The Asymptotics of the Area-Preserving Mean Curvature and the Mullins-Sekerka Flow in Two Dimensions

Abstract

We provide the first general result for the asymptotics of the area preserving mean curvature flow in two dimensions showing that flat flow solutions, starting from any bounded set of finite perimeter, converge with exponential rate to a finite union of equally sized disjoint disks. A similar result is established also for the periodic two-phase Mullins-Sekerka flow.

Stability of the ball under volume preserving fractional mean curvature flow

“…This result provides an improvement of the result in [12] discussed above, see Corollary 3.5, ruling out the possibility of indefinite translations and giving the exponential rate of convergence. Similar results in the local setting date back to [19] for the case of convex initial sets, to [1,16] for nearly spherical initial sets, and more recently to [22] in dimension 2 for weak solutions starting from a general bounded set of finite perimeter.…”

The Mullins–Sekerka problem via the method of potentials

“…We also refer to [21,24,41] where stability issues are investigated and to [8,22,39,45] for numerical studies pertaining to this problem. Finally, we mention the papers [10,11,26,27,42] where weak solutions to the Mullins-Sekerka problem are studied.…”

Long time behaviour of the discrete volume preserving mean curvature flow in the flat torus

“…Regarding the volume preserving mean curvature flow, recent progresses have been made in proving the dynamical stability of strictly stable sets in the flat torus of dimension 3 [35], while older results mainly concern convex sets, balls, or the 2-dimensional setting. The dynamical stability of balls has been proven in the Euclidean setting under various hypoteses on the dimension or on the initial set in [15,19,23,28] (see also the approach based on weak solutions of [24] in R 2 and in [6] in the anisotropic and crystalline setting, for convex initial data). For the surface diffusion flow, most famous results deal with the stability of balls [14,36], infinite cylinders [27], and two-dimensional triple junctions [20], as well double bubbles [1].…”