带乘性噪声的二维Navier-Stokes方程不变测度的唯一性与渐近稳定性
Uniqueness of the invariant measure and asymptotic stability for the 2D Navier Stokes equations with multiplicative noise
AI总结：
本文在有效椭圆设定下，利用广义渐近耦合技术和期望值形式的Foias-Prodi估计，证明了带乘性噪声（有界、次线性或线性增长）的二维Navier-Stokes方程不变测度的唯一性与渐近稳定性。
摘要：
我们建立了由乘性噪声驱动的二维Navier-Stokes方程不变测度的唯一性和渐近稳定性，其中乘性噪声可以是有界的，也可以是次线性或线性增长的。我们在一个有效椭圆设定下进行研究，即要求协方差算子的值域包含不稳定方向。我们利用了Glatt Holtz、Mattingly、Richards（2017）以及Kulik、Scheutzow（2018）的广义渐近耦合技术，这些作者曾将该技术用于带加性噪声的随机Navier-Stokes方程。本文展示了这些方法具有足够的灵活性，同样能够处理乘性噪声。我们的论证中一个关键作用是期望值形式的Foias-Prodi估计，根据乘性噪声的增长条件，该估计具有不同的形式（指数衰减或多项式衰减）。
英文摘要：
We establish the uniqueness and the asymptotic stability of the invariant measure for the two dimensional Navier Stokes equations driven by a multiplicative noise which is either bounded or with a sublinear or a linear growth. We work on an effectively elliptic setting, that is we require that the range of the covariance operator contains the unstable directions. We exploit the generalized asymptotic coupling techniques of Glatt Holtz,Mattingly,Richards(2017) and Kulik,Scheutzow(2018), used by these authors for the stochastic Navier Stokes equations with additive noise. Here we show how these methods are flexible enough to deal with multiplicative noise as well. A crucial role in our argument is played by the Foias Prodi estimate in expected valued, which has a different form (exponential or polynomial decay) according to the growth condition of the multiplicative noise.
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