Some Problems in Stochastic Flows and Random Media

0450756Cranston PI将致力于随机流和随机介质领域的问题。流动中的问题涉及由流动携带的一组被动示踪剂的分散率。这里特别感兴趣的是弹道点的集合，这些弹道点以大大超过扩散速率的线性速率行进。PI将检查这组点的Hausdorff维数以及以超扩散速率行进的这组点的图像的结构。另一个问题是研究在水流下运动的曲线的长度。PI将试图表明，在最高李雅普诺夫指数为正的假设下，长度以大于最高李雅普诺夫指数的速率呈指数增长。PI还建议研究抛物型安德森方程在标量和矢量情况下的解。矢量情况下的问题涉及建立由湍流速度场产生的磁场的指数增长率。在标量情形下，问题涉及到对解的性质有更多的了解。 提出的问题，随机流的动机是在海洋表面或地下水中的污染物的运动。随机流是一个方便的模型，粒子的运动在洋流的影响。抛物型安德森模型的工作源于天体物理学中一个叫做发电机问题的问题。抛物型安德森方程的矢量解模拟了年轻星星的磁场。天体物理学中的一个基本问题是磁场是否以指数速度增长，并确定指数增长率。

0450756Cranston The PI will work on problems in the area of stochastic flows and random media. The problems in flows involve rates of dispersion of a set of passive tracers carried by the flow. Of special interest here are the set of ballistic points, those which travel at a linear rate that greatly exceeds the diffusive rate. The PI will examine the Hausdorff dimension of this set of points and the structure of the image of the set of points which have traveled at a superdiffusive rate. Another problem is the study of the length of a curve moving under the flow. The PI will attempt to show, under the assumption that the top Lyapunov exponent is positive, that the length grows exponentially at a rate greater than the top Lyapunov exponent. The PI also proposes to study solutions of the parabolic Anderson equation in both the scalar and vector case. The problems in the vector case involve establishing exponential growth rates for magnetic fields generated by turbulent velocity fields. In the scalar case, the problems involve gaining more insight into the properties of solutions. The proposed problems on stochastic flows are motivated by the motion of pollutants on the ocean surface or in ground water. A stochastic flow is a convenient model for the motion of particles under the effect of ocean currents. The proposed work on the parabolic Anderson model arises from a problem in astrophysics called the dynamo problem. The solution of a vector version of the parabolic Anderson equation models the magnetic field in a young star. The basic open question in astrophysics which will be approached by the PI is whether the magnetic field grows exponentially fast and to determine the exponential growth rate.