arXiv 2409.176972024-09math.PRmath.AP

随机超粘性Navier-Stokes方程的无粘极限与$\mathbb R^2$上Euler方程的不变测度

Inviscid Limit of the Stochastic Hyperviscous Navier-Stokes Equations and Invariant Measures for the Euler Equations in $\mathbb R^2$

University of York（约克大学）；University of Pavia（帕维亚大学）；University of Milano Bicocca（米兰比可卡大学）

Zdzisław Brzeźniak, Matteo Ferrari

备注：: 64 pages

阅读 PDF

arXiv 页面 ↗HTML 原文 ↗

AI总结：

通过构造带随机噪声的超粘性Navier-Stokes方程的马尔可夫平稳过程，证明其无粘极限收敛到Euler方程的路径平稳解，从而得到Euler方程的不变测度及其矩估计。

摘要：

我们证明了在无界域$\mathbb R^2$上、具有高度正则初始条件的二维确定性Euler方程存在不变测度$\mu$及其矩估计。该结果首先通过证明存在马尔可夫平稳过程来实现，这些过程求解了运动粘度$\nu>0$且附加随机噪声尺度为$\sqrt \nu$的超粘性二维Navier-Stokes方程。然后我们研究无粘极限，并证明当$\nu$趋于$0$时，这些过程在适当的轨迹空间中收敛到Euler方程的路径平稳解。其律即为所求的不变测度$\mu$。

英文摘要：

We prove the existence and some moment estimates for an invariant measure $μ$ for the two-dimensional ($2$D) deterministic Euler equations on the unbounded domain $\mathbb R^2$ and with highly regular initial data. The result is achieved by first showing the existence of Markov stationary processes which solve the hyperviscous $2$D Navier-Stokes equations with kinematic viscosity $ν>0$ and an additive stochastic noise scaling as $\sqrt ν$. We then study the inviscid limit and prove that, as $ν$ tends to $0$, these processes converge, in an appropriate trajectory space, to a pathwise stationary solution to the Euler equations. Its law is the sought invariant measure $μ$.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

AI总结：

摘要：

英文摘要：

相关论文