Zero Viscosity Limit for Analytic Solutions, of the Navier-Stokes Equation on a Half-Space.¶I. Existence for Euler and Prandtl Equations

This is the first of two papers on the zero-viscosity limit for the incompressible Navier-Stokes equations in a half-space. In this paper we prove short time existence theorems for the Euler and Prandtl equations with analytic initial data in either two or three spatial dimensions. The main technical tool in this analysis is the abstract Cauchy-Kowalewski theorem. For the Euler equations, the projection method is used in the primitive variables, to which the Cauchy-Kowalewski theorem is directly applicable. For the Prandtl equations, Cauchy-Kowalewski is applicable once the diffusion operator in the vertical direction is inverted. Received: 5 September 1995 / Accepted: 14 July 1997     

The fluid dynamic limit of the nonlinear boltzmann equation

Solutions of the nonlinear Boltzmann equation are constructed up to the first appearance of shocks in the corresponding fluid dynamics. This construction assumes the knowledge of solutions of the Euler equations for compressible gas flow. The Boltzmann solution is found as a truncated Hilbert expansion with a remainder, and the remainder term solves a weakly nonlinear equation which is solved by iteration. The solutions found have special initial values. They should serve as “outer expansions” to which initial layers, boundary layers and shock layers can be matched.     

Reactivity of β-aluminas with water

The reactivity with air moisture of finely divided crystals of Na β and Li⁺-containing β-aluminas is analyzed with thermogravimetric and X-ray techniques. For Na β-alumina two distinct water intake processes are evident: 1) a fast one, controlled by the elastic interactions among water molecules near the crystal boundaries; 2) a slow one, driven by water diffusion away from the boundaries. For finely divided crystals of Li β-alumina the two processes cannot be distinguished but the slow one is observable in crystals of macroscopic dimensions. The water intake is generally accompanied by an expansion of the c-axis. However, the relationship between water content and c-parameter is highly non-linear and the c-axis length at room temperature depends upon the thermal history of the sample. Our results confirm previous reports suggesting that β-aluminas absorb about one water molecule per cation of the conducting plane.     

Hexasulfanyl analogues of cyclotriveratrylene

The synthesis of four 3,4-di(alkylsulfanyl)benzyl alcohol derivatives is described, in five steps from methyl 3,4-di(hydroxy)benzoate via a Newman–Kwart rearrangement. Incubation of these derivatives in formic acid affords 2,3,7,8,12,13-hexakis(alkylsulfanyl)-10,15-dihydro-5H-tribenzo[a,d,g]cyclononene products, which are hexa-sulfanyl analogues of the well-known supramolecular cavitand host, cyclotriveratrylene (CTV). The yield of this cyclization depends strongly on the alkylsulfanyl substituents present, in the order SMe > SEt ≈ SiPr ? SBn. A crystal structure determination of one of the cyclotrimers shows a mode of self-association that is commonly exhibited by CTV itself.     

In silico discovery of beta-secretase inhibitors

Alzheimer's disease, the most common amyloid-associated disorder, accounts for the majority of the dementia diagnosed after the age of 60. The cleavage of the beta-amyloid precursor protein is initiated by beta-secretase (BACE-1), a membrane-bound aspartic protease, which has emerged as an important but difficult protein target. Here, an in silico screening approach consisting of fragment-based docking, ligand conformational search by a genetic algorithm, and evaluation of free energy of binding was used to identify low-molecular-weight inhibitors of BACE-1. More than 300,000 small molecules were docked and about 15,000 prioritized according to a linear interaction energy model with evaluation of solvation by continuum electrostatics. Eighty-eight compounds were tested in vitro, and 10 of them showed an IC(50) value lower than 100 microM in a BACE-1 enzymatic assay. Interestingly, the 10 active compounds shared a triazine scaffold. Moreover, four of them were active in an assay with mammalian cells (EC(50) < 20 microM), indicating that they are cell-permeable. Therefore, these triazine derivatives are very promising lead candidates for BACE-1 inhibition. The discoveries of this series and two other series of nonpeptidic BACE-1 inhibitors demonstrate the usefulness of our in silico high-throughput screening approach.     

An intramolecularly self-templated synthesis of macrocycles: self-filling effects on the formation of prismarenes

Ethyl- and propyl-prism[6]arenes are obtained in high yields and in short reaction times, independent of the nature and size of the solvent, in the cyclization of 2,6-dialkoxynaphthalene with paraformaldehyde. PrS[6]Et or PrS[6]nPr adopt, both in solution and in the solid state, a folded cuboid-shaped conformation, in which four inward oriented alkyl chains fill the cavity of the macrocycle. On these bases, we proposed that the cyclization of PrS[6]Et or PrS[6]nPr occurs through an intramolecular thermodynamic self-templating effect. In other words, the self-filling of the internal cavity of PrS[6]Et or PrS[6]nPr stabilizes their cuboid structure, driving the equilibrium toward their formation. Molecular recognition studies, both in solution and in the solid state, show that the introduction of guests into the macrocycle cavity forces the cuboid scaffold to open, through an induced-fit mechanism. An analogous conformational change from a closed to an open state occurs during the endo-cavity complexation process of the pentamer, PrS[5]. These results represent a rare example of a thermodynamically controlled cyclization process driven through an intramolecular self-template effect, which could be exploited in the synthesis of novel macrocycles.

Ethyl- and propyl-prism[6]arenes are obtained by an intramolecular thermodynamic self-template effect: the self-filling of the internal cavity stabilizes their cuboid structure, driving the equilibrium toward their formation.