On evolution equations governed by non-autonomous forms

TL;DR

This paper studies evolution equations driven by time-dependent forms, proving that solutions of piecewise affine approximations converge to those of the original problem, thus offering an alternative proof of maximal regularity results.

Contribution

It demonstrates that solutions of approximate non-autonomous problems with piecewise affine forms converge to the original problem's solutions, providing an alternative proof of maximal regularity.

Findings

Solutions of approximate problems converge to the original solutions.

Provides an alternative proof of $L^2$-maximal regularity in $H$.

Shows approximation by piecewise affine forms is effective.

Abstract

We consider a linear non-autonomous evolutionary Cauchy problem \begin{equation} \dot{u} (t)+A(t)u(t)=f(t) \hbox{ for }\ \hbox{a.e. t}\in [0,T],\quad u(0)=u_0, \end{equation} where the operator $A(t)$ arises from a time depending sesquilinear form $a(t,.,.)$ on a Hilbert space $H$ with constant domain $V.$ Recently a result on $L^2$-maximal regularity in $H,$ i.e., for each given $f\in L^2(0,T,H)$ and $u_0 \in V$ the problem above has a unique solution $u\in L^2(0,T,V)\cap H^1(0,T,H),$ is proved in [10] under the assumption that $a$ is symmetric and of bounded variation. The aim of this paper is to prove that the solutions of an approximate non-autonomous Cauchy problem in which $a$ is symmetric and piecewise affine are closed to the solutions of that governed by symmetric and of bounded variation form. In particular, this provide an alternative proof of the result in [10] on $L^2$-maximal regularity in $H.$