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Considering the evolution of a perfect fluid with self-similarity of the second kind, we have found that an initial naked singularity can be trapped by an event horizon due to collapsing matter. The fluid moves along time-like geodesics…

We investigate geometric properties of surfaces given by certain formulae. In particular, we calculate the singular curvature and the limiting normal curvature of such surfaces along the set of singular points consisting of singular points…

Two fluid configurations along a flow are conjugate if there is a one parameter family of geodesics (fluid flows) joining them to infinitesimal order. Geometrically, they can be seen as a consequence of the (infinite dimensional) group of…

Consider the complete n-vertex graph whose edge-lengths are independent exponentially distributed random variables. Simultaneously for each pair of vertices, put a constant flow between them along the shortest path. Each edge gets some…

In this article we study the tangent cones at first time singularity of a Lagrangian mean curvature flow. If the initial compact submanifold is Lagrangian and almost calibrated by Re\Omega in a Calabi-Yau n-fold (M,\Omega), and T>0 is the…

We describe all possible topological structures of typical one-parameter bifurcations of gradient flows on the 2-sphere with holes in the case that the number of singular point of flows is at most six. To describe structures, we separatrix…

We prove that the limit hypersurfaces of converging curvature flows are stable, if the initial velocity has a weak sign, and give a survey of the existence and regularity results.

Pairs of numerically computed trajectories of a chaotic system may coalesce because of finite arithmetic precision. We analyse an example of this phenomenon, showing that it occurs surprisingly frequently. We argue that our model belongs to…

The aim in these lectures is to study singularity formation, nonuniqueness, and topological change in motion by mean curvature.

We show that the mean curvature blows up at the first finite singular time for a closed smooth embedded mean curvature flow in R^3.

We study the multiplicity of the singularity of mean curvature flow with bounded mean curvature and Morse index. For $3\leq n\leq 6$, we show that either the mean curvature or the Morse index blows up at the first singular time for a closed…

The large deviation functional of the density field in the weakly asymmetric exclusion process with open boundaries is studied using a combination of numerical and analytical methods. For appropriate boundary conditions and bulk drives the…

The search of finite-time singularity solutions of Euler equations is considered for the case of an incompressible and inviscid fluid. Under the assumption that a finite-time blow-up solution may be spatially anisotropic as time goes by…

In this paper, we introduce a monotonicity formula for the mean curvature flow. We also apply this monotonicity formula to study the asymptotic behavior of eternal solutions.

We provide sufficient conditions on an initial curve for the area preserving and the length preserving curvature flows of curves in a plane, to develop a singularity at some finite time or converge to an $m$-fold circle as time goes to…

We define two conformal structures on $S^1$ which give rise to a different view of the affine curvature flow and a new curvature flow, the ``$Q$-curvature flow". The steady state of these flows are studied. More specifically, we prove four…

It is shown that a hypersurface of a space form is the initial data for a solution to the mean curvature flow by parallel hypersurfaces if, and only if, it is isoparametric. By solving an ordinary differential equation, explicit solutions…

For each $g\ge 3$, we prove existence of a compact, connected, smoothly embedded, genus-$g$ surface $M_g$ with the following property: under mean curvature flow, there is exactly one singular point at the first singular time, and the…

We investigate the formation of singularities in the incompressible Navier-Stokes equations in $d\geq 2$ dimensions with a fractional Laplacian $|\nabla |^\alpha$. We derive analytically a sufficient but not necessary condition for…

We show that the mean curvature flow of a generic closed surface in $\mathbb{R}^3$ avoids multiplicity one tangent flows that are not round spheres/cylinders. In particular, we show that any non-cylindrical self-shrinker with a cylindrical…