Asymptotic Problems in Parabolic Equations and in Random Transport

抛物方程和随机传输中的渐近问题

• 批准号: 0405152
• 负责人: Leonid Koralov
• 金额: $ 11万
• 依托单位: Princeton University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2004
• 资助国家: 美国
• 起止时间: 2004-09-01 至 2007-09-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0405152&HistoricalAwards=false
• 关键词: Asymptotic; Problems; Parabolic; Equations; Random

0405152Koralov The project concerns several closely related problems in the theory of parabolic partial differential equations and in random transport. The goal of the project is to investigate the behavior of solutions to the parabolic Anderson problem, the solutions to the equation of the evolution of a magnetic field in a random flow, and a variety of probabilistic aspects of transport phenomena. For the Anderson problem the study focuses on the equations with random time-dependent potential. In the scalar case this problem can be looked upon as a scalar model for the equation of the evolution of the magnetic field in a random flow. In the vector case the Anderson model is related to passive transport by random flows, which is also a subject of the current project. The study of transport phenomena is concerned with the long-time behavior of ensembles of points as well as connected sets under the action of a large class of physically relevant flows. Most of the proposed problems arise naturally in the study of various physical phenomena in meteorology, oceanography, and the theory of turbulence. In particular, when studying passive transport, one assumes that certain properties of the media are known (for example, while the temperatures or velocities on the surface of the ocean can not be measured in every single point exactly, certain statistical information is assumed to be available). The problem then consists of trying to predict the long-time behavior of a passive scalar (such as an oil spill carried by the currents on the surface of the ocean) based on the statistical properties of the underlying media. Several such problems can be formulated in relatively simple terms, yet the solutions are very non-trivial, and at times surprising.

0405152Koralov该项目涉及抛物型偏微分方程理论和随机运输中的几个密切相关的问题。该项目的目标是研究抛物型安德森问题的解决方案的行为，在一个随机流的磁场演化方程的解决方案，以及各种概率方面的运输现象。对于安德森问题，主要研究具有随机含时势的方程。在标量的情况下，这个问题可以看作是一个标量模型的磁场在随机流的演变方程。在矢量情况下，安德森模型与随机流的被动输运有关，这也是当前项目的主题。输运现象的研究关注的是在一大类物理相关流的作用下，点系综以及连通集的长时间行为。 大多数提出的问题自然出现在气象学、海洋学和湍流理论中各种物理现象的研究中。特别是，在研究被动输运时，人们假设介质的某些特性是已知的（例如，虽然海洋表面的温度或速度不能在每一个点精确测量，但假设某些统计信息是可用的）。然后，这个问题包括试图根据底层介质的统计特性预测被动标量的长期行为（例如海洋表面海流携带的石油泄漏）。几个这样的问题可以用相对简单的术语来表述，但解决方案是非常重要的，有时令人惊讶。