论文标题
时间的最佳最小型界限，用于不可压缩的Navier-Stokes方程的合奏平均
Optimal minimax bounds for time and ensemble averages for the incompressible Navier-Stokes equations
论文作者
论文摘要
获得给定模型涉及的数量的尖锐估计是建模过程的组成部分。对于动态系统的轨道表现出复杂的，也许是混乱的行为，其目的通常是估算给定数量的时间或集合平均值。例如，在湍流中就是这种情况。在这项工作中，目的是提出一个最小值优化公式，该公式为两维和三维的Navier-Stokes方程提供时间和/或集合平均值的最佳界限。此处介绍的结果是针对2017年烟草，Goluskin和Doering给出的有限维情况结果的无限维度设置的扩展。最佳结果以微型型优化问题的形式出现，并且不需要对解决方案的解决方案（仅需要该系统法律）的知识。最小值优化问题以最大化的形式在系统的相位空间的一部分上出现，并在相位空间上定义的圆柱测试功能制成的辅助函数家族上最小化。要优化的功能是系统定律与辅助函数的导数之间的二元乘积。
Obtaining sharp estimates for quantities involved in a given model is an integral part of the modeling process. For dynamical systems whose orbits display a complicated, perhaps chaotic, behaviour, the aim is usually to estimate time or ensemble averages of given quantities. This is the case, for instance, in turbulent flows. In this work, the aim is to present a minimax optimization formula that yields optimal bounds for time and/or ensemble averages for the two- and three-dimensional Navier-Stokes equations. The results presented here are extensions to the infinite-dimensional setting of a recent result on the finite-dimensional case given by Tobasco, Goluskin, and Doering in 2017. The optimal result occurs in the form of a minimax optimization problem and does not require knowledge of the solutions, only the law of the system. The minimax optimization problem appears in the form of a maximization over a portion of the phase space of the system and a minimization over a family of auxiliary functions made of cylindrical test functionals defined on the phase space. The function to be optimized is the desired quantity plus the duality product between the law of the system and the derivative of the auxiliary function.
