  An $\mathrm{L}_1$-maximal regularity theory for parabolic evolution equations
inspired by the pioneering work of Da Prato and Grisvard is developed. Besides
of its own interest, the approach yields a framework allowing global-in-time
control of the change of Eulerian to Lagrangian coordinates in various problems
related to fluid mechanics. This property is of course decisive for free
boundary problems. This concept is illustrated by the analysis of the free
boundary value problem describing the motion of viscous, incompressible
Newtonian fluids without surface tension and, secondly, the motion of
compressible pressureless gases.
  For this purpose, an endpoint maximal $\mathrm{L}_1$-regularity approach to
the Stokes and Lam\'e systems is developed. It is applied then to establish
global, strong well-posedness results for the free boundary problems described
above in the case where the initial domain coincides with the half-space, and
the initial velocity is small with respect to a suitable scaling invariant
norm.
