Abstract
AbstractWe show short time existence for the evolution of triple junction clusters driven by the surface diffusion flow. On the triple line we use the boundary conditions derived by Garcke and Novick-Cohen as the singular limit of a Cahn-Hilliard equation with degenerated mobility. These conditions are concurrency of the triple junction, angle conditions between the hypersurfaces, continuity of the chemical potentials and a flux-balance. For the existence analysis we first write the geometric problem over a fixed reference surface and then use for the resulting analytic problem an approach in a parabolic Hölder setting.
Highlights
We show short time existence for the evolution of triple junction clusters driven by the surface di usion ow
Motion by surface di usion ows was rstly proposed by Mullins [25] to describe the development of thermal grooves at grain boundaries of heated polycrystalls
The evolution is connected to the Cahn-Hilliard equations as Cahn, Elliott and Novick-Cohen proved via formal asymptotics that the surface di usion ow is its singular limit, cf. [6]
Summary
Motion by surface di usion ows was rstly proposed by Mullins [25] to describe the development of thermal grooves at grain boundaries of heated polycrystalls. This process depends mainly on two physical effects, which are evaporation and surface di usion, i.e., molecular motion on the surface of heated, solid substances. The evolution is connected to the Cahn-Hilliard equations as Cahn, Elliott and Novick-Cohen proved via formal asymptotics that the surface di usion ow is its singular limit, cf [6]. E.g., the mean curvature ow, it is very typical for the surface ow to have a loss of convexity and to develop singularities during the evolution, cf [18], [19].
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