Calculus of Variations and Geometric Measure Theory

A. Cesaroni - C. Muratov - M. Novaga

Front propagation in geometric and phase fi eld models of strati fied media

created by novaga on 09 Oct 2013
modified on 02 Mar 2015


Published Paper

Inserted: 9 oct 2013
Last Updated: 2 mar 2015

Journal: Arch. Rat. Mech. Anal.
Volume: 216
Number: 1
Pages: 153-191
Year: 2015


We study front propagation problems for forced mean curvature flows and their phase field variants that take place in stratified media, i.e., heterogeneous media whose characteristics do not vary in one direction. We consider phase change fronts in infinite cylinders whose axis coincides with the symmetry axis of the medium. Using a recently developed variational approaches, we provide a convergence result relating asymptotic in time front propagation in the diffuse interface case to that in the sharp interface case, for suitably balanced nonlinearities of Allen-Cahn type. The result is established by using arguments in the spirit of $\Gamma$-convergence, to obtain a correspondence between the minimizers of an exponentially weighted Ginzburg-Landau-type functional and the minimizers of an exponentially weighted area-type functional. These minimizers yield the fastest moving traveling waves in the respective models and determine the asymptotic propagation speeds for front-like initial data. We further show that generically these fronts are the exponentially stable global attractors for this kind of initial data and give sufficient conditions under which complete phase change occurs via the formation of the considered fronts.