Some Problems in Stochastic Flows and Random Media

0450756Cranston PI 将致力于解决随机流和随机介质领域的问题。流动中的问题涉及流动所携带的一组被动示踪剂的分散率。这里特别令人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 将检查这组点的豪斯多夫维数以及以超扩散速率传播的这组点的图像结构。另一个问题是研究水流下移动的曲线的长度。 PI 将尝试表明，在顶部 Lyapunov 指数为正的假设下，长度以大于顶部 Lyapunov 指数的速率呈指数增长。 PI 还建议研究抛物线安德森方程在标量和矢量情况下的解。矢量情况中的问题涉及建立由湍流速度场产生的磁场的指数增长率。在标量情况下，问题涉及更深入地了解解决方案的属性。 所提出的随机流问题是由海洋表面或地下水中污染物的运动引起的。随机流是洋流作用下粒子运动的便捷模型。抛物线安德森模型的拟议工作源于天体物理学中称为发电机问题的问题。抛物线安德森方程的矢量解模拟了年轻恒星的磁场。 PI 将要解决的天体物理学中的基本开放问题是磁场是否以指数方式快速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.