Turbulent boundary layer equationsÉquations de la couche limite turbulente

We study a boundary layer problem for the Navier–Stokes-alpha model obtaining a generalization of the Prandtl equations which we conjecture to represent the averaged flow in a turbulent boundary layer. We study the equations for the semi-infinite plate, both theoretically and numerically. Solutions agree with some experimental data in a part of the turbulent boundary layer. To cite this article: A. Cheskidov, C. R. Acad. Sci. Paris, Ser. I 334 (2002) 423–427.     

Electrical transport and magnetic ordering in R 2Ti3Ge4 (R=Dy, Ho and Er) compounds

New R ₂Ti₃Ge₄ (R=Dy, Ho and Er) intermetallic compounds have been synthesized and characterized by X-ray diffraction and low temperature ac magnetic susceptibility, electrical resistivity and thermoelectric power measurements were carried out. The compounds crystallize in the parent, Sm₅Ge₄-type orthorhombic structure (space group Pnma) and lanthanide contraction is observed as one moves along the rare-earth series. The changeover from paramagnetic to antiferromagnetic phase happens at low temperatures and the ordering temperature scales with the de Gennes factor. The electrical resistivity is metallic with a negative curvature above 100 K. Thermopower displays a weak maximum at temperatures less than 50 K signifying the possible phonon and magnon drag effects.     

The Regularity of Weak Solutions of the 3D Navier–Stokes Equations in {B^{-1}_{\infty,\infty}}

We show that if a Leray–Hopf solution u of the three-dimensional Navier–Stokes equation belongs to C((0,T]; B^-1_￥,￥){C((0,T]; B^{-1}_{\infty,\infty})} or its jumps in the B^-1_￥,￥{B^{-1}_{\infty,\infty}}-norm do not exceed a constant multiple of viscosity, then u is regular for (0, T]. Our method uses frequency local estimates on the nonlinear term, and yields an extension of the classical Ladyzhenskaya–Prodi–Serrin criterion.     

Energy Conservation in Two-dimensional Incompressible Ideal Fluids

We define the model of two-dimensional random interlacements using simple random walk trajectories conditioned on never hitting the origin, and then obtain some properties of this model. Also, for a random walk on a large torus conditioned on not hitting the origin up to some time proportional to the mean cover time, we show that the law of the vacant set around the origin is close to that of random interlacements at the corresponding level. Thus, this new model provides a way to understand the structure of the set of late points of the covering process from a microscopic point of view.     

Ab initio spin-orbit calculations on the lowest states of the nickel dimer

Potential energy curves of the lowest electronic states of the Ni(2) dimer are calculated near the equilibrium using the multireference ab initio methods including the spin-orbit interaction. Scalar-relativistic results fully confirm previous qualitative interpretations based on the correlation with atomic limits and the symmetry of vacancies in the atomic 3d(9) shells. Spin-orbit calculations firmly establish the symmetry of the ground state as 0(+)(g) and give the excitation energies 70 ± 30 cm(-1) and 200 ± 80 cm(-1) for the lowest 0(-)(u) and 5(u) states, respectively. The model electronic spectrum of the Ni(2) shows some trends that might be observed in matrix isolation far-infrared and electron spin resonance spectra.     

Elastic properties of the Ta–V system: bcc Ta0.33V0.67 and C15 TaV2

Resonant ultrasound spectroscopy was used to measure the elastic constants of bcc Ta_0.33V_0.67 over the temperature range 3.5–300?K; the results were compared to earlier measurements on C15 TaV₂. The temperature dependence of the polycrystalline shear modulus is completely different in the two phases; that of the bcc phase decreases with temperature whereas that of the C15 phases increases in an anomalous fashion. This difference is consistent with a model involving doubly-degenerate levels at the X point of the Brillouin zone in the C15 phase with the Fermi level lying near the doubly degenerate level. This model accounted for the unusual behaviour of the C15 phase. Debye temperatures were determined from the ultrasonic measurements: 295?K for the C15 phase and 315?K for the bcc phase.     

A Unified Approach to Regularity Problems for the 3D Navier-Stokes and Euler Equations: the Use of Kolmogorov’s Dissipation Range

Motivated by Kolmogorov’s theory of turbulence we present a unified approach to the regularity problems for the 3D Navier-Stokes and Euler equations. We introduce a dissipation wavenumber ${\Lambda(t)}$ that separates low modes where the Euler dynamics is predominant from the high modes where the viscous forces take over. Then using an indifferent to the viscosity technique we obtain a new regularity criterion which is weaker than every Ladyzhenskaya-Prodi-Serrin condition in the viscous case, and reduces to the Beale-Kato-Majda criterion in the inviscid case. In the viscous case we prove that Leray-Hopf solutions are regular provided ${\Lambda \in L^{5/2}}$ , which improves our previous ${\Lambda \in L^\infty}$ condition. We also show that ${\Lambda \in L^1}$ for all Leray-Hopf solutions. Finally, we prove that Leray-Hopf solutions are regular when the time-averaged spatial intermittency is small, i.e., close to Kolmogorov’s regime.     

Uniform global attractors for the nonautonomous 3D Navier–Stokes equations

We obtain the existence and the structure of the weak uniform (with respect to the initial time) global attractor and construct a trajectory attractor for the 3D Navier–Stokes equations (NSE) with a fixed time-dependent force satisfying a translation boundedness condition. Moreover, we show that if the force is normal and every complete bounded solution is strongly continuous, then the uniform global attractor is strong, strongly compact, and solutions converge strongly toward the trajectory attractor. Our method is based on taking a closure of the autonomous evolutionary system without uniqueness, whose trajectories are solutions to the nonautonomous 3D NSE. The established framework is general and can also be applied to other nonautonomous dissipative partial differential equations for which the uniqueness of solutions might not hold. It is not known whether previous frameworks can also be applied in such cases as we indicate in open problems related to the question of uniqueness of the Leray–Hopf weak solutions.