Propagation of chaos for the 2D viscous vortex model
Propagation of chaos for the 2D viscous vortex model
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DOI：
10.4171/jems/465
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发表时间：
2012-12
影响因子：
2.6
通讯作者：
N. Fournier;M. Hauray;S. Mischler
中科院分区：
文献类型：
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作者：
N. Fournier;M. Hauray;S. Mischler
我们考虑一个随机系统的$N$粒子，通常被称为涡的设置，近似的二维Navier-Stokes方程写在涡。假设初始分布的位置和流通的漩涡有有限的（部分）熵和有限的时刻的正序，我们表明，粒子系统的经验措施收敛于法律的唯一的（根据适当的先验估计）解决方案的二维Navier-Stokes方程。我们实际上证明了一个稍微强一些的结果：混沌的传播的随机路径对预期的非线性随机微分方程的解决方案。此外，收敛在一个强意义上成立，通常称为熵（在极限中没有熵的损失）。该结果对粘度参数没有限制（但为正）。主要的困难是存在的奇异Biot-Savart内核的方程。为了克服这个问题，我们使用熵的耗散，它提供了粒子系统Fisher信息的一些（在$N$中均匀）界限，然后广泛使用与Fisher信息的经典和新性质结合在一起的界限。
We consider a stochastic system of $N$ particles, usually called vortices in that setting, approximating the 2D Navier-Stokes equation written in vorticity. Assuming that the initial distribution of the position and circulation of the vortices has finite (partial) entropy and a finite moment of positive order, we show that the empirical measure of the particle system converges in law to the unique (under suitable a priori estimates) solution of the 2D Navier-Stokes equation. We actually prove a slightly stronger result : the propagation of chaos of the stochastic paths towards the solution of the expected nonlinear stochastic differential equation. Moreover, the convergence holds in a strong sense, usually called entropic (there is no loss of entropy in the limit). The result holds without restriction (but positivity) on the viscosity parameter. The main difficulty is the presence of the singular Biot-Savart kernel in the equation. To overcome this problem, we use the dissipation of entropy which provides some (uniform in $N$) bound on the Fisher information of the particle system, and then use extensively that bound together with classical and new properties of the Fisher information.