Curve Shortening and the Banchoff–Pohl Inequality in Symmetric Minkowski Geometries

In this paper the classical Banchoff–Pohl inequality, an isoperimetric inequality for nonsimple closed curves in the Euclidean plane, involving the square of the winding number, is generalized to symmetric Minkowski geometries. The proof uses the well-known curve shortening flow.     

Curve shortening in a Riemannian manifold

In this paper, we study the curve shortening flow in a general Riemannian manifold. We have many results for the global behavior of the flow. In particular, we show the following results: let M be a compact Riemannian manifold. (1) If the curve shortening flow exists for infinite time, and , then for every n > 0, . Furthermore, the limiting curve exists and is a closed geodesic in M. (2) In M × S ¹, if γ₀ is a ramp, then we have a global flow which converges to a closed geodesic in C ^∞ norm. As an application, we prove the theorem of Lyusternik and Fet.      

Two Dimensional Incompressible Ideal Flow Around a Small Curve

We study the asymptotic behavior of solutions of the two dimensional incompressible Euler equations in the exterior of a curve when the curve shrinks to a point. This work links the two previous results [5 Iftimie , D. , Lopes Filho , M.C. , Nussenzveig Lopes , H.J. ( 2003 ). Two dimensional incompressible ideal flow around a small obstacle . Comm. Part. Diff. Eqs. 28 : 349 – 379 .[Taylor & Francis Online], [Web of Science ®] , [Google Scholar], 9 Lacave , C. ( 2009 ). Two dimensional incompressible ideal flow around a thin obstacle tending to a curve . Annales de l'IHP, Anl. 26 : 1121 – 1148 . [Google Scholar]]. The second goal of this work is to complete the previous article, in defining the way the obstacles shrink to a curve. In particular, we give geometric properties for domain convergences in order that the limit flow be a solution of Euler equations.     

Uniqueness of Self-Similar Solutions to the Network Flow in a Given Topological Class

This paper deals with the uniqueness, within a fixed topological class, of “tree-like” solutions to the evolution of networks by curve shortening flow. More precisely, we show that if for a given initial condition, there is a solution to the network flow that is tree-like and regular for positive times, then this solution is unique within its topological class. The result in particular applies to expanding self-similar solutions. The proof is based on the following Allen–Cahn approximation result: every regular tree-like solution to the network flow can be realized as the nodal set of a family of solutions to the Allen–Cahn equation. Then, the main result of this paper follows from the uniqueness of the “ε-level” solutions. The results in this paper deal only with uniqueness of solutions. The existence of solutions for the general class of initial conditions that we consider in this paper is unknown in most cases.     

The Connectedness of Space Curve Invariants

In this paper we will give necessary conditions for a Borel-fixed monomial ideal to be the generic initial ideal of a reduced, irreducible, non-degenerate curve in P³.     

On Gage-Hamilton's entropy formula and Harnack inequality for the curve shortening flow

By the first two derivatives of the Boltzmann entropy of the curvature, which was first studied by Gage and Hamilton for the curve shortening flow in the plane, we define a monotonicity formula which is strictly increasing unless on a shrinking circle. By calculating pointwisely we give an alternate proof of Gage-Hamilton's Harnack inequality.     

Type I Singularities in the Curve Shortening Flow Associated to a Density

We define Type I singularities for the mean curvature flow associated to a density \(\psi \) (\(\psi \)MCF ) and describe the blow-up at any singular time of these singularities. Special attention is paid to the case where the singularity comes from the part of the \(\psi \)-curvature due to the density. We describe a family of curves whose evolution under \(\psi \)MCF (in a Riemannian surface of non-negative curvature with a density that is singular at a geodesic of the surface) produces only Type I singularities and study the limits of their rescalings.     

Evolution Equations of Curvature Tensors Along the Hyperbolic Geometric Flow

The author considers the hyperbolic geometric flow δ2/δt2 g(t) =-2Ricg(t) introduced by Kong and Liu. Using the techniques and ideas to deal with the evolution equations along the Ricci flow by Brendle, the author derives the global forms of evolution equations for Levi-Civita connection and curvature tensors under the hyperbolic geometric flow. In addition, similarly to the Ricci flow case, it is shown that any solution to the hyperbolic geometric flow that develops a singularity in finite time has unbounded Ricci curvature.     

