Uniform approximation of non-autonomous evolution equations

TL;DR

This paper demonstrates that solutions to non-autonomous evolution equations with sesquilinear form coefficients can be uniformly approximated in various function spaces using a subdivision-based approximation method, extending previous regularity results.

Contribution

It introduces an approximation approach for non-autonomous evolution equations that recovers known maximal regularity results and proves uniform convergence of solutions.

Findings

The approximation method achieves $L^2$-maximal regularity for the equations.

Solutions of the approximate equations converge uniformly to the true solution.

The method extends to less regular moduli of continuity under additional assumptions.

Abstract

We study $L^2$-maximal regularity for non-autonomous evolution equations of the form \begin{equation}\label{Abstract equation} \dot u(t)+\mathcal A(t)u(t)=f(t)\ \ t\in[0,T],\ \ u(0)=u_0. \end{equation} where $\mathcal A(t),\ t\in [0,T]$ arise from a non-autonomous sesquilinear forms $a(t,\cdot,\cdot)$ on a Hilbert space $H$ with constant domain $V\subset H.$ $L^2-$maximal regularity result is proved recently in \cite{Ar-Mo15} when $a$ is H\"older continuous of type $\alpha>1/2.$ In this paper we recover the same results by an approximation method developed in \cite{ELLA13}, \cite{LASA14} and \cite{ELLA15}. The method uses an appropriate approximation $\mathcal A_\Lambda(\cdot)$ of $\mathcal A(\cdot)$ for which \begin{equation}\label{Abstract equation approx} \dot u_{\Lambda}(t)+\mathcal A_{\Lambda}(t)u_{\Lambda}(t)=f(t)\ \ t\in[0,T],\ \ u_{\Lambda}(0)=u_0 \end{equation} has $L^2$-maximal regularity where $\Lambda$ is a subdivision of $[0,T].$ Moreover, under a little more assumptions on the modulus of continuity we show that the solutions of (\ref{Abstract equation approx}) converges in $L^2(0,T,V)\cap H^1(0,T,H)\cap C(0,T,V)$ uniformly on the initial datas $(u_0,f)$ to the solution of (\ref{Abstract equation}) as $|\Lambda| \rightarrow 0.$