Scaling theory and morphometrics for a coarsening multiscale surface, via a principle of maximal dissipation

We consider the coarsening dynamics of multiscale solutions to a dissipative singularly perturbed partial differential equation which models the evolution of a thermodynamically unstable crystalline surface. The late-time leading-order behavior of solutions is identified, through the asymptotic expansion of a maximal-dissipation principle, with a completely faceted surface governed by an intrinsic dynamical system. The properties of the resulting piecewise-affine dynamic surface predict the scaling law L(Mu) approximately t(1/3), for the growth in time of a characteristic morphological length scale L(Mu). A novel computational geometry tool which directly simulates a million-facet piecewise-affine dynamic surface is also introduced. Our computed data are consistent with the dynamic scaling hypothesis, and we report a variety of associated morphometric scaling functions.