Melting and freezing rates of the radial interior Stefan problem in two dimension

Abstract

We consider the interior Stefan problem under radial symmetry in two dimension. A water ball surrounded by ice undergoes melting or freezing. We construct a discrete family of global-in-time solutions, both melting and freezing scenarios. The evolution of the free boundary, represented by the radius of the water ball, $\lambda (t)$ exhibits exponential convergence to a limiting radius value ${\lambda }_{\infty }>0$, characterized by the asymptotic expression \[ λ(t) = λ_\infty + (1 - λ_\infty)\, e^{-\frac{λ_k}{λ_\infty^2} t + o_{t \to \infty}(1)}, \] where ${\lambda }_k$ stands for the $k$-th Dirichlet eigenvalue of the Laplacian on the unit disk for any $k\in \mathbb{N}$. Our approach draws inspiration from the research conducted by Hadžić and Raphaël [24] concerning the exterior radial Stefan problem, which involves an ice ball is surrounded by water. In contrast, the bounded geometry in our setting leads to scenario results in a non-degenerate spectrum, leading to distinctly different long-term behavior. These solutions for each $k$ remain stable under perturbations of co-dimension $k-1$.

46 pages