  This work addresses an extension of the Aubin-Lions-Simon compactness result
to generalized Bochner spaces $L^2(0,T;H(t))$, where $H(t)$ is a family of
Hilbert spaces, parameterized by $t$. A compactness result of this type is
needed in the study of the existence of weak solutions to nonlinear evolution
problems governed by partial differential equations defined on moving domains.
We identify the conditions on the regularity of the domain motion in time under
which our extension of the Aubin-Lions-Simon compactness result holds. Concrete
examples of the application of the compactness theorem are presented, including
a classical problem for the incompressible, Navier-Stokes equations defined on
a {\sl given} non-cylindrical domain, and a class of fluid-structure
interaction problems for the incompressible, Navier-Stokes equations, coupled
to the elastodynamics of a Koiter shell. The compactness result presented in
this manuscript is crucial in obtaining constructive existence proofs to
nonlinear, moving boundary problems, using Rothe's method.
