Singularities and statistics in nonlinear PDE

非线性偏微分方程中的奇异性和统计量

• 批准号: 0202531
• 负责人: Peter Constantin
• 金额: $ 19.96万
• 依托单位: University of Chicago
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2002
• 资助国家: 美国
• 起止时间: 2002-07-01 至 2005-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0202531&HistoricalAwards=false
• 关键词: Singularities; statistics; nonlinear; PDE

DMS Award AbstractAward #: 0202531PI: Constantin, Peter Institution: University of ChicagoProgram: Applied MathematicsProgram Manager: Catherine MavriplisTitle: Singularities and Statistics in Nonlinear Partial Differential EquationsA first goal of this proposal is to study mathematically singularities in fluids and plasmas, and their connection to physical processes. Specific problems include free surface pinch-off in fluids, geometric depletion of singularities in active scalars, incompressible Euler and MHD equations related to vortex and magnetic reconnection. A second goal of the proposal is to study statistics of long time behavior of dissipative systems, such as thermal fluid convection. Specific goals include bounds on heat transport in convection, energy cascades, coarse inertial manifolds, and turbulent front propagation.Fluids and plasmas involve many active length and time scales that interact dynamically. The long time behavior of complex systems, such as turbulent convection, involves also many interacting spatial and temporal scales. Mathematical studies are needed to guide computer simulations. What quantities are well behaved, and can be safely predicted with rather inexpensive, coarse computations? What conditions produce rapid generation of small scales? Which of these are important and result in qualitative changes? What are the characteristics of the ensuing dissipative energy transfer? These are fundamental questions for progress in understanding and computation of complex phenomena. This proposal is aimed at a core of such questions.Date: April 12, 2002

DMS 奖摘要奖项编号：0202531PI：Constantin, Peter 机构：芝加哥大学项目：应用数学项目经理：Catherine Mavriplis 标题：非线性偏微分方程中的奇点和统计该提案的第一个目标是研究流体和等离子体中的数学奇点及其与物理过程的联系。 具体问题包括流体中的自由表面夹断、活动标量中奇点的几何损耗、与涡旋和磁重联相关的不可压缩欧拉和 MHD 方程。 该提案的第二个目标是研究耗散系统的长期行为的统计数据，例如热流体对流。具体目标包括对流热传输、能量级联、粗惯性流形和湍流前沿传播的界限。流体和等离子体涉及许多动态相互作用的活动长度和时间尺度。复杂系统的长期行为，例如湍流对流，还涉及许多相互作用的空间和时间尺度。需要数学研究来指导计算机模拟。哪些量表现良好，并且可以通过相当便宜的粗略计算来安全地预测？什么条件会产生小鳞片的快速产生？其中哪些是重要的并且会带来质的变化？随后的耗散能量转移有哪些特征？这些是理解和计算复杂现象的进步的基本问题。这个提案正是针对此类问题的核心。日期：2002年4月12日