GRADUATE STUDENT SEMINAR: Probabilistic solutions to the supercooled Stefan problem, Graeme Baker

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.