On Finite Time Singularity and Global Regularity of an Axisymmetric Model for the 3D Euler Equations

We investigate the large time behavior of an axisymmetric model for the 3D Euler equations. In \cite{HL09}, Hou and Lei proposed a 3D model for the axisymmetric incompressible Euler and Navier-Stokes equations with swirl. This model shares many properties of the 3D incompressible Euler and Navier-Stokes equations. The main difference between the 3D model of Hou and Lei and the reformulated 3D Euler and Navier-Stokes equations is that the convection term is neglected in the 3D model. In \cite{HSW09}, the authors proved that the 3D inviscid model can develop a finite time singularity starting from smooth initial data on a rectangular domain. A global well-posedness result was also proved for a class of smooth initial data under some smallness condition. The analysis in \cite{HSW09} does not apply to the case when the domain is axisymmetric and unbounded in the radial direction. In this paper, we prove that the 3D inviscid model with an appropriate Neumann-Robin boundary condition will develop a finite time singularity starting from smooth initial data in an axisymmetric domain. Moreover, we prove that the 3D inviscid model has globally smooth solutions for a class of large smooth initial data with some appropriate boundary condition.Comment: Please read the published versio

2-Eth­oxy-4-[2-(3-nitro­phen­yl)­hydrazono­meth­yl]phenol

The title Schiff base compound, C15H15N3O4, was prepared from a condensation reaction of 3-eth­oxy-4-hydroxy­benz­aldehyde and 3-nitro­phenyl­hydrazine. The mol­ecule is nearly planar; the dihedral angle between the hydroxy­benzene ring and the nitro­benzene ring is 6.57 (7)°. O—H⋯O, O—H⋯N and C—H⋯O hydrogen bonding helps to stabilize the crystal structure

N′-(But-2-enyl­idene)­iso­nicotino­hydrazide

In the title Schiff base compound, C10H11N3O, the pyridine ring is twisted with respect to the mean plane containing the hydrazine chain, making a dihedral angle of 31.40 (9)°. The NH group inter­acts with the N atom of the pyridine ring through N—H⋯N hydrogen bonds to build up a zigzag chain developing parallel to the (01) plane

Soft corneal contact lens for postoperative comfort in patients with recurrent pterygium function

AIM:To evaluate the effect of soft contact lens on the comfort of postoperative patients with recurrent pterygium. METHODS: For 100 eyes of 98 patients after recurrent pterygium excision, the conjunctival implant with limbal stem cells from the upper part of the cornea was used, and the scleral surface of the operation area was transferred and covered. Fifty eyes were treated with corneal bandage lens for 5d(Group A)and 50 eyes were conventionally ensheathed(Group B). The time of corneal wound healing and various symptoms and signs were observed after operation. RESULTS: Compared with Group B, 1d after pterygium, the symptoms and signs of corneal irritation in Group A were significantly less(PPCONCLUSION: The corneal contact lens can obviously improve the early postoperative corneal irritation in patients with pterygium, promote corneal wound healing and relieve postoperative discomfort