On Maximal Regularity for a Class of Evolutionary Equations

TL;DR

This paper investigates maximal regularity for a class of evolutionary equations in Hilbert spaces, introducing a novel approach based on structural coefficient constraints rather than traditional semigroup or sesquilinear form methods.

Contribution

It develops a new method for establishing maximal regularity by analyzing the operator sum with structural coefficient constraints, extending existing strategies in the Hilbert space setting.

Findings

Operator sum with natural domain is closed under structural constraints

The approach complements and extends known maximal regularity methods

Re-derivation of known results within the new framework

Abstract

The issue of so-called maximal regularity is discussed within a Hilbert space framework for a class of evolutionary equations. Viewing evolutionary equations as a sums of two unbounded operators, showing maximal regularity amounts to establishing that the operator sum considered with its natural domain is already closed. For this we use structural constraints of the coefficients rather than semi-group strategies or sesqui-linear form methods, which would be difficult to come by for our general problem class. Our approach, although limited to the Hilbert space case, complements known strategies for approaching maximal regularity and extends them in a different direction. The abstract findings are illustrated by re-considering some known maximal regularity results within the framework presented.