  In the past few decades maximal regularity theory has successfully been
applied to moving boundary problems. The basic idea is to reduce the system
with varying domains to one in a fixed domain. This is done by a transformation
(the so-called Hanzawa transformation) and yields a typically nonlocal and
nonlinear coupled system of (evolution) equations. Well-posedness results can
then often be established as soon as it is proved that the relevant
linearization is the generator of an analytic semigroup or admits maximal
regularity. To implement this program, it is necessary to somehow parametrize
to space of boundaries/domains (typically the space of compact hypersurfaces
$\Gamma$ in $\mathbb{R}^n$, in the Euclidean setting). This has traditionally
been achieved by means of the already mentioned Hanzawa transformation. The
approach, while successful, requires the introduction of a smooth manifold
$\Gamma_\infty$ close to the manifold $\Gamma_0$ in which one cares to
linearize. This prevents one to use coordinates in which $\Gamma_0$ lies at
their "center". As a result formul{\ae} tend to contain terms that would
otherwise not be present were one able to linearize in a neighborhood emanating
from $\Gamma_0$ instead of from $\Gamma_\infty$. In this paper it is made use
of flows (curves of diffeomorphisms) to obtain a general form of the relevant
linearization in combination with an alternative coordinatization of the
manifold of hypersurfaces, which circumvents the need for the introduction of a
"phantom" reference manifold $\Gamma_\infty$ by, in its place, making use of a
"phantom geometry" on $\Gamma_0$. The upshot is a clear insight into the
structure of the linearization, simplified calculations, and simpler
formul{\ae} for the resulting linear operators, which are useful in
applications.