On a Class of Generalized Curve Flows for Planar Convex Curves

In this paper, the authors consider a class of generalized curve flow for convex curves in the plane. They show that either the maximal existence time of the flow is finite and the evolving curve collapses to a round point with the enclosed area of the evolving curve tending to zero, i.e., limt→T A(t) = 0, or the maximal time is infinite, that is, the flow is a global one. In the case that the maximal existence time of the flow is finite, they also obtain a convergence theorem for rescaled curves at the maximal time.     

On multiquadratic extensions of n-hilbert fields in characteristic 2

We give a quasihomogeneity criterion for Gorenstein curves. For complete intersections, it is related to the first step of Vasconcelos’ normalization algorithm. In the process, we give a simplified proof of the Kunz–Ruppert criterion.     

Area‐preserving curvature flow of one‐dimensional graphs and application to smoothing of dynamic PET data

We design two area‐preserving curvature flows of one‐dimensional graphs and analyse the asymptotic shape of the solutions. An application to the smoothing of curves generated in dynamic positron emission tomography (PET) is also presented.     

关于奇数维流形上规范化的Ricci流的一个注记

要设（Mn,go）（n奇数）是紧Riemannian流形,λ（go）〉0,这里λ（go）是算子-4△go＋R（go）的第一特征值,R（go）是（Mn,go）的数量曲率．设以（Mn,go）为初值的规范化的Ricci流的极大解g（t）满足｜R（g（t））｜≤C和λ（对某个常数C一致成立）．我们证明这个解有子列收敛于一个Ricci收缩孤立子．进一步,当n=3时,条件fM ｜Rm（g（t））＋n/2dμt ≤ C可去．     

Sharp criteria of global existence and blow-up for a type of nonlinear parabolic equations

We study a type of nonlinear parabolic equations. In terms of the variational characterization of the corresponding nonlinear elliptic equations and the invariant flow arguments, we establish the sharp criteria for global existence and blow-up. Furthermore, we also get the instability of the steady states and the global existence with small initial data.     

Two Dimensional Incompressible Ideal Flow Around a Small Obstacle

Abstract

In this article we study the asymptotic behavior of incompressible, ideal, time-dependent two dimensional flow in the exterior of a single smooth obstacle when the size of the obstacle becomes very small. Our main purpose is to identify the equation satisfied by the limit flow. We will see that the asymptotic behavior depends on γ, the circulation around the obstacle. For smooth flow around a single obstacle, γ is a conserved quantity which is determined by the initial data. We will show that if γ = 0, the limit flow satisfies the standard incompressible Euler equations in the full plane but, if γ≠ 0, the limit equation acquires an additional forcing term. We treat this problem by first constructing a sequence of approximate solutions to the incompressible 2D Euler equation in the full plane from the exact solutions obtained when solving the equation on the exterior of each obstacle and then passing to the limit on the weak formulation of the equation. We use an explicit treatment of the Green's function of the exterior domain based on conformal maps, a priori estimates obtained by carefully examining the limiting process and the Div-Curl Lemma, together with a standard weak convergence treatment of the nonlinearity for the passage to the limit.     

二阶流体在旋转坐标系中的三维管道流动

就两个水平板构成的旋转系统,在磁场作用下分析二阶磁流体在其间的流动.下表面是一块可伸展的平面,上面是一块多孔的固体平板.选用合适的变换,将质量和动量的守恒方程,简化为耦合的非线性常微分方程组.应用最强大的分析技术,即同伦分析法(HAM),得到该非线性耦合方程组的级数解.结果用图形给出,并详细地讨论了无量纲参数Re,λ,Ha2,α和K2对速度场的影响.     

On a Logarithmic Type Nonlocal Plane Curve Flow

In this paper the author devotes to studying a logarithmic type nonlocal plane curve flow.Along this flow,the convexity of evolving curve is preserved,the perimeter decreases,while the enclosed area expands.The flow is proved to exist globally and converge to a finite circle in the C∞metric as time goes to infinity.     

Blow up for Systems of Wave Equations in Exterior Domain

In this paper the author studies the initial boundary value problem of semilinear wave systems in exterior domain in high dimensions(n ≥ 3). Blow up result is established and what is more, the author gets the upper bound of the lifespan. For this purpose the test function method is used.