Circulation and Energy Theorem Preserving Stochastic Fluids
Circulation and Energy Theorem Preserving Stochastic Fluids
复制标题
DOI：
10.1017/prm.2019.43
复制
发表时间：
2020-12-01
影响因子：
1.3
通讯作者：
Holm, Darryl D.
中科院分区：
文献类型：
--
作者：
Drivas, Theodore D.;Holm, Darryl D.
不可压缩欧拉方程的光滑解具有绕物质环的循环守恒的性质。这就是开尔文定理。同样，Navier-Stokes的光滑解由Constantin-Iyer（2008）引入的广义开尔文定理表征。在这篇笔记中，我们介绍了一类随机流体方程，其光滑解的特征是它们的确定性对应物的开尔文定理的自然扩展，它在某些嘈杂的流动中成立。这些方程分别称为随机欧拉-庞加莱方程和随机纳维-斯托克斯-庞加莱方程。随机欧拉-庞加莱方程之前是由Holm（2015）从随机变分原理推导出来的，我们简要回顾一下。这些方程的解一般不服从路径能量守恒/耗散。相反，我们还讨论了一类随机流体模型，其解具有能量定理，但一般不保持循环定理。
Smooth solutions of the incompressible Euler equations are characterized by the property that circulation around material loops is conserved. This is the Kelvin theorem. Likewise, smooth solutions of Navier-Stokes are characterized by a generalized Kelvin's theorem, introduced by Constantin-Iyer (2008). In this note, we introduce a class of stochastic fluid equations, whose smooth solutions are characterized by natural extensions of the Kelvin theorems of their deterministic counterparts, which hold along certain noisy flows. These equations are called the stochastic Euler-Poincare and stochastic Navier-Stokes-Poincare equations respectively. The stochastic Euler-Poincare equations were previously derived from a stochastic variational principle by Holm (2015), which we briefly review. Solutions of these equations do not obey pathwise energy conservation/dissipation in general. In contrast, we also discuss a class of stochastic fluid models, solutions of which possess energy theorems but do not, in general, preserve circulation theorems.