On the parabolic Stefan problem for Ostwald ripening with kinetic undercooling and inhomogeneous driving force

Ostwald ripening is the coarsening phenomenon caused by the diffusion and solidification process which occurs in the last stage of a first-order phase transformation. The force that drives the system towards equilibrium is the gradient of the chemical potential that, according to the Gibbs–Thomson condition, on the interface, is proportional to its mean curvature. A quantitative description of Ostwald ripening has been developed by the Lifschitz–Slyozov–Wagner (LSW) theory. We extend the work of Niethammer (2000) [15] which deals with kinetic undercooling in the quasi-static case to the parabolic setting with temporally inhomogeneous driving forces on the solid–liquid interfaces. By means of a priori estimates, local and global existence results for the parabolic Stefan problem, we derive a first order approximation for the dynamical equations for the heat distribution and particle radii and then prove the convergence to a limiting description using a mean-field equation.