On the global regularity of shear thinning flows in smooth domains

In some recent papers we have been pursuing regularity results up to the boundary, in W2,l(Ω) spaces for the velocity, and in W1,l(Ω) spaces for the pressure, for fluid flows with shear dependent viscosity. To fix ideas, we assume the classical non-slip boundary condition. From the mathematical point of view it is appropriate to distinguish between the shear thickening case, p>2, and the shear thinning case, p<2, and between flat-boundaries and smooth, arbitrary, boundaries. The p<2 non-flat boundary case is still open. The aim of this work is to extend to smooth boundaries the results proved in reference [H. Beirão da Veiga, On non-Newtonian p-fluids. The pseudo-plastic case, J. Math. Anal. Appl. 344 (1) (2008) 175-185]. This is done here by appealing to a quite general method, introduced in reference [H. Beirão da Veiga, On the Ladyzhenskaya-Smagorinsky turbulence model of the Navier-Stokes equations in smooth domains. The regularity problem, J. Eur. Math. Soc., in press], suitable for considering non-flat boundaries.     

On the regularity of flows with Ladyzhenskaya Shear‐dependent viscosity and slip or non‐slip boundary conditions

Navier‐Stokes equations with shear dependent viscosity under the classical non‐slip boundary condition have been introduced and studied, in the sixties, by O. A. Ladyzhenskaya and, in the case of gradient dependent viscosity, by J.‐L. Lions. A particular case is the well known Smagorinsky turbulence model. This is nowadays a central subject of investigation. On the other hand, boundary conditions of slip type seems to be more realistic in some situations, in particular in numerical applications. They are a main research subject. The existence of weak solutions u to the above problems, with slip (or non‐slip) type boundary conditions, is well known in many cases. However, regularity up to the boundary still presents many open questions. In what follows we present some regularity results, in the stationary case, for weak solutions to this kind of problems; see Theorems 3.1 and 3.2. The evolution problem is studied in the forthcoming paper [6]; see the remark at the end of the introduction. © 2004 Wiley Periodicals, Inc.     

Differentiability and bifurcation points for a class of monotone nonlinear operators

Sunto Sia P l'operatore definito nello spazio L²(Ω) ponendo u = Pf in(11″). Si dimostra che questo operatore (differenziabile secondo Fréchet nell'origine (vedi teorema 1) e si utilizza questo risultato per caratterizzare i punti di biforcazione per il problema non lineare(25) (vedi teorema 2).

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Entrata in Redazione l'll settembre 1975.     

Sulla hölderianità delle soluzioni di alcune disequazioni variazionali con condizioni unilatere al bordo

Summary In this paper the Author proves the Holder-continuity of the weak solutions of the problems P ₁ , P ₂ and P ₃ (see Introduction) expressed by variational inequalities.

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Lavoro eseguito per conto dell'Istituto per le Applicazioni del Calcolo del C.N.R.

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Entrata in Redazione il 2 aprile 1969.     

3‐Hydro­xy­benz­aldehyde

The title compound, C₇H₆O₂, forms infinite chains where the mol­ecules are hydrogen bonded via the hydroxyl and aldehyde groups, with an O?O distance of 2.719 (3) Å. Interchain interactions are weak. The geometry of the ring differs from the ideal form due to the effect of the substituents. Abinitio (Hartree–Fock self‐consistent field–molecular orbital and density functional theory) calculations for the free mol­ecule reproduce well the observed small distortions of the ring. In the crystal, the geometry deviates from the ideal C_s symmetry of the free mol­ecule, as given by the ab initio calculations. The aldehyde and hydroxyl groups are twisted around the single bonds which join them to the ring as a result of the intermolecular hydrogen‐bond interactions. These are also responsible for an elongation of the hydroxy C—OH bond compared with that calculated for the free mol­ecule.     

On the Smoothness of a Class of Weak Solutions to the Navier--Stokes equations

We improve regularity criteria for weak solutions to the Navier-Stokes equations stated in references [1], [3] and [12], by using in the proof given in [3], a new idea introduced by H. O. Bae and H. J. Choe in [1]. This idea allows us, in one of the main hypothesis (see eq. (1.7)), to replace the velocity u by its projection [`(u)] \bar u into an arbitrary hyperplane of \Bbb Rⁿ {\Bbb R}^n ; see Theorem A. For simplicity, we state our results for space dimension n ￡ 4 n \le 4 , since if n 3 5 n \ge 5 the proofs become more technical and additional hypotheses are needed. However, for the interested reader, we will present the formal calculations for arbitrary dimension n.     

Smooth driving of M?ssbauer electromechanical transducers

Quality of Mössbauer spectra is strongly related to the performance of source velocity modulator. Traditional electromechanical driving techniques demand hard-edged square or triangular velocity waveforms that introduce long settling times and demand careful driver tuning. For this work, the behavior of commercial velocity transducers and drive units was studied under different working conditions. Different velocity reference waveforms in constant-acceleration, constant-velocity and programmable-velocity techniques were tested. Significant improvement in spectrometer efficiency and accuracy was achieved by replacing triangular and square hard edges with continuous smooth-shaped transitions. A criterion for best waveform selection and synchronization is presented and attainable enhancements are evaluated. In order to fully exploit this driving technique, a compact microprocessor-based architecture is proposed and a suitable data acquisition system implementation is presented. System linearity and efficiency characterization are also shown.     

On a family of results concerning direction of vorticity and regularity for the Navier–Stokes equations

These notes concern existence, and suitable formulation, of meaningful conditions on the direction of the vorticity which guarantee the regularity of the solutions to the evolution Navier–Stokes equations. A main concern here is to compare the different situations which appear in considering slip and no-slip boundary conditions. The paper reviews mainly results obtained in some of the references cited.     

Mucuna pruriens Administration Minimizes Neuroinflammation and Shows Anxiolytic,Antidepressant and Slimming Effects in Obese Rats

This study evaluated the effect of Mucuna pruriens (MP) administration on neuroinflammation and behavioral and murinometric parameters in obese rats. Proximate composition, oligosaccharide and phenolic compound profile of MP were determined. Wistar adult male rats were randomized into healthy (HG) and obese group (OG). The HG consumed a control chow diet while OG consumed a cafeteria diet for eight weeks. Then, they were subdivided into: Healthy (HG); Healthy with MP administration (HGMP); Obese (OG); Obese with MP administration (OGMP), with the consumption of the respective diets remaining for another eight weeks, in addition to gavage with MP extract to supplemented groups (750 mg/kg weight). MP presented a composition rich in proteins and phenolic compounds, especially catechin, in addition to 1-kestose and levodopa. Supplementation reduced food intake, body weight, and thoracic and abdominal circumferences in obese rats. MP showed anxiolytic and antidepressant effects and reduced morphological damage and expression of interleukin 6 in the hippocampus of obese rats. MP treatment showed satietogenic, slimming, anxiolytic and antidepressant effects, besides to minimizing hippocampal neuroinflammation in obese rats. Our results demonstrated the potential anti-obesity of MP which are probably related to the high content of bioactive compounds present in this plant extract.