Related papers: A convergent algorithm for forced mean curvature f…
We estimate from above the rate at which a solution to the rescaled mean curvature flow on a closed hypersurface may converge to a limit self-similar solution, i.e. a shrinker. Our main result implies that any solution which converges to a…
The paper develops a hybrid method for solving a system of advection--diffusion equations in a bulk domain coupled to advection--diffusion equations on an embedded surface. A monotone nonlinear finite volume method for equations posed in…
We prove that if a rescaled mean curvature flow is a global graph over the round cylinder with small gradient and converges super-exponentially fast, then it must coincide with the cylinder itself. We also show that the result is sharp with…
In a rotationally symmetric space $\oM$ around an axis A (whose precise definition includes all real space forms), we consider a domain $G$ limited by two equidistant hypersurfaces orthogonal to A. Let $M \subset \oM$ be a revolution…
Diffusion processes arise in many fields, and so simulating the path of a diffusion is an important problem. It is usually necessary to make some sort of approximation via model-discretization, but a recently introduced class of algorithms,…
We are interested in the kernel of one-dimensional diffusion equations with continuous coefficients as evaluated by means of explicit discretization schemes of uniform step $h>0$ in the limit as $h\to0$. We consider both semidiscrete…
Geometric flows, where an immersed manifold evolves in time according to its own geometry, exhibit important structural properties. For example, surface diffusion dissipates surface area while conserving volume; it is desirable to preserve…
In an ambient space with rotational symmetry around an axis (which include the Hyperbolic and Euclidean spaces), we study the evolution under the volume-preserving mean curvature flow of a revolution hypersurface M generated by a graph over…
We investigate the motion of a family of closed curves evolving according to the geometric evolution law on a given two dimensional manifold which is embedded or immersed in the three-dimensional Euclidean space. We derive a system of…
We study evolution of a closed embedded plane curve with the normal velocity depending on the curvature, the orientation and the position of the curve. We propose a new method of tangential redistribution of points by curvature adjusted…
We consider a finite element approximation for a system consisting of the evolution of a closed planar curve by forced curve shortening flow coupled to a reaction-diffusion equation on the evolving curve. The scheme for the curve evolution…
Discrete element method simulations of confined bidisperse granular shear flows elucidate the balance between diffusion and segregation that can lead to either mixed or segregated states, depending on confining pressure. Results indicate…
In this article, we introduce a variational algorithm, in the spirit of the minimizing movements scheme, to model the volume-preserving anisotropic mean curvature flow in 2D. We show that this algorithm can be used to prove the existence of…
In this paper, we introduce a parameterized discrete curvature ($\alpha$-curvature) for piecewise linear metrics on polyhedral surfaces, which is a generalization of the classical discrete curvature. A discrete uniformization theorem is…
We study graphical mean curvature flow of complete solutions defined on subsets of Euclidean space. We obtain smooth long time existence. The projections of the evolving graphs also solve mean curvature flow. Hence this approach allows to…
We consider the volume preserving geometric evolution of the boundary of a set under fractional mean curvature. We show that smooth convex solutions maintain their fractional curvatures bounded for all times, and the long time asymptotics…
We consider the graphical mean curvature flow of maps ${\bf f}:\mathbb{R}^m\to\mathbb{R}^n$, $m\ge 2$, and derive estimates on the growth rates of the evolved graphs, based on a new version of the maximum principle for properly immersed…
We prove that the mean curvature of a smooth surface in $\mathbb{R}^n$, $n\geq 2$, arises as the limit of a sequence of functions that are intrinsically related to the difference between an $n$- and $1$-dimensional fractional Laplacian of a…
Models that involve coupled dynamics in a mixed-dimensional geometry are of increasing interest in several applications. Here, we describe the development of a simulation model for flow in fractured porous media, where the fractures and…
Advection and mean curvature flow is used as a model of bone microarchitecture adaptation. It is an equivalent geometric flow to prescribed mean curvature flow with an additional rate term. In order to validate numerical methods for…