On a phase field model for solid-liquid phase transitions

TL;DR

This paper introduces a generalized phase field model for solid-liquid transitions with nonlinear diffusion, demonstrating its asymptotic convergence to a Stefan-like model and establishing local well-posedness for the PDE system.

Contribution

It presents a novel phase field model with nonlinear diffusion terms and analyzes its asymptotic limit and well-posedness, extending existing models like Caginalp's.

Findings

The model converges to a Stefan-like sharp interface limit.

It derives a generalized Gibbs-Thomson relation with curvature and velocity effects.

The initial-boundary value problem is locally well-posed for smooth data.

Abstract

A new phase field model is introduced, which can be viewed as nontrivial generalisation of what is known as the Caginalp model. It involves in particular nonlinear diffusion terms. By formal asymptotic analysis, it is shown that in the sharp interface limit it still yields a Stefan-like model with: 1) a (generalized) Gibbs-Thomson relation telling how much the interface temperature differs from the equilibrium temperature when the interface is moving or/and is curved with surface tension; 2) a jump condition for the heat flux, which turns out to depend on the latent heat and on the velocity of the interface with a new, nonlinear term compared to standard models. From the PDE analysis point of view, the initial-boundary value problem is proved to be locally well-posed in time (for smooth data).