Related papers: Lower bounds for boundary roughness for droplets i…
We exhibit a concentration-collapse decomposition of singularities of fourth order curvature flows, including the $L^2$ curvature flow and Calabi flow, in dimensions $n \leq 4$. The proof requires the development of several new a priori…
In this note, we investigate Bernoulli oriented bond percolation with parameter $p$ on $\mathbb{Z}^2$. In addition to the standard edges, which are open with probability $p$, we introduce diagonal edges each open with probability…
In this paper, we propose nonlocal diffusion models with Dirichlet boundary. These nonlocal diffusion models preserve the maximum principle and also have corresponding variational form. With these good properties, we can prove the…
Assuming a lower bound on the Ricci curvature of a complete Riemannian manifold, for $p< 1/2$ we show the existence of bounds on the local $L^p$ norm of the Ricci curvature that depend only on the dimension and which improve with volume…
The main focus of this article concerns the strongly percolative regime of the vacant set of random interlacements on $ \mathbb{Z}^d$, with $d \ge 3$. We investigate the occurrence in a large box of an excessive fraction of sites that get…
When two drops of radius $R$ touch, surface tension drives an initially singular motion which joins them into a bigger drop with smaller surface area. This motion is always viscously dominated at early times. We focus on the early-time…
Marangoni self-contracted droplets are formed by a mixture of two liquids, one of larger surface tension and larger evaporation rate than the other. Due to evaporation, the droplets contract to a stable contact angle instead of spreading on…
The process of breaking of inviscid incompressible flows along a rigid body with slipping boundary conditions is studied. Such slipping flows are compressible, which is the main reason for the formation of a singularity for the gradient of…
In this paper we give a geometric argument for bounding the diameter of a connected compact surface (with boundary) of arbitrary codimension in Euclidean space in terms of Topping's diameter bound for closed surfaces (without boundary). The…
The maximum volume ($\Omega$) of a droplet that can remain attached to a horizontal fiber defines the stability limit of droplet-fiber interactions, phenomena common in nature and critical to diverse engineering applications. Existing…
Leptons, quarks and gauge bosons are assumed to be described by local field theory. Stringent bounds on nonlocal scale can be derived from the high- presision LEP measurements. We find a bound on nonlocal scale $\Lambda_{nl} > 1020\: Gev$,…
When a droplet hits a surface fast enough, droplet splashing can occur: smaller secondary droplets detach from the main droplet during impact. While droplet splashing on smooth surfaces is by now well understood, the surface roughness also…
This paper is concerned with the incompressible limit of the compressible hydrodynamic flow of liquid crystals with periodic boundary conditions in R^N(N = 2, 3). It is rigorously shown that the local (and global) strong solution of the…
The variance of first-passage percolation admits a decomposition into Fourier levels indexed by the order of environment derivatives. These Fourier levels capture how local perturbations of different orders contribute to global…
The optimal exponentials of the thickness in the geometry rigidity inequality of shells represent the geometry rigidity of the shells. We obtain that the lower bounds of the optimal exponentials are $4/3,$ $3/2,$ and $1,$ for the hyperbolic…
We investigate locality of the supercritical regime for Bernoulli percolation on transitive graphs with polynomial growth, by which we mean the following. Take a transitive graph of polynomial growth $\mathscr{G}$ satisfying…
Motivated by lubrication problems, we consider a micropolar uid ow in a 2D domain with a rough and free boundary. We assume that the thickness and the roughness are both of order 0 < " << 1. We prove the existence and uniqueness of a…
The nonlocal models of peridynamics have successfully predicted fractures and deformations for a variety of materials. In contrast to local mechanics, peridynamic boundary conditions must be defined on a finite volume region outside the…
We consider the two-dimensional Euler equations in non-smooth domains with corners. It is shown that if the angle of the corner $\theta$ is strictly less than $\pi/2$, the Lipschitz estimate of the vorticity at the corner is at most single…
We consider the three-dimensional Euler equations in a domain with a free boundary with no surface tension. We construct unique local-in-time solutions in the Lagrangian setting for $u_0 \in H^{2.5+\delta }$ such that the Rayleigh-Taylor…