Mullins-Sekerka as the Wasserstein flow of the perimeter

The Hele-Shaw flow as the sharp interface limit of the Cahn-Hilliard equation with disparate mobilities

“…Our approach is different and inspired by the work of Simon and one of the authors [LS18] who derive the sharp-interface limit of a system of Allen-Cahn equations, a second-order version of our problem here. On a conceptual level, our proof of the second main result is also similar to the work by Chambolle and one of the authors [CL21] who showed that the implicit time discretization of the Hele-Shaw flow produces varifold solutions which are slightly weaker than the solutions considered here. Jacobs, Meszaros, and Kim [JKM21] introduced a thresholding-type scheme, similar to this implicit time discretization and proved its convergence to a weak solution under an energy convergence assumption.…”

The asymptotics of the area-preserving mean curvature and the Mullins–Sekerka flow in two dimensions

“…We remark that one could also consider the one-phase model for the Mullins-Sekerka as in [9] in the whole R 2 and expect the above convergence to hold also in this case. We also expect the convergence of the sets in Theorem 1.3 to hold with respect to Hausdorff distance but we do not prove it here.…”

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

“…We remark that one could also consider the one-phase model for the Mullins-Sekerka as in [8] in the whole R 2 and expect the above convergence to hold also in this case. We also expect the convergence of the sets in Theorem 1.3 to hold with respect to Hausdorff distance but we do not prove it here.…”