  Let $M$ be a compact, real analytic manifold and $G$ be the Lie group of all
real-analytic diffeomorphisms of $M$, which is modelled on the space
${\mathfrak g}$ of real-analytic vector fields on $M$. We study flows of
time-dependent real-analytic vector fields on $M$ which are integrable
functions in time, and their dependence on the time-dependent vector field.
Notably, we show that the Lie group $G$ is $L^1$-regular in the sense that each
$[\gamma]$ in $L^1([0,1],{\mathfrak g})$ has an evolution which is an
absolutely continuous $G$-valued function on $[0,1]$ and depends smoothly on
$[\gamma]$. As tools for the proof, we develop new results concerning
$L^1$-regularity of infinite-dimensional Lie groups, and new results concerning
the continuity and complex analyticity of non-linear mappings on locally convex
direct limits.
