Maximal regularity estimates for the abstract Cauchy problems

TL;DR

This paper extends the theory of maximal regularity estimates for sectorial operators, providing new results on interpolation spaces, $L^1$-estimates, and a novel framework for $L^p$-maximal regularity, with applications to fluid mechanics.

Contribution

It introduces new maximal regularity results in interpolation spaces, characterizes weighted $L^1$-estimates, and develops a new interpolation framework for $L^p$-maximal regularity.

Findings

Extended Da Prato-Grisvard theory to new interpolation spaces.

Established new homogeneous and inhomogeneous $L^1$-maximal regularity estimates.

Reformulated $L^1$-maximal regularity characterization without semigroup reliance.

Abstract

In this work, we extend the Da Prato-Grisvard theory of maximal regularity estimates for sectorial operators in interpolation spaces. Specifically, for any generator $-A$ of an analytic semigroup on a Banach space $X$, we identify the interpolation spaces between $X$ and the domain $D_{A}$ of $A$ in which the part of $A$ satisfies certain maximal regularity estimates. We also establish several new results concerning both homogeneous and inhomogeneous $L^1$-maximal regularity estimates, extending and completing recent findings in the literature. These results are motivated not only by applications to problems in areas such as fluid mechanics but also by the intrinsic theoretical interest of the subject. In particular, we address the optimal choice of data spaces for the Cauchy problem associated with $A$, ensuring the existence of strong solutions with global-in-time control of their derivatives. This control is measured via the homogeneous parts of the interpolation norms in the spatial variable and weighted Lebesgue norms over the time interval. Furthermore, we characterize weighted $L^1$-estimates and establish their relationship with unweighted estimates. Additionally, we reformulate the characterization condition for $L^1$-maximal regularity due to Kalton and Portal in a priori terms that do not rely on semigroup operators. Finally, we introduce a new interpolation framework for $L^p$-maximal regularity estimates, where $p \in (1, \infty)$, within interpolation spaces generated by non-classical interpolation functors.