Convergence rate to equilibrium for conservative scattering models on the torus: a new tauberian approach

TL;DR

This paper introduces a new tauberian approach to analyze the long-time behavior of conservative kinetic models on the torus, establishing explicit convergence rates to equilibrium under general scattering conditions.

Contribution

It develops a systematic tauberian method to quantify convergence rates for kinetic semigroups with degenerate collision frequencies, including criteria for invariant densities.

Findings

Derived explicit polynomial decay rates for semigroup convergence.

Provided new criteria for existence of invariant densities.

Achieved sharp subgeometric convergence rates for Markov semigroups.

Abstract

The object of this paper is to provide a new and systematic tauberian approach to quantitative long time behaviour of $C_{0}$-semigroups $\left(\mathcal{V}(t)\right)_{t \geq0}$ in $L^{1}(\mathbb{T}^{d}\times \mathbb{R}^{d})$ governing conservative linear kinetic equations on the torus with general scattering kernel $ {k}(v,v')$ and degenerate (i.e. not bounded away from zero) collision frequency $\sigma(v)=\int_{\mathbb{R}^{d}} {k}(v',v)m(\mathrm{d} v')$, (with $m(\mathrm{d} v)$ being absolutely continuous with respect to the Lebesgue measure). We show in particular that if $N_{0}$ is the maximal integer $s \geq0$ such that $$\frac{1}{\sigma(\cdot)}\int_{\mathbb{R}^{d}}k(\cdot,v)\sigma^{-s}(v)m(\mathrm{d} v) \in L^{\infty}(\mathbb{R}^{d})$$ then, for initial datum $f$ such that $\mathrm{d} s\int_{\mathbb{T}^{d}\times \mathbb{R}^{d}}|f(x,v)|\sigma^{-N_{0}}(v)\mathrm{d} x m(\mathrm{d} v) <\infty$ it holds $$\left\|\mathcal{V}(t)f-\varrho_{f}\Psi\right\|_{L^{1}}=\dfrac{{\epsilon}_{f}(t)}{(1+t)^{N_{0}-1}}, \qquad \varrho_{f}:= \int_{\mathbb{R}^{d}}f(x,v)m(\mathrm{d} v)$$ where $\Psi$ is the unique invariant density of $\left(\mathcal{V}(t)\right)_{t \geq0}$ and $\lim_{t\to\infty}{\epsilon}_{f}(t)=0$. We in particular provide a new criteria of the existence of invariant density. The proof relies on the explicit computation of the time decay of each term of the Dyson-Phillips expansion of $\mathcal{V}(t)$ and on suitable smoothness and integrability properties of the trace on the imaginary axis of Laplace transform of remainders of large order of this Dyson-Phillips expansion. Our construction resorts also on collective compactness arguments and provides various technical results of independent interest. Finally, as a by-product of our analysis, we derive essentially sharp ``subgeometric'' convergence rate for Markov semigroups associated to general transition kernels.