Maximal regularity for non-autonomous equations with measurable dependence on time

TL;DR

This paper establishes maximal $L^p$-regularity for non-autonomous evolution equations with merely measurable time dependence, introducing new conditions for vector-valued singular integrals and applying them to elliptic operators with time-dependent coefficients.

Contribution

It develops a novel abstract operator-theoretic approach to maximal regularity under minimal time regularity assumptions, extending results to quasilinear PDEs with measurable coefficients.

Findings

New sufficient condition for $L^p$-boundedness of vector-valued singular integrals.

Established $L^p(L^q)$-theory for elliptic operators with measurable time coefficients.

Extended well-posedness results to time-dependent quasilinear equations.

Abstract

In this paper we study maximal $L^p$-regularity for evolution equations with time-dependent operators $A$. We merely assume a measurable dependence on time. In the first part of the paper we present a new sufficient condition for the $L^p$-boundedness of a class of vector-valued singular integrals which does not rely on H\"ormander conditions in the time variable. This is then used to develop an abstract operator-theoretic approach to maximal regularity. The results are applied to the case of $m$-th order elliptic operators $A$ with time and space-dependent coefficients. Here the highest order coefficients are assumed to be measurable in time and continuous in the space variables. This results in an $L^p(L^q)$-theory for such equations for $p,q\in (1, \infty)$. In the final section we extend a well-posedness result for quasilinear equations to the time-dependent setting. Here we give an example of a nonlinear parabolic PDE to which the result can be applied.