Sharpening of decay rates in Fourier based hypocoercivity methods

Anton Arnold (1) , Jean Dolbeault (2) , Christian Schmeiser (3) , Tobias Wöhrer (1)

1 TU Wien - Vienna University of Technology = Technische Universität Wien
2 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision
3 Universität Wien = University of Vienna

This paper is dealing with two L2 hypocoercivity methods based on Fourier decomposition and mode-by-mode estimates, with applications to rates of convergence or decay in kinetic equations on the torus and on the whole Euclidean space. The main idea is to perturb the standard L2 norm by a twist obtained either by a nonlocal perturbation build upon diffusive macroscopic dynamics, or by a change of the scalar product based on Lyapunov matrix inequalities. We explore various estimates for equations involving a Fokker-Planck and a linear relaxation operator. We review existing results in simple cases and focus on the accuracy of the estimates of the rates. The two methods are compared in the case of the Goldstein-Taylor model in one-dimension.

Mots clés: Hypocoercivity; linear kinetic equations; entropy-entropy production inequalities; Goldstein-Taylor model; Fokker-Planck operator; linear relaxation operator; linear BGK operator; transport operator; Fourier modes decomposition; pseudo-differential operators; Nash's inequality

Domaines: Equations aux dérivées partielles [math.AP]

Anton Arnold, Jean Dolbeault, Christian Schmeiser, Tobias Wöhrer. Sharpening of decay rates in Fourier based hypocoercivity methods. 2020. ⟨hal-03078698v1⟩