  We prove that the Cauchy problem is well-posed in a strong sense and in a
general setting. Our main result is the construction of an abstract semi-flow
for the Hele-Shaw problem within general fluid domains (enabling, for instance,
changes in the topology of the fluid domain) and which satisfies several
properties: We provide simple comparison arguments, establish a new stability
estimate and derive several consequences, including monotonicity and continuity
results for the solutions, along with many Lyapunov functionals. We establish
an eventual analytic regularity result for any arbitrary initial data. We also
study numerous qualitative properties, including global regularity for initial
data in sub-critical Sobolev spaces, well-posedness in a strong sense for
initial data with barely a modulus of continuity, as well as waiting-time
phenomena for Lipschitz solutions, in any dimension.
