Regularity criteria for suitable weak solutions of the Navier-Stokes equations near the boundary

We present some new regularity criteria for ``suitable weak solutions'' of the Navier-Stokes equations near the boundary in dimension three. We prove that suitable weak solutions are H\"older continuous up to the boundary provided that the scaled mixed norm $L^{p,q}_{x,t}$ with $3/p+2/q\leq 2, 2<q\le \infty$, $(p,q) \not = (3/2,\infty)$, is small near the boundary. Our methods yield new results in the interior case as well. Partial regularity of weak solutions is also analyzed under some conditions of the Prodi-Serrin type

(E)-2-[4-(2-Chloro­phen­yl)but-3-en-2-yl­idene]malononitrile

There are two independent but virtually identical mol­ecules in the asymmetric unit of the title compound, C13H19ClN2. Each mol­ecular skeleton displays an approximately planar structure except for the methyl group [the r.m.s. deviations for all 16 non-H atoms are 0.039 (mol­ecule 1) and 0.056 Å (mol­ecule 2)]. An E configuration is found about each of the C=C bonds. The crystal packing is stabilized by C—H⋯N inter­actions that connect the independent mol­ecules into supra­molecular chains along the c-axis direction

Diffusion through porus media in transition region

Chemical engineering is the design of industrial processes in which raw materials are under physical and/or chemical treatment. Catalysis is very importment in these processes. Therefore, a knowledge of the movement of molecules of fluids into a catalyst particle, usually in pores, is important. The diffusion problem in the bulk phase has been well developed. However, when the gas molecules, henceforth called particles, diffuse through a porous medium, the mechanism of the diffusion is no longer described by particle - particle collisions. The collisions between particles and the wall of the porous medium become more and more important when the mean free path is equal to or greater than the mean pore diameter. The properties of porous solids, such as pore size distribution, tortuosity, porosity, etc. then influence in the diffusion mechanism. This study begins with a discussion of ordinary diffusion because ordinary diffusion often occurs in porous media. The theory of diffusion through capillaries is then surveyed. Many models of diffusion in porous media based on the capillary model. A modified model is suggested by the author . Experimental data are collected from published papers. The effective diffusion coefficients of gases in fifteen porous pellets were obtained by experiments. These data are then described by parallel pore models bidisperse model, tridisperse model, and revised bidisperse model. These result are discussed and compared. The methods to measure and characterize the properties of porous solids are reviewed. The apparatus of the diffusion measurements are also discussed. Several recommendations are made in the last chapter of this work.--Introduction

A Constraint Handling Strategy for Bit-Array Representation GA in Structural Topology Optimization

In this study, an improved bit-array representation method for structural topology optimization using the Genetic Algorithm (GA) is proposed. The issue of representation degeneracy is fully addressed and the importance of structural connectivity in a design is further emphasized. To evaluate the constrained objective function, Deb's constraint handling approach is further developed to ensure that feasible individuals are always better than infeasible ones in the population to improve the efficiency of the GA. A hierarchical violation penalty method is proposed to drive the GA search towards the topologies with higher structural performance, less unusable material and fewer separate objects in the design domain in a hierarchical manner. Numerical results of structural topology optimization problems of minimum weight and minimum compliance designs show the success of this novel bit-array representation method and suggest that the GA performance can be significantly improved by handling the design connectivity properly.Singapore-MIT Alliance (SMA

The local relaxation and correlation production in the quantum Ising model

Isolated quantum systems follow the unitary evolution, which guarantees the full many body state always keeps a constant entropy as its initial one. Here we consider the local dynamics of a quantum Ising model with a finite size. It turns out, for both strong and weak coupling situations, the dynamics of local observables exhibits similar relaxation behavior as the macroscopic thermodynamics, which is called the local relaxation; after a certain typical time, the relaxation behavior suddenly changes and appears random, which is referred as a recurrence. Besides, we find that the total correlation entropy of this system approximately exhibit a monotonic increasing envelope in both strong and weak coupling cases, which is quite similar as the irreversible entropy production in the standard macroscopic thermodynamics.Comment: 6 pages, 2 figure

(E)-2-[1-(4-Fluoro­phen­yl)pent-1-en-3-yl­idene]malononitrile

The title mol­ecule, C14H11FN2, is approximately planar except the ethyl group, the maximum atomic deviation being 0.105 (5) Å. The fluoro­phenyl ring and 2-propyl­idene­malononitrile unit are located on the opposite sides of the C=C double bond, showing an E configuration