauche
2 Leipzig University / Universität Leipzig
3 WMI - Warwick Mathematics Institute

This paper is about the rate of convergence to equilibrium for hypocoercive linear kinetic equations. We look for the spatially dependent jump rate which yields the fastest decay rate of perturbations. For the Goldstein-Taylor model, we show (i) that for a locally optimal jump rate the spectral bound is determined by multiple, possibly degenerate, eigenvectors and (ii) that globally the fastest decay is obtained with a spatially homogeneous jump rate. Our proofs rely on a connection to damped wave equations and a relationship to the spectral theory of Schrödinger operators.

Mots clés: wave equation; optimal control; Hypocoercivity; spatial weight; optimal design of damping set; Goldstein-Taylor model

Domaines: Equations aux dérivées partielles [math.AP]; Optimisation et contrôle [math.OC]

Helge Dietert, Josephine Evans. Finding the jump rate for fastest decay in the Goldstein-Taylor model. Journal of Statistical Physics, 2022, ⟨10.1007/s10955-022-02925-3⟩. ⟨hal-03171498v4⟩