Besov-Morrey spaces: Function space theory and applications to non-linear PDE

This paper is devoted to the analysis of function spaces modeled on Besov spaces and their applications to non-linear partial differential equations, with emphasis on the incompressible, isotropic Navier-Stokes system and semi-linear heat equations. Specifically, we consider the class, introduced by Hideo Kozono and Masao Yamazaki, of Besov spaces based on Morrey spaces, which we call Besov-Morrey or BM spaces. We obtain equivalent representations in terms of the Weierstrass semigroup and wavelets, and various embeddings in classical spaces. We then establish pseudo-differential and para-differential estimates. Our results cover non-regular and exotic symbols. Although the heat semigroup is not strongly continuous on Morrey spaces, we show that its action defines an equivalent norm. In particular, homogeneous BM spaces belong to a larger class constructed by Grzegorz Karch to analyze scaling in parabolic equations. We compare Karch's results with those of Kozono and Yamazaki and generalize them by obtaining short-time existence and uniqueness of solutions for arbitrary data with subcritical regularity. We exploit pseudo-differential calculus to extend the analysis to compact, smooth, boundaryless, Riemannian manifolds. BM spaces are defined by means of partitions of unity and coordinate patches, and intrinsically in terms of functions of the Laplace operator.

     

The liquid-crystalline properties of bis[N-[[4-[4-(alkoxy)benzoyloxy]2-hydroxyphenyl]methylene]alkanamino] complexes of Cu(II), Pd(II) and Ni(II). A general view

The thermotropic liquid-crystalline phase behaviour of a homologous set of bis[N-[[4-[4-(alkoxy)benzoyloxy]2-hydroxyphenyl]methylene]alkanamino] copper(II) complexes is examined. New data are reported and taken into account in addition to those previously reported in this Journal. The influence of alkoxy and alkanamine groups on the nature and stability of the mesophase is underlined. As a general trend, with some notable exception, smectic (C type) mesomorphism is favoured by longer alkoxy and alkanamine chains. Data concerning some Pd(II) and Ni(II) homologous complexes are also reported. The mesophase stability (nematic phase) is higher for palladium and nickel complexes than for the copper containing homologues.     

Chemometric approaches in environmental problems concerning PCDD and PCDF. Data interpretation and source correlation. Mechanisms of formation and destruction in MSW combustion process

Summary The role of multivariate analysis methods in evaluating, rationalizing, and working out complex environmental problems is discussed. The discussion is organized in two sections; a literature analysis of the application of chemometric methods to PCDD/PCDF data interpretation and source correlation and a review of the role of chemometric methods in analysing the results obtained by the Authors studying PCDD/PCDF formation and destruction mechanisms in MSW combustion processes.     

Generation of analytic semigroups by degenerate elliptic operators

We consider the generation of analytic semigroups by degenerate elliptic operators satisfying a uniform H?rmander condition under homogeneous Dirichlet boundary conditions. Received January 12, 1996     

The behaviour of alkylcobalt carbonyls under “oxo” conditions

Both normal and secondary alkylcobalt carbonyls are shown not to undergo isomerization under catalytic “oxo” conditions, and the relevance of this obervation for the mechanism of the “oxo” reaction is discussed.     

Acid-base equilibria in organic solvents : Part 2. Cyclic voltammetry in the study of hydrogen-bond formation

The cyclic voltammetric technique is used to study hydrogen-bond formation in some polar organic solvents (S) of electroanalytical interest (1,2-dimethoxy, tetrahydrofuran, dimethylformamide, dimethyl sulphoxide and pyridine). The cathodic shift of the proton reduction caused by stepwise addition of the solvent investigated to a solution of anhydrous perchloric acid in acetonitrile is utilized. The theoretical treatment applied produced evidence that HS⁺ and HS⁺₂ are the only acidic species involved, so that the relevant formation constants can be evaluated. The data obtained mostly compare well with those available in the literature. The features that condition the tendency to hydrogen-bonding and the effect of hydrogen bonding on solvent basicity are discussed.     

Asymmetric synthesis by chiral ruthenium complexes : VI. Enantiomer-discrimination in the hydrogen transfer from racemic alcohols to prochiral ketones in the presence of H4Ru4(CO)8[(−)-diop]2

Hydrogen transfer from racemic alcohols to prochiral ketones in the presence of H₄Ru₄(CO)₈[(?)-DIOP]₂ has been examined. The enantiomer-discrimination is influenced by the structure of the reactants, temperature and the excess of phosphine present.     

Photophysical behaviour of azaphenanthrenes

Nitrogen position and internal heavy atom effects on the radiative and radiationless transitions from the lowest excited states of the isomeric azaphenanthrenes and some of their methyl, chlorine and bromine derivatives have been studied in E.P.A. solutions at 77 K. The nitrogen position affects the fluorescence and S₁-T₁ intersystem crossing rates more than the phosphorescence and T₁-S₀ intersystem crossing rates. Small differences in the behaviour of 9-azaphenanthrene are enhanced in non-hydroxylic solvents and at room temperature, and it is inferred that (n, π*) states play a more important role in the photophysical behaviour of this isomer. Halogen, substitution in all the isomers increases the phosphorescence rate, induces a smaller increase in the T₁-S₀ intersystem crossing rates and has a negligible effect on the fluorescence rate.     

Solvent and temperature effects on the fluorescence and competitive photoreactions of cis-9-styrylanthracene

A fluorimetric and photochemical study of cis-9-styrylanthracene as a function of temperature has been carried out in different solvents with the aim of answering some open questions about the behavior of its first excited singlet state, S₁. In non-polar solvents, a parallel photoreactive pathway, leading to a cyclization adduct, was found to compete with fluorescence and isomerization already at 200 K, its contribution increasing markedly with temperature. The cis » trans photoisomerization occurs prevalently by a triplet mechanism, a detectable contribution of diabatic and adiabatic isomerization in S₁ being operative in these solvents from room temperature upwards. In polar solvents, the main deactivation pathway competitive with fluorescence is isomerization to trans, which occurs prevalently through a mixed singlet mechanism with a major diabatic and a minor adiabatic components.     

On the capacity of boolean graph formulæ

In a recent paper we have solved several well-known combinatorial problems treating them as special cases of our generalization of Shannon's notion of graph capacity. We present a new simple formalism to deal with all such problems in a unified manner, considering graphs or families of graphs as special formulæ the variables of which are pairs of vertices of their common vertex sets. In all of these problems, the maximum size of a set ofn-length sequences from a fixed alphabet is to be determined under various restrictions on the element pairs present in the same coordinate of any two sequences from the set. For sufficiently homogeneous formulæ capacity becomes computable.New applications include generalizations of our result on the maximum number of pairwise qualitatively independentk-partitions of ann-set from independence to various forms of qualitative dependence.Work partially supported by the Italian Ministry of the University and Scientific Research, Project: Algoritmi, Modelli di Calcolo e Strutture Informative.