Non-autonomous maximal regularity in Hilbert spaces

We consider non-autonomous evolutionary problems of the form u′(t)+A(t)u(t)=f(t),u(0)=u0,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u' (t)+A(t)u(t) = f(t), \quad u(0) = u_{0},$$\end{document}on L2([0,T];H)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${L^{2}([0, T]; H)}$$\end{document}, where H is a Hilbert space. We do not assume that the domain of the operator A(t) is constant in time t, but that A(t) is associated with a sesquilinear form a(t)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathfrak{a}(t)}$$\end{document}. Under sufficient time regularity of the forms a(t)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathfrak{a}(t)}$$\end{document}, we prove well-posedness with maximal regularity in L2([0,T];H)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${L^{2}([0, T]; H)}$$\end{document}. Our regularity assumption is significantly weaker than those from previous results inasmuch as we only require a fractional Sobolev regularity with arbitrary small Sobolev index.