The Stefan problem with surface tension in the three dimensional case with spherical symmetry: nonexistence of the classical solution

Abstract

The Stefan problem with surface tension in the three-dimensional case with spherical symmetry is considered. We first establish the existence and uniqueness of the classical solution with surface tension and kinetic undercooling effects for all time, and then pass to the limit as the kinetic undercooling tends to zero. The limiting solution is the global-in-time weak solution and the local-in-time classical solution for the Stefan problem with surface tension. This solution cannot be the global-in-time classical solution. If S(t) is the radius of a solid ball in a supercooled liquid, then (1) there exists at least one point t* of discontinuity of the function S(t):or (2) the continuous function S(t) cannot be absolutely continuous, and it maps some zero-measure set of (t*, T*) onto some set of Ω with a strictly positive measure.