Asymptotic Problems in Parabolic Equations and in Random Transport

0405152 Koralov该项目涉及抛物型偏微分方程组理论和随机运输中几个密切相关的问题。该项目的目标是研究抛物型Anderson问题解的行为，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Anderson问题，研究的重点是具有随机时变势的方程。在标量情况下，这个问题可以看作是随机流中磁场演化方程的标量模型。在向量情况下，安德森模型与随机流的被动运输有关，这也是当前项目的一个主题。输运现象的研究是指在一大类物理相关流的作用下，点集合和连通集合的长期行为。大多数提出的问题都是在气象学、海洋学和湍流理论中对各种物理现象的研究中自然产生的。特别是，在研究被动运输时，人们假设介质的某些性质是已知的(例如，虽然海洋表面的温度或速度不能准确地在每一个点上测量，但假设可以获得某些统计信息)。然后，问题包括试图基于底层介质的统计特性来预测被动标量(例如海洋表面的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，但解决方案非常重要，有时甚至令人惊讶。

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.