We discuss the issue of maximal regularity for evolutionary equations with non-autonomous coefficients. Here evolutionary equations are abstract partial-differential algebraic equations considered in Hilbert spaces. The catch is to consider time-dependent partial differential equations in an exponentially weighted Hilbert space. In passing, one establishes the time derivative as a continuously invertible, normal operator admitting a functional calculus with the Fourier–Laplace transformation providing the spectral representation. Here, the main result is then a regularity result for well-posed evolutionary equations solely based on an assumed parabolic-type structure of the equation and estimates of the commutator of the coefficients with the square root of the time derivative. We thus simultaneously generalise available results in the literature for non-smooth domains. Examples for equations in divergence form, integro-differential equations, perturbations with non-autonomous and rough coefficients as well as non-autonomous equations of eddy current type are considered.

我们讨论了具有非自治系数的进化方程的最大正则性问题。这里的进化方程是在希尔伯特空间中考虑的抽象偏微分代数方程。问题是在指数加权的希尔伯特空间中考虑时间相关的偏微分方程。顺便说一下，人们将时间导数建立为一个连续可逆的正规算子，它允许使用傅立叶-拉普拉斯变换提供谱表示的泛函演算。在这里，主要结果是完全基于方程的假定抛物线型结构的适定演化方程的正则性结果，以及具有时间导数平方根的系数对易子的估计。因此，我们同时概括了文献中非光滑域的可用结果。考虑了发散形式的方程、积分微分方程、具有非自治和粗糙系数的扰动以及涡流类型的非自治方程的示例。