Related papers: Jellyfish exist
Let $(M,J,\Omega)$ be a closed polarized complex manifold of K\"ahler type. Let $G$ be the maximal compact subgroup of the automorphism group of $(M,J)$. On the space of K\"ahler metrics that are invariant under $G$ and represent the…
We develop a geometric mechanism to prove the existence of orbits that drift along a prescribed sequence of cylinders, under some general conditions on the dynamics. This mechanism can be used to prove the existence of Arnold diffusion for…
Truncated Taylor expansions of smooth flow maps are used in Hamilton's principle to derive a multiscale Lagrangian particle representation of ideal fluid dynamics. Numerical simulations for scattering of solutions at one level of truncation…
We introduce variational approximations for curve evolutions in two-dimensional Riemannian manifolds that are conformally flat, i.e.\ conformally equivalent to the Euclidean space. Examples include the hyperbolic plane, the hyperbolic disk,…
The free motion of charged colloids within ionic solutions and in the vicinity of charged boundaries, is a phenomenon that occurs in various natural, biological and industrial settings. Here, we develop an electrohydrodynamic lubrication…
For inviscid fluid flow in any n-dimensional Riemannian manifold, new conserved vorticity integrals generalizing helicity, enstrophy, and entropy circulation are derived for lower-dimensional surfaces that move along fluid streamlines.…
We prove the existence of a large class of global-in-time expanding solutions to vacuum free boundary compressible Euler flows without relying on the existence of an underlying finite-dimensional family of special affine solutions of 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…
We develop new variational principles to study stability and equilibrium of axisymmetric flows. We show that there is an infinite number of steady state solutions. We show that these steady states maximize a (non-universal) $H$-function. We…
We give biLipschitz models for the Ricci flow on some 4-manifolds (minimal surfaces of general type), exhibiting a combination of expanding and static behavior.
We investigate several closely related "homothety conjectures" for convex bodies on a plane. Using the modern language of differential geometry, we systematically derive the fundamental properties of bodies of flotation, bodies of buoyancy,…
In this paper, we prove that finite-dimensional topological flows without fixed points and having a countable number of periodic orbits, have the small flow boundary property. This enables us to answer positively a question of Bowen and…
A mechanism is proposed for the growth of vesicles dispersed in a liquid solvent and a size distribution function is obtained for the vesicles, both from the first principles calculations. This distribution function is shown to be…
Streets and Tian introduced pluriclosed flow and symplectic curvature flow in recent years. Here we construct a curvature flow to unify these two flows. We show the short time existence of our flow and exhibit an obstruction to long time…
Streamer electrical discharges are often investigated with computer simulations of density models (also called drift-diffusion-reaction models). We review these models, detailing their physical foundations, their range of validity and the…
We exhibit orbits of the geodesic flow on a hyperbolic surface with at least one cusp such that every tubular neighborhood contains uncountably many distinct geodesic flow orbits. The proof relies on new phenomena, namely the existence of…
We study the Ricci flow on complete Kaehler metrics that live on the complement of a divisor in a compact complex manifold. In earlier work, we considered finite-volume metrics which, at spatial infinity, are transversely hyperbolic. In…
We show that every convex ancient solution of mean curvature flow with Type I curvature growth is either spherical, cylindrical, or planar. We then prove the corresponding statement for flows by a natural class of curvature functions which…
The equations for the three-dimensional incompressible flow of liquid crystals are considered in a smooth bounded domain. The existence and uniqueness of the global strong solution with small initial data are established. It is also proved…
The dynamics along the particle trajectories for the 3D axisymmetric Euler equations in an infinite cylinder are considered. It is shown that if the inflow-outflow is highly oscillating in time, the corresponding Euler flow cannot keep the…