Long Time Dynamics in Combustion, Mixing, and Fluids Models

这个项目的目的是更好地理解几个重要的现实世界现象的数学模型，包括反应过程的传播，流体的湍流运动，以及通过这种运动来混合物质。反应性过程(例如森林火灾)经常发生在不同环境(树木、灌木丛、草地)的拼凑中，过程通过这些环境以不同的速度传播。均质化理论旨在更好地理解大尺度(长距离和时间)过程的动态如何依赖于环境中较小的尺度变化。该项目这一部分的主要目标是证明，由于对大区域进行平均，可以在很大程度上自信地预测过程的长期行为，而不需要关于环境组成的过于详细的信息。通过流体运动进行混合在合金等材料的生产以及化学反应等过程的强化中至关重要。因此，量化流动的混合效率并确定在这方面表现突出的流动是非常重要的。PI最近建造了通用搅拌机，这种搅拌机在混合方面特别有效。它们的结构相对简单，但对于实际应用来说可能太不规则了。该项目这一部分的主要目标是确定更常规的通用混合器，以及研究这种快速混合如何影响扩散过程(例如，热传输)。即使不考虑它对物质混合的影响，我们对流体运动的理解也远远不能令人满意。研究流体中湍流的开始以及与之相关的小尺度结构的形成在数学、物理和工程中都是重要的，尽管近年来它已经取得了很大的进展，但许多基本问题仍然没有得到解决。在项目的这一部分，PI建议研究流体中湍流结构的发展速度和严重程度，包括在流体的主体和规则和不规则边界(流体容器的壁)附近。燃烧、混合和流体湍流等物理过程，其研究动机是线性和非线性偏微分方程组，包括反应扩散方程、传输方程、漂移扩散方程和流体动力学方程。这个由三个部分组成的项目的主要焦点是研究这些方程的解的长期动力学以及它们在有限时间内可能形成的奇性。该项目第一部分的目的是了解在非均质介质中传播的反应过程的大尺度行为，包括获得多维随机介质中这些模型的满意的齐次化理论。该项目第二部分的目的是研究流动的混合效率，并寻找足够规则的通用混合器，即无论后者的初始配置如何，都能高效地混合它们所流出的物质的流动。我们还将研究流动诱导混合对漂移扩散方程中扩散的强化作用。该项目第三部分的目的是研究二维欧拉中的湍流和相关的流体动力学方程，包括流体主体中解的梯度的快速增长，以及在更奇异的模型中可能形成有限时间奇点。欧拉方程在具有不规则边界的平面区域中的适定性问题也将被解决。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

The aim of this project is a better understanding of mathematical models of several important real world phenomena, including propagation of reactive processes, turbulent motion of fluids, and mixing of substances by such motion. Reactive processes (e.g., forest fires) frequently occur in a patchwork of different environments (trees, brush, grass), through which the process propagates at different rates. The theory of homogenization aims to better understand how the dynamics of the process over large scales (long distances and times) depends on smaller scale variations in the environment. The main goal of this part of the project is to demonstrate that thanks to averaging over large regions, the long term behavior of the process can be predicted with a large degree of confidence without the need for overly detailed information about the composition of the environment. Mixing via fluid motion is crucial in the production of materials such as alloys, as well as in enhancement of processes such as chemical reactions. Quantifying mixing efficiency of flows and identifying those that stand out in this respect is therefore of great importance. The PI recently constructed universal mixers, flows that are particularly efficient in mixing. These have a relatively simple structure but may be too irregular for practical applications. The main goal of this part of the project is to identify more regular universal mixers, as well as to study how such fast mixing affects diffusive processes (e.g., heat transport). Even without considering its effects on mixing of substances, our understanding of the motion of fluids is still far from satisfactory. The study of onset of turbulence and the associated creation of small scale structures in fluids is important in mathematics as well as in physics and engineering, and while it has seen great progress in recent years, many fundamental questions remain unresolved. In this part of the project, the PI proposes to study how fast the development of turbulent structures in fluids can be and how severe this turbulence can become, both in the bulk of the fluid and in the vicinity of regular as well as irregular boundaries (walls of the fluid container).Physical processes such as combustion, mixing, and fluid turbulence, whose study motivates this project, are modeled by linear and nonlinear partial differential equations, including reaction-diffusion equations, transport equations, drift-diffusion equations, and equations of fluid dynamics. The primary focus of this project, which consists of three parts, is the study of long term dynamics of the solutions of these equations as well as their possible formation of singularities in finite time. The aim of the first part of the project is the understanding of large scale behavior of reactive processes spreading through heterogeneous media, including obtaining a satisfactory homogenization theory for these models in multi-dimensional random media. The aim of the second part of the project is the study of mixing efficiency of flows and the search for sufficiently regular universal mixers, flows that are highly efficient in mixing of substances advected by them regardless of the initial configuration of the latter. Enhancement of diffusion in drift-diffusion equations via flow-induced mixing will also be studied. The aim of the third part of the project is the study of turbulence in two-dimensional Euler and related equations of fluid dynamics, including rapid growth of gradients of solutions in the bulk of the fluid and possible formation of finite time singularities in more singular models. The question of well-posedness of Euler equations in planar domains with irregular boundaries will also be addressed.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.