Regularity and Critical Thresholds Phenomena in Nonlinear Balance Laws

非线性平衡定律中的规律性和临界阈值现象

• 批准号: 0407704
• 负责人: Eitan Tadmor
• 金额: $ 41.97万
• 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: 2004
• 资助国家: 美国
• 起止时间: 2004-08-01 至 2007-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0407704&HistoricalAwards=false
• 关键词: Regularity; Critical; Thresholds; Phenomena; Nonlinear

The overall goal of this project is the study of critical regularity associated with different nonlinear balance laws. We focus on borderline cases, of interest in various applications, where intrinsic features of the solutions such as smoothness vs. generic finite-time breakdown, time decay, etc., hinge on a delicate balance between nonlinear convection and a variety of (possibly nonlinear) forcing mechanisms. Often this balance is maintained by global invariants of the flow. These include spectral invariants, which in turn lead to critical threshold phenomena, or borderline invariant regularity spaces. A main focal topic of this project is the study of balance laws governed by Eulerian dynamics. Such models show up in different contexts dictated by different forcing. Previous research developed a precise framework for studying the critical threshold phenomena for such Eulerian dynamics. Here we seek extensions to include more realistic models driven by global and non-isotropic forcing. In particular, this project investigates global regularity for multidimensional Euler-Poisson equations with sub-critical initial data and persistence of minimal regularity for Euler and Navier-Stokes equations with initial configurations in borderline regularity spaces.This research project pursues the development of mathematical tools for the understanding of fundamental properties of fluid flow and other dynamical processes. The results will have potential application for the modeling and numerical simulation of a variety of physical phenomena, including the turbulent flow of fluids, that are governed by the type of partial differential equations under study.

该项目的总体目标是研究与不同非线性平衡定律相关的临界规律。 我们关注各种应用中感兴趣的边界情况，其中解决方案的内在特征（例如平滑性与通用有限时间击穿、时间衰减等）取决于非线性对流和各种（可能是非线性）强迫机制之间的微妙平衡。 通常，这种平衡是由流量的全局不变量来维持的。 其中包括谱不变量，这反过来又导致临界阈值现象，或边界不变量正则空间。 该项目的一个主要焦点主题是研究欧拉动力学控制的平衡定律。 这些模型出现在由不同强迫决定的不同背景中。 先前的研究开发了一个精确的框架来研究这种欧拉动力学的临界阈值现象。 在这里，我们寻求扩展以包括由全局和非各向同性强迫驱动的更现实的模型。 特别是，该项目研究具有亚临界初始数据的多维欧拉-泊松方程的全局正则性，以及具有边界正则空间中初始配置的欧拉和纳维-斯托克斯方程的最小正则性的持久性。该研究项目致力于开发数学工具，以理解流体流动和其他动态过程的基本属性。 研究结果将潜在地应用于各种物理现象的建模和数值模拟，包括受所研究的偏微分方程类型控制的流体湍流。