论文标题
双空间随机吸引子，随机型定理和整个空间上随机navier-stokes方程的真实性
Bi-spatial random attractors, a stochastic Liouville type theorem and ergodicity for stochastic Navier-Stokes equations on the whole space
论文作者
论文摘要
本文涉及众所周知的数学模型，Navier-Stokes方程的随机动力学和渐近分析。我们考虑由\ textsl {线性乘法白噪声}驱动的二维随机Navier-Stokes方程（SNSE）。首先，我们证明非自主2D SNSE会生成双空间$（\ mathbb {l}^2（\ Mathbb {r}^2），\ Mathbb {h}^1（\ Mathbb {r}^2）由于与SNSE相关的随机共生的双空间连续性属性，我们表明，如果初始数据位于$ \ Mathbb {l}^2（\ Mathbb {r}^2）$中，则存在一个唯一的双空间$（\ MATHBB {l}^2（\ Mathbb {r}^2），\ Mathbb {h}^1（\ Mathbb {r}^2））$ - 非自主snse的回拔随机吸引子，它是紧凑且不仅在$ \ m i \ mathbb {l Mathbb {2 $ -Norm中，而且在$ \ mathbb {h}^1 $ -Norm。接下来，由于存在回调随机吸引子，我们证明了由2D非自治SNSE产生的非自主随机动力学系统的不变样品度量的存在。此外，我们表明，不变样本量度的家族满足随机的liouville型定理。最后，我们讨论了与2D自主snSE相关的随机合过程的不变度度量的存在。我们通过使用噪声系数的线性乘法结构和解决方案的指数稳定性的线性乘法结构，证明了$ \ boldsymbol {f} = \ mathbf {0} $的不变度唯一性。在$ \ mathbb {r}^2 $上定义的SNSE的上述结果是全新的，尤其是双空间随机吸引子和随机Liouville Type type type type type typer type type type type snse，该定理首次以任何类型的域获得具有线性乘法噪声的2D SNSE。
This article concerns the random dynamics and asymptotic analysis of the well known mathematical model, the Navier-Stokes equations. We consider the two-dimensional stochastic Navier-Stokes equations (SNSE) driven by a \textsl{linear multiplicative white noise of Itô type} on the whole space $\mathbb{R}^2$. Firstly, we prove that the non-autonomous 2D SNSE generates a bi-spatial $(\mathbb{L}^2(\mathbb{R}^2),\mathbb{H}^1(\mathbb{R}^2))$-continuous random cocycle. Due to the bi-spatial continuity property of the random cocycle associated with SNSE, we show that if the initial data is in $\mathbb{L}^2(\mathbb{R}^2)$, then there exists a unique bi-spatial $(\mathbb{L}^2(\mathbb{R}^2),\mathbb{H}^1(\mathbb{R}^2))$-pullback random attractor for non-autonomous SNSE which is compact and attracting not only in $\mathbb{L}^2$-norm but also in $\mathbb{H}^1$-norm. Next, as a consequence of the existence of pullback random attractors, we prove the existence of a family of invariant sample measures for non-autonomous random dynamical system generated by 2D non-autonomous SNSE. Moreover, we show that the family of invariant sample measures satisfies a stochastic Liouville type theorem. Finally, we discuss the existence of an invariant measure for the random cocycle associated with 2D autonomous SNSE. We prove the uniqueness of invariant measures for $\boldsymbol{f}=\mathbf{0}$ and for any $ν>0$ by using the linear multiplicative structure of the noise coefficient and exponential stability of solutions. The above results for SNSE defined on $\mathbb{R}^2$ are totally new, especially the results on bi-spatial random attractors and stochastic Liouville type theorem for 2D SNSE with linear multiplicative noise are obtained in any kind of domains for the first time.
