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 C_{0} -semigroups ( V ( t ) ) t ⩾ 0 \left (\mathcal {V}(t)\right )_{t \geqslant 0} in L 1 ( T d × R d ) L^{1}(\mathbb {T}^{d}\times \mathbb {R}^{d}) governing conservative linear kinetic equations on the torus with general scattering kernel k ( v , v ′ ) \boldsymbol {k}(v,v’) and degenerate (i.e. not bounded away from zero) collision frequency σ ( v ) = ∫ R d k ( v ′ , v ) m ( d v ′ ) \sigma (v)=\int _{\mathbb {R}^{d}}\boldsymbol {k}(v’,v)\boldsymbol {m}(\mathrm {d}v’) , (with m ( d v ) \boldsymbol {m}(\mathrm {d}v) being absolutely continuous with respect to the Lebesgue measure). We show in particular that if N 0 N_{0} is the maximal integer s ⩾ 0 s \geqslant 0 such that 1 σ ( ⋅ ) ∫ R d k ( ⋅ , v ) σ − s ( v ) m ( d v ) ∈ L ∞ ( R d ) , \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 f such that ∫ T d × R d | f ( x , v ) | σ − N 0 ( v ) d x m ( d v ) > ∞ \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 ‖ V ( t ) f − ϱ f Ψ ‖ L 1 ( T d × R d ) = ε f ( t ) ( 1 + t ) N 0 − 1 , ϱ f ≔ ∫ R d f ( x , v ) d x m ( d v ) , \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 Ψ \Psi is the unique invariant density of ( V ( t ) ) t ⩾ 0 \left (\mathcal {V}(t)\right )_{t \geqslant 0} and lim t → ∞ ε f ( t ) = 0 \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 ( V ( t ) ) t ⩾ 0 \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.