Related papers: Jellyfish exist
We construct new expanders for mean curvature flow that are smoothly asymptotic to cones arising from certain shrinkers. For each such cone, we prove the existence of expanders of arbitrarily large genus. Thus, for a fixed incoming…
In this paper, we study self-expanding solutions to a large class of parabolic inverse curvature flows by homogeneous symmetric functions of principal curvatures in Euclidean spaces. These flows include the inverse mean curvature flow and…
An existence and uniqueness result, up to fattening, for crystalline mean curvature flows with forcing and arbitrary (convex) mobilities, is proven. This is achieved by introducing a new notion of solution to the corresponding level set…
We establish various existence and uniqueness results for the Yang-Mills flow on cylindrical end 4-manifolds. We also show long-time existence and infinite-time convergence under certain hypotheses on the underlying data.
We examine the L^2-gradient flow of Euler's elastic energy for closed curves in hyperbolic space and prove convergence to the global minimizer for initial curves with elastic energy bounded by 16. We show the sharpness of this bound by…
We consider geometric flows of hypersurfaces expanding by a function of the extrinsic curvature and we show that the homothethic sphere is the unique solution of the flow which converges to a point at the initial time. The result does not…
In this paper we establish the short-time existence and uniqueness theorem for hyperbolic geometric flow, and prove the nonlinear stability of hyperbolic geometric flow defined on the Euclidean space with dimension larger than 4. Wave…
Autonomous locomotion is a ubiquitous phenomenon in biology and in physics of active systems at microscopic scale. This includes prokaryotic, eukaryotic cells (crawling and swimming) and artificial swimmers. An outstanding feature is the…
Given a planar crystalline anisotropy, we study the crystalline elastic flow of immersed polygonal curves, possibly also unbounded. Assuming that the segments evolve by parallel translation (as it happens in the standard crystalline…
We prove that every $C^1$ generic three-dimensional flow has either infinitely many sinks, or, infinitely many hyperbolic or singular-hyperbolic attractors whose basins form a full Lebesgue measure set. We also prove in the orientable case…
We study positivity-preserving properties for the elastic flow of non-compact, complete curves in Euclidean space. Despite the fact that the canonical elastic energy is infinite in this context, we extend our recent work based on the…
We consider a wide class of approximate models of evolution of singular distributions of vorticity in three dimensional incompressible fluids and we show that they have global smooth solutions. The proof exploits the existence of suitable…
We study the length-preserving elastic flow of curves in arbitrary codimension with free boundary on hypersurfaces. This constrained gradient flow is given by a nonlocal evolution equation with nonlinear higher-order boundary conditions. We…
An existence and uniqueness result, up to fattening, for a class of crystalline mean curvature flows with natural mobility is proved. The results are valid in any dimension and for arbitrary, possibly unbounded, initial closed sets. The…
We prove short-time existence of \phi-regular solutions to the planar anisotropic curvature flow, including the crystalline case, with an additional forcing term possibly unbounded and discontinuous in time, such as for instance a white…
In this note we establish exponentially fast smooth convergence for global curve diffusion flows, and discuss open problems relating embeddedness to global existence (Giga's conjecture) and the shape of Type I singularities (Chou's…
This article discusses a relatively new geometric flow, called the hypersymplectic flow. In the first half of the article we explain the original motivating ideas for the flow, coming from both 4-dimensional symplectic topology and…
We develop a global bifurcation theory for two classes of nonlinear elastic materials. It is supposed that they are subjected to anti-plane shear deformation and occupy an infinite cylinder in the reference configuration. Curves of…
A new family of simple, analytic solutions of self-similarly expanding fireballs is found for systems with ellipsoidal symmetry and a direction dependent, generalized Hubble flow. Gaussian, shell like or oscillating density profiles emerge…
Propagation of elastic waves along the axis of cylindrical shells is of great current interest due to their ubiquitous presence and technological importance. Geometric imperfections and spatial variations of properties are inevitable in…