Weak solutions of the stationary Navier–Stokes equations for a viscous incompressible fluid past an obstacle

Consider the stationary Navier–Stokes equations in an exterior domain $\varOmega \subset \mathbb{R }^3 $ with smooth boundary. For every prescribed constant vector $u_{\infty } \ne 0$ and every external force $f \in \dot{H}_2^{-1} (\varOmega )$ , Leray (J. Math. Pures. Appl., 9:1–82, 1933) constructed a weak solution $u $ with $\nabla u \in L_2 (\varOmega )$ and $u - u_{\infty } \in L_6(\varOmega )$ . Here $\dot{H}^{-1}_2 (\varOmega )$ denotes the dual space of the homogeneous Sobolev space $\dot{H}^1_{2}(\varOmega ) $ . We prove that the weak solution $u$ fulfills the additional regularity property $u- u_{\infty } \in L_4(\varOmega )$ and $u_\infty \cdot \nabla u \in \dot{H}_2^{-1} (\varOmega )$ without any restriction on $f$ except for $f \in \dot{H}_2^{-1} (\varOmega )$ . As a consequence, it turns out that every weak solution necessarily satisfies the generalized energy equality. Moreover, we obtain a sharp a priori estimate and uniqueness result for weak solutions assuming only that $\Vert f\Vert _{\dot{H}^{-1}_2(\varOmega )}$ and $|u_{\infty }|$ are suitably small. Our results give final affirmative answers to open questions left by Leray (J. Math. Pures. Appl., 9:1–82, 1933) about energy equality and uniqueness of weak solutions. Finally we investigate the convergence of weak solutions as $u_{\infty } \rightarrow 0$ in the strong norm topology, while the limiting weak solution exhibits a completely different behavior from that in the case $u_{\infty } \ne 0$ .     

Mathematics on the threshold

Lowering the dropout rate of incoming mathematics and science students, and enhancing the provision of mathematics support for freshmen are two important aims of the University of Amsterdam. The approach recently adopted to support first year students is to set up a diagnostic pretest and posttest and use these tests to identify students being at risk of failing their mathematics courses and other modules in the first year. Follow-up procedures are implemented and computer algebra based assessment and practise of mathematics skills play an important role in it. In this paper we describe this approach and its success.     

Second‐Order Nonlinear Optical Activity of Dipolar Chromophores Based on Pyrrole‐Hydrazono Donor Moieties

Donor1+donor2→acceptor : The second‐order NLO molecular properties of a class of dipolar chromophores that incorporate the following design elements are investigated: 1) a substituted hydrazono moiety as a strong donor; 2) a pyrrole ring as an auxiliary donor; 3) strong acceptor groups (see figure). Their first hyperpolarisabilities show good promise for use in electro‐optical devices.

     

Influence of Magnetic Confinement on the Yellow Excitons in Cuprous Oxide Subject to an Electric Field

We study the spectrum of the yellow exciton series in crossed electric and magnetic fields. The electric field, applied along the optical axis, tilts the Coulomb potential between electron and hole, so that at sufficiently high fields exciton dissociation becomes possible, roughly when the electric dipole interaction energy exceeds the binding energy of an exciton state with principal quantum number n. For an applied voltage of U = 20 V all excitons above n = 6 are dissociated. Additional application of a magnetic field normal to the optical axis introduces magnetic confinement, due to which above a threshold field strength around B = 2.5 T the exciton lines re-emerge. The complex dispersion with increasing fields suggests quantum chaotic behavior in this crossed field configuration, so that the search for exceptional points may be promising.     

ESR measurements on hydrogenated Zr2Ni

X-band magnetic resonance measurements have been carried out on Zr₂NiHx (x=0–5). The room temperature results indicate the presence of Ni atom clusters which form during the hydrogenation process. The intensity of the resonance signal depends on the hydrogenation treatment.     

The activation energy for γ-Fe precipitates inCuFe

The precipitation of the fcc γ-phase of iron has been studied by a systematic series of isothermal and isochronal (0–48 h) heat treatments (300–800°C) on a supersaturated solution ofCuFe containing 3 at% Fe. The optimal conditions (450–600°C) for precipitation of the maximum fraction (90%) of iron attainable within 48 h in the form of γ-Fe have been delineated from analysis of the room temperature spectra. The time dependence for formation of γ-Fe precipitates is well described by the equation for long term annealing. An activation energyE _a ≈0.6 eV for the formation of γ-Fe in Cu is obtained. It indicates short range rather than long range diffusion in theCuFe sample studied.     

New MCRs: The first 4-component reaction leading to 2,4-disubstituted thiazoles

New organic reactions allow chemical transformations which were previously not possible. Therefore, new reactions are important contributions to the progress in the field of organic synthesis. In this series we describe the design, scope, and limitations of newly-discovered multi-component reactions (MCRs). Herein, a first example of a MCR is introduced which allows general access to the class of 2,4-di- and 2,4,5-trisubstituted thiazoles.     

Rotational states in 151Nd populated through thermal neutron capture

The level scheme in the nucleus ¹⁵¹ Nd has been studied following neutron capture by observing the γ-rays and conversion electrons with curved-crystal spectrometers, a Ge(Li) detector and a β-spectrometer. A comprehensive level scheme up to ? 1 MeV was established. It is shown that the present level scheme energies differ from those previously found in a ¹⁵⁰Nd(d, p) reaction study by a systematic shift of 27 keV. The binding energy of the last neutron in ¹⁵¹Nd was deduced. For most levels spin and parity values are proposed and an identification of bands with Nilsson-model configurations is given.