On the Surface Diffusion Flow with Triple Junctions in Higher Space Dimensions

Abstract

We 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.

Weak-strong uniqueness for the mean curvature flow of double bubbles

“…However, here we ask for higher regularity of the interfaces Īi,j and higher compatibility along the triple line Γ. The estimate away from the triple line corresponds to standard interior Schauder estimates while the behavior close to the triple line can be achieved by an adaptation of the argument in [3] similar to the one by Garcke and Gößwein [6], who derived these higher-order compatibility conditions in the context of the surface diffusion flow, a fourth-order geometric evolution equation. Of course one needs to assume that the initial conditions satisfy the corresponding regularity and the higher compatibility along the triple line Γ0 .…”

A structure‐preserving finite element approximation of surface diffusion for curve networks and surface clusters

“…A model for surface diffusion of a network of curves has been introduced in [27] for $$$ d=2 $$$ and generalized to arbitrary space dimensions in [28, 29]. Well‐posedness was shown in [30] for $$$ d=2 $$$ and in [31] for higher space dimensions. We will present the precise mathematical formulation of this evolution law in Section 2 below.…”

A structure-preserving finite element approximation of surface diffusion for curve networks and surface clusters

“…A model for surface diffusion of a network of curves has been introduced in [41] for d = 2 and generalized to arbitrary space dimensions in [17,32]. Well-posedness was shown in [1] for d = 2 and in [37] for higher space dimensions. We will present the precise mathematical formulation of this evolution law in Section 2 below.…”