发表机构
University of California, Santa Barbara（加州大学圣巴巴拉分校）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文通过最优控制框架研究不可压缩流体混合，提出几何方法，推导最优性条件、测地线与梯度流，并指出涡量度量优于动能，连接至EPDiff方程。
AI 中文摘要
不可压缩流体混合问题已被长期研究，但许多问题仍未解决。本文旨在回答“高效的混合流场是什么样的，它们如何表现？”这一问题。受最优质量输运的动态与几何方法启发，我们将流体混合问题的一种形式表述为一个用于被动标量输运的最优控制问题，其中动力学由连续性方程及不可压缩约束给出。然后，我们围绕该问题发展了一个与经典最优输运平行的几何框架：该最优控制问题在可达状态集上诱导出一个度量（类似于Wasserstein度量），并且当最优控制目标由内积诱导时，该度量具有形式上的黎曼结构。利用这一结构，我们将问题分解为一系列更基本的几何问题，并推导出最优性的（形式上的）必要条件、测地线方程以及混合的梯度流。将控制目标特化为动能时，可恢复不可压缩Euler方程，从而将我们的框架与Arnold的几何流体动力学联系起来。然而，由此产生的问题似乎是适定性的，这表明动能是混合努力的一种不自然度量。相反，特化为涡量（我们认为它更适合混合）时，则得到体积保持微分同胚群上右不变齐次$\dot{H}^1$度量的Euler-Poincaré（EPDiff）测地线方程，并且我们在此情况下展示了测地线和梯度流的数值模拟。在此过程中，我们推广了经典的Helmholtz向量场分解，重新推导了不可压缩流的Clebsch表示，并发展了与最优输运的Otto演算平行的理论。
英文摘要
The problem of incompressible fluid mixing is long-studied, yet many questions remain open. This paper aims to address the question ``what do efficient flow fields for mixing look like, and how do they behave?'' Inspired by the dynamic and geometric approach to optimal mass transport, we formulate a version of the fluid mixing problem as an optimal control problem for transport of a passive scalar, in which the dynamics are given by the continuity equation together with an incompressibility constraint. We then develop a geometric framework around this problem which parallels that of classical optimal transport: the optimal control problem induces a metric on the set of reachable states (analogous to the Wasserstein metric), and this metric carries a formal Riemannian structure when the optimal control objective is induced by an inner product. Using this structure, we decompose our problem into a series of more fundamental geometric problems and derive (formal) necessary conditions for optimality, geodesic equations, and gradient flows for mixing. Specializing the control objective to the kinetic energy recovers the incompressible Euler equation, connecting our framework to the geometric hydrodynamics of Arnold. The resulting problem appears to be ill-posed, however, suggesting that the kinetic energy is an unnatural measure of mixing effort. Specializing instead to the enstrophy, which we argue is better suited to mixing, yields the Euler-Poincaré (EPDiff) geodesic equation for the right-invariant homogeneous $\dot{H}^1$ metric on the group of volume-preserving diffeomorphisms, and we present numerical simulations of both geodesics and gradient flows in this case. Along the way, we generalize the classical Helmholtz decomposition of vector fields, rederive the Clebsch representation of incompressible flows, and develop a parallel to the Otto calculus of optimal transport.
Comments54 pages, 6 figures
