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.

我们将使用现代数学工具辅以新颖的计算模拟来检查正则化效应、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此类方程中的正则化效应？ (ii) 欧拉动力学：旋转和压力强迫之间的竞争如何决定整体稳定性？ (iii) 趋化性和相关生物相关的传输扩散模型：我们应该如何解释超出其关键时间的解决方案？ (iv) 图像的层次分解：逆尺度空间如何适应数据以进行有效的图像处理？首席研究员和他的合作者计划开发新的分析和计算工具来探索传输扩散模型，这有望有助于理解各种应用中现实模型的动力学。该项目的目标是研究非线性传输扩散方程中全局特征的持久性，这些方程出现在各种应用中。 例子包括具有简并扩散的非线性守恒定律，它对沉积、交通流和图像处理中的数据驱动应用进行建模；无处不在的欧拉动力学控制着从小规模的半导体到最大规模的恒星形成的一系列现象；和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。 主要关注的是关键规律性问题，这些问题取决于非线性对流机制、非线性扩散过程以及驱动此类系统的可能的非线性强迫之间的边界平衡。