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) = u(0), on , 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 . Under sufficient time regularity of the forms , we prove well-posedness with maximal regularity in . 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.