Besov Space Regularity Conditions for Weak Solutions of the Navier–Stokes Equations

Consider a bounded domain ${{\Omega \subseteq \mathbb{R}^3}}$ with smooth boundary, some initial value ${{u_0 \in L^2_{\sigma}(\Omega )}}$ , and a weak solution u of the Navier–Stokes system in ${{[0,T) \times\Omega,\,0 < T \le \infty}}$ . Our aim is to develop regularity and uniqueness conditions for u which are based on the Besov space $$B^{q,s}(\Omega ):=\left\{v\in L^2_{\sigma}(\Omega ); \|v\|_{B^{q,s}(\Omega )} := \left(\int\limits^{\infty}_0 \left\|e^{-\tau A}v\right\|^s_q {\rm d} \tau\right)^{1/s}<\infty \right\}$$ with ${{2 < s < \infty,\,3 < q <\infty,\,\frac2{s}+\frac{3}{q} = 1}}$ ; here A denotes the Stokes operator. This space, introduced by Farwig et al. (Ann. Univ. Ferrara 55:89–110, 2009 and J. Math. Fluid Mech. 14: 529–540, 2012), is a subspace of the well known Besov space ${{{\mathbb{B}}^{-2/s}_{q,s}(\Omega )}}$ , see Amann (Nonhomogeneous Navier–Stokes Equations with Integrable Low-Regularity Data. Int. Math. Ser. pp. 1–28. Kluwer/Plenum, New York, 2002). Our main results on the regularity of u exploits a variant of the space ${{B^{q,s}(\Omega )}}$ in which the integral in time has to be considered only on finite intervals (0, δ ) with ${{\delta \to 0}}$ . Further we discuss several criteria for uniqueness and local right-hand regularity, in particular, if u satisfies Serrin’s limit condition ${{u\in L^{\infty}_{\text{loc}}([0,T);L^3_{\sigma}(\Omega ))}}$ . Finally, we obtain a large class of regular weak solutions u defined by a smallness condition ${{\|u_0\|_{B^{q,s}(\Omega )} \le K}}$ with some constant ${{K=K(\Omega, q)>0}}$ .     

The transition from spark to arc discharge and its implications with respect to nanoparticle production

Indium (III) phthalocyanine (InPc) was encapsulated into nanoparticles of PEGylated poly(d,l-lactide-co-glycolide) (PLGA-PEG) to improve the photobiological activity of the photosensitizer. The efficacy of nanoparticles loaded with InPc and their cellular uptake was investigated with MCF-7 breast tumor cells, and compared with the free InPc. The influence of photosensitizer (PS) concentration (1.8–7.5 μmol/L), incubation time (1–2 h), and laser power (10–100 mW) were studied on the photodynamic effect caused by the encapsulated and the free InPc. Nanoparticles with a size distribution ranging from 61 to 243 nm and with InPc entrapment efficiency of 72 ± 6 % were used in the experiments. Only the photodynamic effect of encapsulated InPc was dependent on PS concentration and laser power. The InPc-loaded nanoparticles were more efficient in reducing MCF-7 cell viability than the free PS. For a light dose of 7.5 J/cm² and laser power of 100 mW, the effectiveness of encapsulated InPc to reduce the viability was 34 ± 3 % while for free InPc was 60 ± 7 %. Confocal microscopy showed that InPc-loaded nanoparticles, as well as free InPc, were found throughout the cytosol. However, the nanoparticle aggregates and the aggregates of free PS were found in the cell periphery and outside of the cell. The nanoparticles aggregates were generated due to the particles concentration used in the experiment because of the small loading of the InPc while the low solubility of InPc caused the formation of aggregates of free PS in the culture medium. The participation of singlet oxygen in the photocytotoxic effect of InPc-loaded nanoparticles was corroborated by electron paramagnetic resonance experiments, and the encapsulation of photosensitizers reduced the photobleaching of InPc.     

A functional equation characterizing the second derivative

Consider an operator T:C²(R)→C(R) and isotropic maps A₁,A₂:C¹(R)→C(R) such that the functional equation

     

Characterizing the derivative and the entropy function by the Leibniz rule

Consider an operator T:C¹(R)→C(R) satisfying the Leibniz rule functional equation

     

Investigation of plasmas produced by laser ablation using single and double pulses for food analysis demonstrated by probing potato skins

We report on investigations of plasmas produced by laser ablation of fresh potatoes using infrared nanosecond laser radiation. A twin laser system consisting of two Nd:YAG oscillators was used to generate single or double pulses of adjustable interpulse delay. The potatoes were irradiated under ambient air with moderate pulse energies of about 10 mJ. The expansion dynamics of the ablation plume was characterized using fast imaging with a gated camera. In addition, time-resolved optical emission spectroscopy was applied to study the spectral line emission of the various plasma species. The electron density was deduced from Stark broadening, and the plasma temperature was inferred from the relative emission intensities of spectral lines. The relative concentrations of metals were estimated from the comparison of the measured emission spectra to the spectral radiance computed for a plasma in local thermal equilibrium. It is shown that the plasma produced by double pulses has a larger volume and a lower density. These properties lead to an increase of the signal-to-noise ratio by a factor of 2 and thus to an improved measurement sensitivity.     

