标题： 不可压缩最优传输及其在流体混合中的应用 显示英文标题

标题： Incompressible Optimal Transport and Applications in Fluid Mixing

摘要： 不可压缩流体混合的问题在许多工程应用中出现，并且多年来已经得到了充分的研究，但仍然存在许多未解的问题。 本文旨在解决这样一个问题：“高效的混合流场看起来是什么样的，它们是如何行为的？” 我们通过开发一个框架来解决这个问题，该框架受到最优质量传输的动力学和几何方法的启发。 具体来说，我们将流体混合问题表述为一个最优控制问题，其中动力学由连续性方程和不可压缩性约束给出。 我们证明，在这个框架下，可达的流体配置集可以形式上赋予无限维黎曼流形的结构，该结构由控制努力所诱导，而混合效率最高的流场则对应于这个黎曼空间中的测地线。 显示英文摘要

摘要： The problem of incompressible fluid mixing arises in numerous engineering applications and has been well-studied over the years, yet many open questions remain. This paper aims to address the question "what do efficient flow fields for mixing look like, and how do they behave?" We approach this question by developing a framework which is inspired by the dynamic and geometric approach to optimal mass transport. Specifically, we formulate the fluid mixing problem as an optimal control problem where the dynamics are given by the continuity equation together with an incompressibility constraint. We show that within this framework, the set of reachable fluid configurations can formally be endowed with the structure of an infinite-dimensional Riemannian manifold, with a metric which is induced by the control effort, and that flow fields which are maximally efficient at mixing correspond to geodesics in this Riemannian space.