Minimizing movements for forced anisotropic mean curvature flow of partitions with mobilities

De Giorgi's inequality for the thresholding scheme with arbitrary mobilities and surface tensions

We provide a new convergence proof of the celebrated Merriman-Bence-Osher scheme for multiphase mean curvature flow. Our proof applies to the new variant incorporating a general class of surface tensions and mobilities, including typical choices for modeling grain growth. The basis of the proof are the minimizing movements interpretation of Esedoğlu and Otto and De Giorgi's general theory of gradient flows. Under a typical energy convergence assumption we show that the limit satisfies a sharp energy-dissipation relation.

液滴受迫各向异性曲率流的极小化移动

Minimizing movements for forced anisotropic curvature flow of droplets

本文研究非均匀水平超平面上液滴的受迫各向异性曲率流，建立了光滑流和1/2-赫尔德连续极小化移动解的存在性，并探讨了其比较原理、一致有界性及与光滑流的一致性。

Existence of solutions for a class of fourth order cross-diffusion systems of gradient flow type

This article is concerned with the existence and the long time behavior of weak solutions to certain coupled systems of fourth-order degenerate parabolic equations of gradient flow type. The underlying metric is a Wasserstein-like transportation distance for vector-valued functions, with nonlinear mobilities in each component. Under the hypothesis of (flat) convexity of the driving free energy functional, weak solutions are constructed by means of the variational minimizing movement scheme for metric gradient flows. The essential regularity estimates are derived by variational methods.

The variational approach to $s$-fractional heat flows and the limit cases $s\to 0^+$ and $s\to 1^-$

This paper deals with the limit cases for $s$-fractional heat flows in a cylindrical domain, with homogeneous Dirichlet boundary conditions, as $s\to 0^+$ and $s\to 1^-$\,. To this purpose, we describe the fractional heat flows as minimizing movements of the corresponding Gagliardo seminorms, with respect to the $L^2$ metric. First, we provide an abstract stability result for minimizing movements in Hilbert spaces, with respect to a sequence of $Γ$-converging uniformly $λ$-convex energy functionals. Then, we provide the $Γ$-convergence analysis of the $s$-Gagliardo seminorms as $s\to 0^+$ and …

A two-phase two-fluxes degenerate Cahn-Hilliard model as constrained Wasserstein gradient flow

We study a non-local version of the Cahn-Hilliard dynamics for phase separation in a two-component incompressible and immiscible mixture with linear mobilities. In difference to the celebrated local model with nonlinear mobility, it is only assumed that the divergences of the two fluxes --- but not necessarily the fluxes themselves --- annihilate each other. Our main result is a rigorous proof of existence of weak solutions. The starting point is the formal representation of the dynamics as a constrained gradient flow in the Wasserstein metric. We then show that time-discrete approximations by…

各向异性与晶体曲率流的存在性与唯一性

Existence and uniqueness for anisotropic and crystalline mean curvature flows

本文研究了带有强迫项和任意凸动力学的晶体曲率流的存在性和唯一性，通过引入新的解定义，证明了在任意维度和初始闭集的情况下，解满足比较原理和稳定性，且处理了时间依赖的有界强迫项。