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传输中的几个密切相关的问题。该项目的目标是研究抛物线安德森问题的解的行为、随机流中磁场演化方程的解以及输运现象的各种概率方面。对于安德森问题，研究重点是具有随机时间相关势的方程。在标量情况下，这个问题可以看作随机流中磁场演化方程的标量模型。在矢量情况下，安德森模型与随机流的被动运输相关，这也是当前项目的主题。传输现象的研究涉及点集合以及连接集在一大类物理相关流的作用下的长期行为。 大多数提出的问题在气象学、海洋学和湍流理论中的各种物理现象的研究中自然出现。特别是，在研究被动传输时，人们假设介质的某些属性是已知的（例如，虽然无法在每个点精确测量海洋表面的温度或速度，但假设某些统计信息是已知的）可用的）。然后，问题包括尝试根据底层介质的统计特性来预测被动标量（例如海洋表面洋流携带的石油泄漏）的长期行为。几个这样的问题可以用相对简单的术语来表述，但解决方案却非常重要，有时甚至令人惊讶。