Non-Autonomous Maximal Regularity for Forms of Bounded Variation

TL;DR

This paper establishes conditions under which solutions to non-autonomous evolution equations exhibit maximal regularity in the Hilbert space H, specifically when the associated form has bounded variation, ensuring better regularity and continuity properties.

Contribution

The paper proves that maximal regularity in H holds for forms of bounded variation, extending previous results and including certain perturbations of the operator.

Findings

Maximal regularity in H is achieved when the form has bounded variation.

Solutions are continuous with values in V under these conditions.

The results extend to specific perturbations of the operator.

Abstract

We consider a non-autonomous evolutionary problem \[ u' (t)+\mathcal A (t)u(t)=f(t), \quad u(0)=u_0, \] where $V, H$ are Hilbert spaces such that $V$ is continuously and densely embedded in $H$ and the operator $\mathcal A (t)\colon V\to V^\prime$ is associated with a coercive, bounded, symmetric form $\mathfrak{a}(t,.,.)\colon V\times V \to \mathbb{C}$ for all $t \in [0,T]$. Given $f \in L^2(0,T;H)$, $u_0\in V$ there exists always a unique solution $u \in MR(V,V'):= L^2(0,T;V) \cap H^1(0,T;V')$. The purpose of this article is to investigate when $u \in H^1(0,T;H)$. This property of maximal regularity in $H$ is not known in general. We give a positive answer if the form is of bounded variation; i.e., if there exists a bounded and non-decreasing function $g \colon [0,T] \to \mathbb{R}$ such that \begin{equation*} \lvert\mathfrak{a}(t,u,v)- \mathfrak{a}(s,u,v)\rvert \le [g(t)-g(s)] \lVert u \rVert_V \lVert v \rVert_V \quad (s,t \in [0,T], s \le t). \end{equation*} In that case, we also show that $u(.)$ is continuous with values in $V$. Moreover we extend this result to certain perturbations of $\mathcal A (t)$.