Calculus of Variations and Geometric Measure Theory

A. Braides - M. Solci

Motion of discrete interfaces through mushy layers

created by braidesa on 12 Jun 2015
modified on 21 Jul 2016


Published Paper

Inserted: 12 jun 2015
Last Updated: 21 jul 2016

Journal: J. Nonlinear Sci.
Volume: 26
Pages: 1031-1053
Year: 2016
Doi: 10.1007/s00332-016-9297-6


We study the geometric motion of sets in the plane derived from the homogenization of discrete ferromagnetic energies with weak inclusions. We show that the discrete sets are composed by a `bulky' part and an external `mushy region' composed only of weak inclusions. The relevant motion is that of the bulky part, which asymptotically obeys to a motion by crystalline mean curvature with a forcing term, due to the energetic contribution of the mushy layers, and pinning effects, due to discreteness. From an analytical standpoint it is interesting to note that the presence of the mushy layers imply only a weak and not strong convergence of the discrete motions, so that the convergence of the energies does not commute with the evolution. From a mechanical standpoint it is interesting to note the geometrical similarity of some phenomena in the cooling of binary melts.

Keywords: crystalline curvature, Geometric motion, discrete-to-continuum, minimizing movement, fluid mechanics, mushy layer