Navier-斯托克斯方程的随机拉格朗日粒子系统
A Stochastic Lagrangian particle system for the Navier-Stokes equations
摘要：
本文的工作是基于P. Constantin和G.Iyer提出的不可压缩Navier-斯托克斯方程的公式，其中粘性不可压缩流体的速度场被写成随机过程的期望值。如果我们取上述过程的$N$副本（每个副本都基于独立的维纳过程），并将期望值替换为经验平均值，则可以证明Navier-斯托克斯方程的粒子系统不会将其所有能量耗散为$t \to \infty$。与真正的（非受迫）Navier-斯托克斯方程相反，它将所有能量耗散为$t \to \infty$。这篇短文的目的是描述一个消除这一缺陷的复位过程。我们证明，如果我们重复这个重置过程足够频繁，那么新的粒子系统的Navier-斯托克斯方程耗散其所有的能量。
英文摘要：
This work is based on a formulation of the incompressible Navier-Stokes equations developed by P. Constantin and G.Iyer, where the velocity field of a viscous incompressible fluid is written as the expected value of a stochastic process. If we take $N$ copies of the above process (each based on independent Wiener processes), and replace the expected value with the empirical mean, then it was shown that the particle system for the Navier-Stokes equations does not dissipate all its energy as $t \to \infty$. In contrast to the true (unforced) Navier-Stokes equations, which dissipate all of its energy as $t \to \infty$. The objective of this short note is to describe a resetting procedure that removes this deficiency. We prove that if we repeat this resetting procedure often enough, then the new particle system for the Navier-Stokes equations dissipates all its energy.
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