Regularization by Kinetic Undercooling of Blow-up in the Ill-posed Stefan Problem
John R. King, Jonathan D. Evans · SIAM Journal on Applied Mathematics · 2005
It is well known that the one-dimensional supercooled Stefan problem possesses solutions that blow up in finite time. The asymptotics of such solutions have been analyzed by Herrero and Velazquez [European J. Appl. Math., 7 (1996), pp. 119--150]. Here we consider the effect of kinetic undercooling as a regularizing mechanism to prevent the formation of such singularities and study the continuation of the solution through the "near blow-up" regime. The asymptotics of solutions and interfaces are described for small values of the kinetic undercooling parameter. It is shown that, in this limit, the interface jumps over an interval determined by the latent heat and by the initial data. Specifically, in dimensionless variables, if the temperature profile at blow-up is denoted by u(x,t_c^-)$, where $t_c$ is the finite blow-up time, then the interface jumps over the interval in which $u(x,t_c^-) < -\lambda$, where $\lambda$ is the latent heat.