Title: Probabilistic solutions to the supercooled Stefan problem
Abstract: The supercooled Stefan problem is a free boundary partial
differential equation, which for general initial conditions may
exhibit finite-time blowup of the derivative of the free boundary and
admit multiple solutions; such a problem is deemed ill-posed.
In recent years, a probabilistic reformulation of this problem has
given rise to a new notion of solution:
these so-called 'physical solutions' satisfy a McKean-Vlasov equation
with interaction through a hitting time, and the size of any possible
discontinuity of the free boundary is governed by the law of the
associated McKean-Vlasov diffusion. Following this discovery,
'minimal' solutions to this problem were introduced, which happen to
also be physical solutions when the initial condition is integrable.
Herein, we give an introduction to the above solution concepts,
indicating their similarities and possible differences. Moreover, we
address the problem of well-posedness for minimal solutions by
investigating the sensitivity of these solutions to changes in the
initial data. We show continuous dependency when the data is shifted
to the right, but possible discontinuity for shifts to the left.