Related papers: Minimum curvature flow and martingale exit times

The aim of this manuscript is to study the influence of the vorticity on the existence time in fluid systems for which global smoothness and decay is known in the case of small irrotational data. We focus on two examples: the Euler-Korteweg…

I consider the problem of vortex tunneling in a two-dimensional superconductor. The vortex dynamics is governed by the Magnus force and the Ohmic friction force. Under-barrier motion in the vicinity of the saddle point of the pinning…

We introduce a geometric evolution equation for 3-manifolds with sectional curvature of one sign which is in some sense dual to the Ricci flow. On a closed 3-manifold with negative sectional curvature, we establish short time existence and…

We describe the variation of the number $N(t)$ of spatial critical points of smooth curves (defined as a scalar distance $r$ from a fixed origin $O$) evolving under curvature-driven flows. In the latter, the speed $v$ in the direction of…

This short note studies the collapsing behavior of the K\"ahler-Ricci flow on a compact K\"ahler manifold X admitting a holomorphic submersion X -> B where B is a K\"ahler manifold of lower dimension than X. We give cohomological and…

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 prove new pinching estimate for the inverse curvature flow of strictly convex hypersurfaces in the space form $N$ of constant sectional curvature $K_N$ with speed given by $F^{-\alpha}$, where $\alpha\in (0,1]$ for $K_N=0,-1$ and…

The conformal flow of metrics [2] has been used to successfully establish a special case of the Penrose inequality, which yields a lower bound for the total mass of a spacetime in terms of horizon area. Here we show how to adapt the…

The quadratic system provided by the Time of Arrival technique can be solved analytically or by optimization algorithms. In practice, a combination of both methods is used. An important problem in quadratic optimization is the possible…

We show the consistency of a threshold dynamics type algorithm for the anisotropic motion by fractional mean curvature, in the presence of a time dependent forcing term. Beside the consistency result, we show that convex sets remain convex…

Rigorous assessment of the uncertainty is crucial to the utility of numerical simulation of Turbulent flow. The Turbulent flows are often stationary and ergodic, after some initial transient time. Therefore, the time averaged of a quantity…

We discuss recent results on minimal surfaces and mean curvature flow, focusing on the classification and structure of embedded minimal surfaces and the stable singularities of mean curvature flow. This article is dedicated to Rick Schoen.

The effects of global flow rotation and curvature on the subcritical transition to turbulence in shear flows are examined. The relevant time-scales of the problem are identified by a decomposition of the flow into a laminar and a deviation…

In this paper, we prove that the mean curvature blows up at the same rate as the second fundamental form at the first singular time $T$ of any compact, Type I mean curvature flow. For the mean curvature flow of surfaces, we obtain similar…

We consider elastic flows of closed curves in Euclidean space. We obtain optimal energy thresholds below which elastic flows preserve embeddedness of initial curves for all time. The obtained thresholds take different values between…

We prove that for any complete three-manifold with a lower Ricci curvature bound and a lower bound on the volume of balls of radius one, a solution to the Ricci flow exists for short time. Actually our proof also yields a (non-canonical)…

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…

In this paper, we consider turbulence from a geometric perspective based on the vorticity equations for incompressible viscous fluid flows. We derive several quantitative statements about the statistics of turbulent flows. In particular we…

We construct a new example of an immortal mean curvature flow of smooth embedded connected surfaces in $\mathbb R^3$, which converges to a plane with multiplicity $2$ as time approaches infinity.

Viscous streaming refers to the rectified, steady flows that emerge when a liquid oscillates around an immersed microfeature, typically a solid body or a bubble. The ability of such features to locally concentrate stresses produces strong…