Maximal Regularity for Non-Autonomous Evolutionary Equations

TL;DR

This paper establishes maximal regularity results for non-autonomous evolutionary equations in Hilbert spaces, extending existing theory to equations with non-smooth coefficients and complex structures.

Contribution

It provides a new regularity theorem for well-posed non-autonomous equations based on parabolic structure and commutator estimates, generalizing prior results.

Findings

Regularity results for divergence form equations

Extension to integro-differential and eddy current equations

Handling non-smooth, time-dependent coefficients

Abstract

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.