Regularity and Critical Thresholds in Nonlinear Transport-Diffusion Equations

非线性传输扩散方程的规律性和临界阈值

• 批准号: 0707949
• 负责人: Eitan Tadmor
• 金额: $ 39.81万
• 依托单位: University of Maryland, College Park
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2007
• 资助国家: 美国
• 起止时间: 2007-07-01 至 2010-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0707949&HistoricalAwards=false
• 关键词: Regularity; Critical; Thresholds; Nonlinear; Transport

We will use modern mathematical tools complemented by novel computational simulations to examine the phenomena of regularizing effects, critical thresholds, time decay, entropy stability, and scaling. The project will address the following questions: (i) Transport-diffusion equations: How do diffusion and entropy dissipation dictate the regularizing effect in such equations? (ii) Eulerian dynamics: How does the competition between rotation and pressure forcing determine the overall stability? (iii) Chemotaxis and related bio-related transport-diffusion models: How should we interpret the solutions beyond their critical time? (iv) Hierarchical decompositions of images: How can the inverse scale space be adapted to the data for effective image processing? The principal investigator and his collaborators plan to pursue the development of new analytical and computational tools to explore transport-diffusion models, which are expected to contribute to understanding of the dynamics of realistic models in a variety of applications. The goal of this project is to study the persistence of global features in nonlinear transport-diffusion equations, which arise in a wide variety of applications. Examples include nonlinear conservation laws with degenerate diffusion, which model sedimentation, traffic flows, and data-driven applications in image processing; the ubiquitous Eulerian dynamics governing a range of phenomena from the small scale of semi-conductors through the largest scale of star formation; and chemotaxis models found in biological applications. We focus our attention on the unifying mathematical content of the underlying transport-diffusion equations. Of primary interest are problems with critical regularity properties that hinge on a borderline balance between the nonlinear convection mechanisms, the nonlinear diffusion processes, and the possibly nonlinear forcing driving such a system.

我们将使用新型计算模拟补充的现代数学工具来检查正规效果，临界阈值，时间衰减，熵稳定性和缩放的现象。 该项目将解决以下问题：（i）传输扩散方程：扩散和熵耗散如何决定这种方程式的正则效果？ （ii）欧拉动力学：旋转与压力强迫之间的竞争如何确定整体稳定性？ （iii）趋化性及相关的与生物相关的运输扩散模型：我们应该如何解释解决方案以外的关键时期？ （iv）图像的层次分解：如何将反向尺度空间适应有效图像处理的数据？首席研究员及其合作者计划追求开发新的分析和计算工具来探索运输扩散模型，这些模型有望有助于理解各种应用中现实模型的动态。该项目的目的是研究非线性传输扩散方程中全球特征的持久性，这些方程在各种应用中产生。 示例包括具有退化扩散的非线性保护定律，该法律模拟了图像处理中的沉积，交通流和数据驱动的应用；无处不在的Eulerian动力学，涉及一系列现象，从半导体的小规模到恒星形成的最大规模；以及在生物应用中发现的趋化模型。 我们将注意力集中在基础传输扩散方程的统一数学内容上。 主要感兴趣的是具有关键规则性能的问题，这些问题在非线性对流机制，非线性扩散过程和可能的非线性强迫驱动这种系统之间取决于边界平衡。