论文标题
在圆环上保守散射模型的平衡速率：一种新的陶拜式方法
Convergence rate to equilibrium for conservative scattering models on the torus: a new tauberian approach
论文作者
论文摘要
本文的目的是提供一种新的，系统的Tauberian方法，以定量长时间行为的$ C_ {0} $ - semigroups $ \ left（\ Mathcal {v}（t）（t）\ right）_ {t \ geq0} $ in $ \ Mathbb {r}^{d}）$ goning在圆环上的保守线性动力学方程，一般散射核$ {k}（v，v，v，v，v，v，v，v，v，v，即不限制在零）碰撞频率$ colly频率$σ（v）= \ int = \ int _ { {k}（v'，v）m（\ mathrm {d} v'）$，（使用$ m（\ mathrm {d} v）$相对于lebesgue Measure绝对连续）。我们特别表明，如果$ n_ {0} $是最大整数$ s \ geq0 $，以至于$ \ frac {1} {σ（\ cdot）} \ int _ {\ int {\ mathbb {r}^r}^{d}^{d}}}} k（\ cdot，v） \ in L^{\ infty}（\ Mathbb {r}^{d}）$$，对于初始datum $ f $ \ Mathbb {r}^{d}} | f（x，v）|σ^{ - n_ {0}}}（v）（v）\ mathrm {d} x m（\ mathrm {d} v）<\ infty $ $$ \ left \ | \ Mathcal {v}（t）f- \ varrho_ {f}ψ\ right \ | _ {l^{1}} = \ dfrac {ε______{f}（f}（f}（t）） \ varrho_ {f}：= \ int _ {\ mathbb {r}^{d}} f（x，x，v）m（\ mathrm {d} v）$ψ$是$ψ$是$ \ weft（\ satercal {v} $ ge} us of und nount var的密度$ \ lim_ {t \ to \ infty}ε_{f}（t）= 0 $。我们特别提供了一个不变密度存在的新标准。证明依赖于$ \ Mathcal {v}（t）$的dyson-Phillips扩展的每个项的明确计算，以及痕迹的合适的平滑度和集成性属性，这些痕迹在拉普拉斯（Laplace）的假想轴上，这是dyson-phillips扩展的剩余量的拉普拉斯（Laplace）剩余者。我们的建筑措施也基于集体紧凑的论点，并提供了独立兴趣的各种技术结果。最后，作为我们分析的副产品，我们基本上得出了与一般过渡内核相关的马尔可夫半群的``子几幅''收敛速率。
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.
