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 $\left (\mathcal {V}(t)\right )_{t \geqslant 0}$ in $L^{1}(\mathbb {T}^{d}\times \mathbb {R}^{d})$ governing conservative linear kinetic equations on the torus with general scattering kernel $\boldsymbol {k}(v,v’)$ and degenerate (i.e. not bounded away from zero) collision frequency $\sigma (v)=\int _{\mathbb {R}^{d}}\boldsymbol {k}(v’,v)\boldsymbol {m}(\mathrm {d}v’)$, (with $\boldsymbol {m}(\mathrm {d}v)$ being absolutely continuous with respect to the Lebesgue measure). We show in particular that if $N_{0}$ is the maximal integer $s \geqslant 0$ such that \begin{equation*} \frac {1}{\sigma (\cdot )}\int _{\mathbb {R}^{d}}\boldsymbol {k}(\cdot ,v)\sigma ^{-s}(v)\boldsymbol {m}(\mathrm {d}v) \in L^{\infty }(\mathbb {R}^{d}), \end{equation*} then, for initial datum $f$ such that $\displaystyle \int _{\mathbb {T}^{d}\times \mathbb {R}^{d}}|f(x,v)|\sigma ^{-N_{0}}(v)\mathrm {d}x\boldsymbol {m}(\mathrm {d}v) <\infty$ it holds \begin{equation*} \left \|\mathcal {V}(t)f-\varrho _{f}\Psi \right \|_{L^{1}(\mathbb {T}^{d}\times \mathbb {R}^{d})}=\dfrac {{\varepsilon }_{f}(t)}{(1+t)^{N_{0}-1}}, \qquad \varrho _{f}≔\int _{\mathbb {R}^{d}}f(x,v)\mathrm {d}x\boldsymbol {m}(\mathrm {d}v), \end{equation*} where $\Psi$ is the unique invariant density of $\left (\mathcal {V}(t)\right )_{t \geqslant 0}$ and $\lim _{t\to \infty }{\varepsilon }_{f}(t)=0$. 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 $\left (\mathcal {V}(t)\right )_{t \geqslant 0}$ 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.