Weak asymptotic solution of the phase field system in the case of confluence of free boundaries in the Stefan problem with undercooling

TL;DR

This paper constructs a smooth approximation for the Stefan problem with undercooling, capturing the classical solution before boundary contact and the heat conduction solution after contact, addressing free boundary confluence.

Contribution

It introduces a weak asymptotic solution approach for the Stefan problem with free boundary confluence, bridging classical and heat conduction solutions.

Findings

Successfully approximates the classical solution before contact

Provides a smooth transition to heat conduction solution after contact

Addresses free boundary confluence in phase transition modeling

Abstract

We assume that the Stefan problem with undercooling has a classical solution until the moment of contact of free boundaries and the free boundaries have finite velocities until the contact. Under these assumptions, we construct a smooth approximation of the global solution of the Stefan problem with undercooling, which, until the contact, gives the classical solution mentioned above and, after the contact, gives a solution which is the solution of the heat conduction equation.