Hypocoercivity without confinement

TL;DR

This paper demonstrates that linear kinetic equations without confinement exhibit algebraic decay rates similar to the heat equation, using hypocoercivity methods with novel approaches based on Fourier analysis and Nash's inequality.

Contribution

It introduces two new analytical approaches for hypocoercivity without confinement, extending decay results to broader function spaces and improving decay rates with initial moment cancellations.

Findings

Solutions decay algebraically at the heat-equation rate

Two alternative analytical methods are developed

Decay rates are improved with initial moment cancellations

Abstract

In this paper, hypocoercivity methods are applied to linear kinetic equations with mass conservation and without confinement, in order to prove that the solutions have an algebraic decay rate in the long-time range, which the same as the rate of the heat equation. Two alternative approaches are developed: an analysis based on decoupled Fourier modes and a direct approach where, instead of the Poincar\'e inequality for the Dirichlet form, Nash's inequality is employed. The first approach is also used to provide a simple proof of exponential decay to equilibrium on the flat torus. The results are obtained on a space with exponential weights and then extended to larger function spaces by a factorization method. The optimality of the rates is discussed. Algebraic rates of decay on the whole space are improved when the initial datum has moment cancellations.