ABSTRACT

In this paper, we establish the existence and uniqueness of invariant measures of the 3D stochastic magnetohydrodynamic-α model (MHD-α) driven by degenerate additive noise. We firstly study the Feller property of solutions and establish the existence of invariant measures by utilizing the classical Krylov–Bogoliubov theorem. Then, we prove the uniqueness of invariant measures for the corresponding transition semigroup by utilizing the notion of asymptotic strong Feller proposed by Hairer and Mattingly [Ergodicity of the 2D Navier–Stokes equations with degenerate stochastic forcing. Ann Math (2). 2006;164(3):993–1032]. The proof not only requires the investigation of degenerate noise, but also the study of highly nonlinear, unbounded drifts.

摘要

在本文中，我们确定了由退化加性噪声驱动的 3D 随机磁流体动力学-α模型 (MHD- α ) 的不变测度的存在性和唯一性。我们首先研究了解的 Feller 性质，并利用经典的 Krylov-Bogoliubov 定理建立了不变测度的存在性。然后，我们利用 Hairer 和 Mattingly 提出的渐近强 Feller 概念证明了相应过渡半群的不变测度的唯一性[具有退化随机强迫的二维 Navier-Stokes 方程的遍历性。安数学（2）。2006;164(3):993–1032]。证明不仅需要研究简并噪声，还需要研究高度非线性、无界的漂移。