Related papers: On the evolution by fractional mean curvature
We study the provenance of singularity formation under mean curvature flow and volume preserving mean curvature flow in an axially symmetric setting. We prove that if the mean curvature is uniformly bounded on any finite time interval, then…
We provide a mean curvature flow method for numerical cosmology and test it on cases of inhomogenous inflation. The results show (in a proof of concept way) that the method can handle even large inhomogeneities that result from different…
Fracture functions and their evolution equations are reviewed. Some phenomenological applications are briefly discussed.
We provide a novel existence result for energy-variational solutions to a general class of evolutionary partial differential equations. Compared to previous works on this solution concept, the generalization is mainly twofold: a relaxation…
Evolution equations describe the effect of consecutively integrating out all quantum fluctuations with momenta larger than some infrared cutoff scale k. We develop a formalism for the introduction of collective degrees of freedom at some…
We study some properties of mean curvature flow solitons in general Riemannian manifolds and in warped products, with emphasis on constant curvature and Schwarzschild type spaces. We focus on splitting and rigidity results under various…
Fourth order curvature driven interface evolution equations frequently appear in the natural sciences. Often axisymmetric geometries are of interest, and in this situation numerical computations are much more efficient. We will introduce…
Biodiversity and extinction are central issues in evolution. Dynamical balance among different species in ecosystems is often described by deterministic replicator equations with moderate success. However, fluctuations are inevitable,…
For an ancient solution of the mean curvature flow, we show that each time slice M_t is contained in an affine subspace with dimension bounded in terms of the density and the dimension of the evolving submanifold. Recall that an ancient…
A self-similar solution for time evolution of quasi-spherical, self-gravitating accretion flows is obtained under the assumption that the generated heat by viscosity is retained in the flow. The solutions are parameterized by the ratio of…
We address surface gradient flows which allow for energy dissipation by evolving the surface and a scalar quantity on it, simultaneously. A proper choice of the time derivative and the gauge of surface independence guarantees energy…
An analysis of a fractional cubic differential equation is presented, which is a generalization of different versions of fractional logistic equations, in order to obtain simpler numerical methods that globalize and extend the results…
We study fractional configurations in gravity theories and Lagrange mechanics. The approach is based on Caputo fractional derivative which gives zero for actions on constants. We elaborate fractional geometric models of physical…
We consider fractional partial differential equations posed on the full space $\R^d$. Using the well-known Caffarelli-Silvestre extension to $\R^d \times \R^+$ as equivalent definition, we derive existence and uniqueness of weak solutions.…
We introduce and study a conformal heat flow of harmonic maps defined by an evolution equation for a pair consisting of a map and a conformal factor of metric on the two-dimensional domain. This flow is designed to postpone finite time…
The dynamics of the Reynolds stress tensor for turbulent flows is described with an evolution equation coupling both geometric effects and turbulent source terms. The effects of the mean flow geometry are shown up when the source terms are…
The mean curvature flow describes the evolution of a surface (a curve) with normal velocity proportional to the local mean curvature. It has many applications in mathematics, science and engineering. In this paper, we develop a numerical…
This paper considers and proposes some algorithms to compute the mean curvature flow under topological changes. Instead of solving the fully nonlinear partial differential equations based on the level set approach, we propose some…
This note corrects a mistake in the original book in the evolution equations of total curvature for the curve-shrinking flow in an ambient Ricci Flow. The resulting upper bound for the evolution of total curvature is an exponential bound in…
We show that any smooth solution to the mean curvature flow equations coming out of a rotationally symmetric double cone is also rotationally symmetric.