有界区域 $2D$ 和 $3D$ 中一类随机三阶流体方程的不变测度
Invariant measures for a class of stochastic third grade fluid equations in $2D$ and $3D$ bounded domains
AI总结：
本文研究了有界区域中乘性噪声下一类三阶流体方程的适定性与遍历不变测度，通过随机紧致性与单调性论证证明鞅解存在，利用稳定性获得强解，并基于 Feller 性质与 Krylov-Bogoliubov 定理证明了遍历不变测度的存在性。
摘要：
本研究旨在探讨在乘性噪声存在的情况下，有界区域 $D\subset\mathbb{R}^d,d=2,3$ 中一类三阶流体方程的适定性与遍历不变测度的存在性。首先，我们通过结合随机紧致性与单调性论证，证明了鞅解的存在性。接着，我们证明了一个稳定性结果，该结果给出了路径唯一性，从而证明了强概率解的存在性。其次，我们利用该稳定性结果证明相关半群是 Feller 的，并通过使用“Krylov-Bogoliubov 定理”获得了不变概率测度的存在性。最后，我们证明所有不变测度均集中于 $L^2$ 的一个紧子集上，从而导出遍历不变测度的存在性。
英文摘要：
This work aims to investigate the well-posedness and the existence of ergodic invariant measures for a class of third grade fluid equations in bounded domain $D\subset\mathbb{R}^d,d=2,3,$ in the presence of a multiplicative noise. First, we show the existence of a martingale solution by coupling a stochastic compactness and monotonicity arguments. Then, we prove a stabilty result, which gives the pathwise uniqueness of the solution and therefore the existence of strong probabilistic solution. Secondly, we use the stability result to show that the associated semigroup is Feller and by using "Krylov-Bogoliubov Theorem" we get the existence of an invariant probability measure. Finally, we show that all the invariant measures are concentrated on a compact subset of $L^2$, which leads to the existence of an ergodic invariant measure.
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