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

Abstract

The object of this paper is to provide a new and systematic tauberian approach to quantitative long time behaviour of $C_0$-semigroups ${\left(\mathcal{V}(t)\right)}_{t\ge 0}$ in $L^1({\mathbb{T}}^d\times {\mathbb{R}}^d)$ governing conservative linear kinetic equations on the torus with general scattering kernel $k(v,v^{\prime })$ and degenerate (i.e. not bounded away from zero) collision frequency $\ddot{I}ƒ(v)={\int }_{{\mathbb{R}}^d}k(v^{\prime },v)m(\mathrm{d}{v}^{\prime })$, (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 $s\ge 0$ such that $$\frac{1}{\ddot{I}ƒ(\cdot )}{\int }_{{\mathbb{R}}^d}k(\cdot ,v)\ddot{I}{ƒ}^{-s}(v)m(\mathrm{d}v)\in L^{\infty }({\mathbb{R}}^d)$$ then, for initial datum $f$ such that $\mathrm{d}s{\int }_{{\mathbb{T}}^d\times {\mathbb{R}}^d}|f(x,v)|\ddot{I}{ƒ}^{-N_0}(v)\mathrm{d}xm(\mathrm{d}v)<\infty$ it holds $${\Vert \mathcal{V}(t)f-{\varrho }_f\hat{I}¨\Vert }_{L^1}=\frac{\hat{I}{\mu }_f(t)}{(1+t{)}^{N_0-1}},{\varrho }_f:={\int }_{{\mathbb{R}}^d}f(x,v)m(\mathrm{d}v)$$ where $\hat{I}¨$ is the unique invariant density of ${\left(\mathcal{V}(t)\right)}_{t\ge 0}$ and ${\lim }_{t\to \infty }\hat{I}{\mu }_f(t)=0$. 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.