标题： 非自治多尺度SPDE的强平均化原理，具有完全局部单调和几乎周期系数 显示英文标题

标题： Strong averaging principle for nonautonomous multi-scale SPDEs with fully local monotone and almost periodic coefficients

摘要： 在本文中，我们考虑一类具有完全局部单调系数的非自治多尺度随机偏微分方程。 通过引入时变非齐次马尔可夫半群的测度演化系统，我们研究了此类系统的平均化原理。 具体而言，我们首先证明多尺度随机系统中的慢分量强收敛于一个平均方程的解，该方程的系数保留了缩放参数的依赖性。 此外，如果系数满足一致几乎周期条件，我们建立慢分量强收敛于另一个平均方程的解，该方程的系数与缩放参数无关。 本文的主要贡献将 Cheng 和 Liu 在 [11] 中研究的基本非自治框架扩展到完全耦合框架，同时将 Liu 等人 [27] 探索的自治框架扩展到更一般的非自治框架。 此外，我们将 [11,27] 中讨论的局部单调系数改进为完全局部单调系数，因此我们的结果可以应用于非线性非自治随机偏微分方程中的广泛情况，例如多尺度随机 Cahn-Hilliard-heat 方程和多尺度随机液晶-多孔介质方程。 显示英文摘要

摘要： In this paper, we consider a class of nonautonomous multi-scale stochastic partial differential equations with fully local monotone coefficients. By introducing the evolution system of measures for time-inhomogeneous Markov semigroups, we study the averaging principle for such kind of system. Specifically, we first prove the slow component in the multi-scale stochastic system converges strongly to the solution of an averaged equation, whose coefficients retain the dependence of the scaling parameter. Furthermore, if the coefficients satisfy uniformly almost periodic conditions, we establish that the slow component converges strongly to the solution of another averaged equation, whose coefficients are independent of the scaling parameter. The main contribution of this paper extends the basic nonautonomous framework investigated by Cheng and Liu in [11] to a fully coupled framework, as well as the autonomous framework explored by Liu et al. in [27] to the more general nonautonomous framework. Additionally, we improve the locally monotone coefficients discussed in [11,27] to the fully local monotone coefficients, thus our results can be applied to a wide range of cases in nonlinear nonautonomous stochastic partial differential equations, such as multi-scale stochastic Cahn-Hilliard-heat equation and multi-scale stochastic liquid-crystal-porous-media equation.