Sharp Entropy Dissipation Bounds and Explicit Rate of Trend to Equilibrium for the Spatially Homogeneous Boltzmann Equation
Sharp Entropy Dissipation Bounds and Explicit Rate of Trend to Equilibrium for the Spatially Homogeneous Boltzmann Equation
复制标题
空间齐次玻尔兹曼方程的锐熵耗散界和显式平衡趋势率
DOI：
10.1007/s002200050631
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发表时间：
1999
影响因子：
2.4
通讯作者：
C. Villani
中科院分区：
文献类型：
--
作者：
G. Toscani;C. Villani
摘要：导出了空间齐次玻尔兹曼方程熵耗散的一个新的下界。这个界限是用相对于平衡的相对熵来表示的，从而产生了一个微分不等式，它证明了相对熵向平衡的收敛，具有明确的速率。我们的结果对Carlen和Carvalho[9,10]的类似估计给出了相当大的改进，在很少的额外假设下。我们的证明利用了玻尔兹曼碰撞算子关于张量积的结构，以及它与福克-普朗克方程和朗道方程的联系。讨论了几种变体。
Abstract:We derive a new lower bound for the entropy dissipation associated with the spatially homogeneous Boltzmann equation. This bound is expressed in terms of the relative entropy with respect to the equilibrium, and thus yields a differential inequality which proves convergence towards equilibrium in relative entropy, with an explicit rate. Our result gives a considerable refinement of the analogous estimate by Carlen and Carvalho [9, 10], under very little additional assumptions. Our proof takes advantage of the structure of Boltzmann's collision operator with respect to the tensor product, and its links with Fokker–Planck and Landau equations. Several variants are discussed.