Many time-dependent linear partial differential equations of mathematical physics and continuum mechanics can be phrased in the form of an abstract evolutionary system defined on a Hilbert space. In this paper we discuss a general framework for homogenization (periodic and stochastic) of such systems. The method combines a unified Hilbert space approach to evolutionary systems with an operator-theoretic reformulation of the well-established periodic unfolding method in homogenization. Regarding the latter, we introduce a well-structured family of unitary operators on a Hilbert space that allows to describe and analyze differential operators with rapidly oscillating (possibly random) coefficients. We illustrate the approach by establishing periodic and stochastic homogenization results for elliptic partial differential equations, Maxwell's equations, and the wave equation.

中文翻译：

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抽象线性时间相关偏微分方程的两尺度同质化

数学物理和连续介质力学的许多与时间相关的线性偏微分方程可以用在希尔伯特空间上定义的抽象进化系统的形式来表述。在本文中，我们讨论了此类系统均质化（周期性和随机）的一般框架。该方法将进化系统的统一希尔伯特空间方法与均质化中完善的周期性展开方法的算子理论重构相结合。关于后者，我们在希尔伯特空间上引入了一个结构良好的酉算子族，它允许描述和分析具有快速振荡（可能是随机）系数的微分算子。我们通过为椭圆偏微分方程 Maxwell' 建立周期性和随机均匀化结果来说明该方法