Singularities and statistics in nonlinear PDE

DMS奖摘要奖#：0202531PI：康斯坦丁，彼得研究所：芝加哥大学项目：应用数学项目经理：凯瑟琳·马夫里普利斯题目：非线性偏微分方程式中的奇点和统计这项建议的第一个目标是研究流体和等离子体中的数学奇点，以及它们与物理过程的联系。具体问题包括流体中的自由面夹断、有源标量中奇点的几何耗尽、与涡旋和磁重联有关的不可压缩欧拉方程和MHD方程。该提案的第二个目标是研究耗散系统的长时间行为的统计数据，例如热流体对流。具体目标包括对流、能量级联、粗惯性流形和湍流锋传播中的热传输的界限。流体和等离子体涉及许多动态相互作用的有效长度和时间尺度。复杂系统的长时间行为，如湍流对流，也涉及到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，并且可以用相当便宜的粗略计算来安全地预测？是什么条件产生了小尺度的快速生成？其中哪些是重要的，并会带来质的变化？随之而来的耗散能量转移的特征是什么？这些都是在理解和计算复杂现象方面取得进展的基本问题。这项提议就是针对这些问题的核心。日期：2002年4月12日

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