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，以及由这种运动引起的物质混合。反应过程（例如，森林火灾）经常发生在不同环境（树木、灌木、草地）的拼凑中，在这些环境中，反应过程以不同的速率传播。均质化理论旨在更好地理解大尺度（长距离和时间）过程的动力学如何取决于环境中较小尺度的变化。这部分项目的主要目标是证明，由于对大区域进行平均，过程的长期行为可以在很大程度上有信心地预测，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对于合金等材料的生产以及化学反应等过程的增强至关重要。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最近建造了通用混合器，在混合中特别有效的流动。它们的结构相对简单，但在实际应用中可能过于不规则。这部分项目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仍然远远不能令人满意。研究流体中湍流的发生和相关的小尺度结构的产生在数学、物理和工程中都是重要的，虽然近年来取得了很大的进展，但许多基本问题仍未解决。在项目的这一部分中，PI建议研究流体中湍流结构的发展速度有多快，以及这种湍流在流体中以及在规则和不规则边界（流体容器的壁）附近会变得有多严重。燃烧、混合和流体湍流等物理过程的研究激发了本项目，这些物理过程采用线性和非线性偏微分方程建模，包括反应扩散方程、输运方程、漂移扩散方程和流体动力学方程。这个项目的主要重点，由三个部分组成，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点。项目第一部分的目的是了解反应过程在异质介质中传播的大规模行为，包括在多维随机介质中为这些模型获得令人满意的均质化理论。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，无论后者的初始配置如何，都能高效混合由它们平流的物质的流动。通过流动诱导混合增强漂移扩散方程中的扩散也将被研究。该项目第三部分的目的是研究二维欧拉方程和流体动力学相关方程中的湍流，包括在大部分流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的有限时间奇点。讨论了具有不规则边界的平面域上欧拉方程的适定性问题。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值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.