  Supercooled Stefan problems describe the evolution of the boundary between
the solid and liquid phases of a substance, where the liquid is assumed to be
cooled below its freezing point. Following the methodology of Delarue,
Nadtochiy and Shkolnikov, we construct solutions to the one-phase
one-dimensional supercooled Stefan problem through a certain McKean-Vlasov
equation, which allows to define global solutions even in the presence of
blow-ups. Solutions to the McKean-Vlasov equation arise as mean-field limits of
particle systems interacting through hitting times, which is important for
systemic risk modeling. Our main contributions are: (i) we prove a general
tightness theorem for the Skorokhod M1-topology which applies to processes that
can be decomposed into a continuous and a monotone part. (ii) We prove
propagation of chaos for a perturbed version of the particle system for general
initial conditions. (iii) We prove a conjecture of Delarue, Nadtochiy and
Shkolnikov, relating the solution concepts of so-called minimal and physical
solutions, showing that minimal solutions of the McKean-Vlasov equation are
physical whenever the initial condition is integrable.