SR-XRD and SR-FTIR study of the alteration of silver foils in medieval paintings

Altarpieces and polychrome carved wood from the fifteenth century AD usually exhibit golden and silvery areas by the application of a very thin foil of metal. The metal foils were normally protected from the atmosphere by a varnish or resin which maybe either preserved or absent. Moreover, they were glued to the background surface by adhesive substances (egg yolk, drying oil or animal glue). The high proportion of the glueing substances often renders the development of reaction compounds. With time, silver alters blacken or simply disappear completely. In this paper, we study the alterations to metal foils from a selection of fifteenth century artworks showing different glueing agents, organic coatings and several degrees of conservation of the organic coatings and metal leafs. The submillimetric layered structure and the high variability and low amount of most of the compounds present in the different layers, as well as their differing nature (organic and inorganic) make the use of micron-sensitive high-resolution techniques essential for their study. In particular, the high resolution, high brilliance and small footprint renders synchrotron radiation most adequate for their study. SR-XRD was performed to identify the reaction compounds formed in the different layers; μFTIR was used at to identify the silver protecting organic coatings, the metal foil glueing layers and the corresponding reaction compounds. The results obtained suggest that atmospheric corrosion is the dominant mechanism, and therefore that the degree of corrosion of the metal foils is mainly related to the conservation state of the protecting coatings.     

Open problems in renewal, coupling and Palm theory

This note presents four sets of problems. The first suggests the possibility of a limit theory for null-recurrent renewal processes similar to the theory in the positive recurrent case. The second concerns exact coupling of random walks on the line with step-lengths that are neither discrete nor spread out. The third concerns the coupling characterization of setwise convergence of distributions of stochastic processes to a stationary limit. The fourth concerns characterizations of mass-stationarity, a concept formalizing the intuitive idea that the origin is a typical location in the mass of a random measure. Mass-stationarity is an intrinsic characterization of Palm versions with respect to stationary random measures.     

Sensing carbohydrate-protein interactions at picomolar concentrations using cantilever arrays

Carbohydrates are important mediators of many biological processes that underlie cellular communication and disease mechanisms. Therapeutic agents include carbohydrate-based vaccines and the potent anti-viral protein Cyanovirin-N (CV-N). CV-N acts by specifically binding the carbohydrate structures decorating the cell surface of deadly viruses including human immunodeficiency virus (HI-V) or Ebola. In search for new carbohydrate-binding proteins and the development of sensors that exploit carbohydrate-protein interactions the label-free cantilever array technique can provides a fast, parallel and low-cost approach.     

Ion speciation driving chirality transfer in imidazolium-based camphorsulfonate ionic liquid solutions

The underlying principle of the chirality transfer in imidazolium-based camphorsulfonate ionic liquids is rationalized by linking catalytic results from the hydrogenation of [N-(3'-oxobutyl)-N-methylimidazolium] [(+)-camphorsulfonate] to [N-(3'-hydroxybutyl)-N-methylimidazolium] [(+)-camphorsulfonate] in tetrahydrofuran with electrolyte theory by the help of dielectric relaxation spectroscopy. Using this approach we are able to explain why the maximum of the enantiomeric excess of the hydrogenation reaction in tetrahydrofuran is found at a medium concentration of 0.15 mol L(-1), whereas it declines at both, lower and higher concentrations. Dielectric spectra in the concentration range between 0.05 and 1.0 mol L(-1) reveal a solute mode due to dipolar ion pairs and larger dipolar ion clusters. They verify that at very low concentrations the ionic liquid ions are fully solvated with an increasing tendency to form neutral ion pairs with increasing concentration. Already at 0.025 mol L(-1) the degree of dissociation reaches a minimum reflecting a maximum of neutral ion pair formation. With increasing ionic liquid concentration ordered ion clusters are formed by two and more ion pairs. At high concentrations these clusters collapse by dilution in the excess ionic liquid and the defined ion contact necessary for the chirality transfer is lost to a great extent.     

Al-RUB-41: a shape-selective zeolite catalyst from a layered silicate

A new zeolite catalyst, Al-RUB-41, was synthesized for the first time. It was tested as a catalyst in methanol amination, and showed a shape-selective performance that results in a highly favorable product distribution. The shape-selective nature was also evidenced by using Pt-Al-RUB-41 as a bifunctional catalyst for decane hydroconversion. With its unique pore architecture and remarkable shape-selective character, Al-RUB-41 presents a significant commercial potential in industrial catalysis.