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

非线性平衡定律中的临界正则性、选择动力学和凝聚

基本信息

批准号：
1812666
负责人：
Hailiang Liu
金额：
$ 24.5万
依托单位：
Iowa State University
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2018
资助国家：
美国
起止时间：
2018-08-01 至 2023-07-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1812666&HistoricalAwards=false
关键词：
Critical Regularity Selection Dynamics Condensation

项目摘要

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.

在物理现实的许多数学模型中，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、量子力学等领域自然出现。例子包括在湍流中形成涡流和相干结构，在气体中形成激波，或在人口动力学中的自然选择，或在量子力学中形成玻色-爱因斯坦凝聚（一种新的物质状态）。当前项目的目标是开发新型数学工具和数值算法，用于分析此类竞争效应如何实现动态平衡或导致临界解行为。数学结果预计将是基本的，并有助于一个身体的理解，承诺是有用的研究人员在一系列学科。临界规律性是流体力学中的一个基本问题，对它的研究可以使人们更深入地了解在更广泛的应用中出现的临界阈值现象。对光子凝聚体的理解可以产生可能适用于设计新型光源的方法。本研究的重点是在强烈的应用和偏微分方程理论的动机领域的动力学行为的研究，从非线性平衡律的临界规律性，选择动力学特征结构的人口模型中的光子散射的能量传输的研究目标。这些数学模型有一个共同的显著特征：潜在的耗散不足以防止有限时间奇点的形成，而动力学行为的持续性取决于相互竞争的力量之间的微妙平衡。这些解决方案的功能取决于跨越与初始配置和/或相互作用内核相关联的临界阈值。高阶数值算法将被开发来解决这些问题，与一个统一的方法，使几个内在的解决方案结构被保留在离散水平。该奖项反映了NSF的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。