Convergence rate to equilibrium for conservative scattering models on the torus: a new tauberian approach
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 $σ(v)=\int_{\mathbb{R}^{d}} {k}(v',v)m(\mathrm{d} v')$, (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 $$\frac{1}{σ(\cdot)}\int_{\mathbb{R}^{d}}k(\cdot,v)σ^{-s}(v)m(\mathrm{d} v) \in L^{\infty}(\mathbb{R}^{d})$$ then, for initial datum $f$ such that $\mathrm{d} s\int_{\mathbb{T}^{d}\times \mathbb{R}^{d}}|f(x,v)|σ^{-N_{0}}(v)\mathrm{d} x m(\mathrm{d} v) <\infty$ it holds $$\left\|\mathcal{V}(t)f-\varrho_{f}Ψ\right\|_{L^{1}}=\dfrac{ε_{f}(t)}{(1+t)^{N_{0}-1}}, \qquad \varrho_{f}:= \int_{\mathbb{R}^{d}}f(x,v)m(\mathrm{d} v)$$ where $Ψ$ is the unique invariant density of $\left(\mathcal{V}(t)\right)_{t \geq0}$ and $\lim_{t\to\infty}ε_{f}(t)=0$. 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.