相关论文: A new version of Brakke's local regularity theorem
Introducing a moment map whose zero locus is the group of symplectomorphisms of the real four-dimensional torus, we exhibit a gradient flow that can be made into a strictly parabolic flow by mean of a DeTurck trick (famously known for its…
We prove that if a one-parameter family of varifolds has an $L^2$ normal velocity $v$ in the sense of Brakke, and if the family is represented as the graph of a continuous function $f$ with continuous spatial derivative $\nabla f$, then $f$…
In this paper, we consider smooth, properly immersed hypersurfaces evolving by mean curvature in some open subset of $\mathbb{R}^{n+1}$ on a time interval $(0, t_0)$. We prove that $p$ - integrability with $p\ge 2$ for the second…
An important consequence of the Hahn-Banach Theorem says that on any locally convex Hausdorff topological space $X$, there are sufficiently many continuous linear functionals to separate points of $X$. In the paper, we establish a `local'…
The classical Prandtl-Batchelor theorem (Prandtl 1904; Batchelor 1956) states that in the regions of steady 2D flow where viscous forces are small and streamlines are closed, the vorticity is constant. In this paper, we extend this theorem…
We establish the well-posedness of the nonlocal mean curvature flow of order ${\alpha\in(0,1)}$ for periodic graphs on $\mathbb{R}^n$ in all subcritical little H\"older spaces ${\rm h}^{1+\beta}(\mathbb{T}^n)$ with $\beta\in(0,1)$.…
In this article, we study regularity criteria for the 3D micropolar fluid equations in terms of one partial derivative of the velocity. It is proved that if \begin{equation*}…
In the pseudo-Euclidean space $\mathbb{R}^{n+1,k}$, we consider the mean curvature flow of $n$-dimensional spacelike submanifolds with spacelike codimension one and arbitrary timelike codimension $k$. We show that if the initial submanifold…
In this paper we relax the current regularity theory for the eikonal equation by using the recent theory of { set-valued} iterated Lie brackets. We give sufficient conditions for small time local attainability of general, symmetric,…
This paper contains a new proof of the short-time existence for the flow by curvature of a network of curves in the plane. Appearing initially in metallurgy and as a model for the evolution of grain boundaries, this flow was later treated…
In this paper, we will derive a small energy regularity theorem for the mean curvature flow of arbitrary dimension and codimension. It says that if the parabolic integral of $|A|^2$ around a point in space-time is small, then the mean…
In this paper we show that a nonlocal minimal surface which is a graph outside a cylinder is in fact a graph in the whole of the space. As a consequence, in dimension~$3$, we show that the graph is smooth. The proofs rely on convolution…
In this paper, we introduce a new method to establish existence of geometric flows with surgery. In contrast to all prior constructions of flows with surgery in the literature our new approach does not require any a priori estimates in the…
We prove the Multiplicity One Conjecture for mean curvature flows of surfaces in $\mathbb{R}^3$. Specifically, we show that any blow-up limit of such mean curvature flows has multiplicity one. This has several applications. First, combining…
A stochastic flow of homeomorphisms of the real line previously studied by Bass and Burdzy is shown to arise in describing a Brownian motion conditional on knowing its local times on hitting a fixed level. This makes it possible to connect…
It is investigated a possibility of physical interpretation of vector fields (dynamic flows) in Euclidean spaces of higher dimension. There are analyzed the methods of measurements of dynamic flows, the characteristics of dynamic flow and…
We continue our analysis of the thresholding scheme from the variational viewpoint and prove a conditional convergence result towards Brakke's notion of mean curvature flow. Our proof is based on a localized version of the minimizing…
We develop a theory of surfaces with boundary moving by mean curvature flow. In particular, we prove a general existence theorem by elliptic regularization, and we prove boundary regularity at all positive times under very mild hypotheses.
This paper extends previous work from arxiv:1702.05223, which shows that the main theorem of Morse theory holds for a large class of functions on singular spaces, where the function and the underlying singular space are required to satisfy…
In this work the evolution of a fluid droplet in vacuum is considered. This means that the surface tension and the fluid forces are in equilibrium at the free boundary. The fluid is governed by the incompressible quasi-steady Stokes…