标题： 关于玻尔兹曼方程的费舍尔信息的单调性 显示英文标题

标题： On the monotonicity of the Fisher information for the Boltzmann equation

摘要： 我们证明了在空间齐次玻尔兹曼方程的解中，费舍尔信息量随时间单调递减，这适用于覆盖所有由粒子系统导出的经典相互作用的碰撞核的大类。 对于一般的碰撞核，沿流的费舍尔信息量的单调性的一个充分条件与球面上函数的积分微分不等式的最佳常数有关，该不等式属于对数-索博列夫不等式家族。 作为结果，我们在主要感兴趣的情况下建立了空间齐次玻尔兹曼方程全局光滑解的存在性，这种情况之前并不为人所知，即非常软势能的范围。 这为完成由Carleman发起的空间齐次玻尔兹曼方程定性研究的经典计划，以及由McKean发起的利用费舍尔信息量研究玻尔兹曼方程的计划铺平了道路。 从证明和讨论中 emerges 了一个加强的关于动能理论、信息论和对数-索博列夫不等式之间联系的图景。 显示英文摘要

摘要： We prove that the Fisher information is monotone decreasing in time along solutions of the space-homogeneous Boltzmann equation for a large class of collision kernels covering all classical interactions derived from systems of particles. For general collision kernels, a sufficient condition for the monotonicity of the Fisher information along the flow is related to the best constant for an integro-differential inequality for functions on the sphere, which belongs in the family of the Log-Sobolev inequalities. As a consequence, we establish the existence of global smooth solutions to the space-homogeneous Boltzmann equation in the main situation of interest where this was not known, namely the regime of very soft potentials. This is opening the path to the completion of both the classical program of qualitative study of space-homogeneous Boltzmann equation, initiated by Carleman, and the program of using the Fisher information in the study of the Boltzmann equation, initiated by McKean. From the proofs and discussion emerges a strengthened picture of the links between kinetic theory, information theory and log-Sobolev inequalities.