Critical Regularity, Selection Dynamics, and Condensation in Nonlinear Balance Laws

在许多物理现实的数学模型中，持久的模式是由竞争影响的平衡形成和维持的。这种现象在流体动力学、进化生物学、量子力学和其他学科中自然出现。例如，从湍流中形成漩涡和连贯结构到气体中的激波形成，或种群动力学中的自然选择，或量子力学中玻色-爱因斯坦凝聚（物质的一种新状态）的形成。当前项目的目标是开发新的数学工具和数值算法来分析这些竞争效应如何实现动态平衡或导致临界解行为。预计数学结果将是基本的，并有助于对一系列学科的研究人员有用的理解体系。临界正则性是流体力学中的一个基本问题，对其研究可以为更广泛应用中出现的临界阈值现象提供更深入的理解。对光子凝聚的理解可以为设计新型光源提供潜在的方法。本研究的重点是在应用和偏微分方程理论强烈驱动的领域中研究动态行为，研究目标包括非线性平衡定律中的临界规律，性状结构种群模型中的选择动力学以及光子散射中的能量输运。这些数学模型有一个显著的共同特征：潜在的耗散不足以防止有限时间奇点的形成，而动力行为的持续取决于相互竞争的力之间的微妙平衡。这些解决方案特性取决于是否跨越与初始配置和/或交互内核相关的关键阈值。将开发高阶数值算法来解决这些问题，采用统一的方法，以便在离散水平上保留几个固有的解结构。保存结构的算法在长时间模拟中捕捉正确的物理现象是很重要的。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。

In many mathematical models of physical reality, persistent patterns are formed and maintained by a balance of competing influences. Such phenomena arise naturally in fluid dynamics, evolutionary biology, quantum mechanics, and others. Examples range from forming eddies and coherent structures in turbulent flows to shock formation in gases, or natural selection in population dynamics, or formation of Bose-Einstein condensate (a new state of matter) in quantum mechanics. The goal of the current project is to develop novel mathematical tools and numerical algorithms for analyzing how such competing effects achieve dynamic balance or lead to critical solution behavior. The mathematical results are expected to be fundamental, and contribute to a body of understanding that promises to be useful to researchers across a range of disciplines. Critical regularity is a fundamental problem in fluid dynamics, the study of which can provide a deeper understanding of critical threshold phenomena occurring in wider applications. The understanding of photon condensate can result in methods which may potentially be suitable for designing novel light sources. This investigation focuses on the study of dynamic behavior in areas strongly motivated by applications and the theory of partial differential equations, with research objectives ranging from critical regularity in nonlinear balance laws, selection dynamics in trait-structured population models to energy transport in photon scattering. The mathematical models have one striking feature in common: the underlying dissipation is insufficient to prevent finite time singularity formation, and the persistence of the dynamic behavior hinges on a delicate balance among competing forces. These solution features depend on crossing a critical threshold associated with the initial configuration and/or the interaction kernels. High order numerical algorithms will be developed to solve these problems, with a unified approach so that several intrinsic solution structures are retained at the discrete level. Structure-preserving algorithms as such are important in capturing the correct physics over long time simulations.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.