Existence and homogenization of the Rayleigh-B\'enard problem

The Navier-Stokes equation driven by heat conduction is studied. As a prototype we consider Rayleigh-B\'enard convection, in the Boussinesq approximation. Under a large aspect ratio assumption, which is the case in Rayleigh-B\'enard experiments with Prandtl number close to one, we prove the existence of a global strong solution to the 3D Navier-Stokes equation coupled with a heat equation, and the existence of a maximal B-attractor. A rigorous two-scale limit is obtained by homogenization theory. The mean velocity field is obtained by averaging the two-scale limit over the unit torus in the local variable

Some generic aspects of bosonic excitations in disordered systems

We consider non-interacting bosonic excitations in disordered systems, emphasising generic features of quadratic Hamiltonians in the absence of Goldstone modes. We discuss relationships between such Hamiltonians and the symmetry classes established for fermionic systems. We examine the density \rho(\omega) of excitation frequencies \omega, showing how the universal behavior \rho(\omega) ~ \omega^4 for small \omega can be obtained both from general arguments and by detailed calculations for one-dimensional models

Adhesive coatings based on aligned arrays of carbon nanostructures

This work was financially supported by Russian Foundation for Basic Research (projects 16-29-14023 and 18-32-00652) and Internal grant of the Southern Federal University (project VnGr-07/2017-26)

Asymptotics of Eigenvalues and Eigenfunctions for the Laplace Operator in a Domain with Oscillating Boundary. Multiple Eigenvalue Case

We study the asymptotic behavior of the solutions of a spectral problem for the Laplacian in a domain with rapidly oscillating boundary. We consider the case where the eigenvalue of the limit problem is multiple. We construct the leading terms of the asymptotic expansions for the eigenelements and verify the asymptotics

Comment: Superconducting transition in Nb nanowires fabricated using focused ion beam

In a recent paper Tettamanzi et al (2009 Nanotechnology \bf{20} 465302) describe the fabrication of superconducting Nb nanowires using a focused ion beam. They interpret their conductivity data in the framework of thermal and quantum phase slips below $T_c$. In the following we will argue that their analysis is inappropriate and incomplete, leading to contradictory results. Instead, we propose an interpretation of the data within a SN proximity model.Comment: 3 pages, 1 figure accepted in Nanotechnolog

Study of the dependence of Young's modulus of vertically aligned carbon nanotubes on their aspect ratio

The reported study was funded by RFBR according to the research projects No.16-29-14023 ofi_m, No.18-32-00652 and by grant of the Southern Federal University (project No. VnGr-07/2017-26)

Vibrational instability, two-level systems and Boson peak in glasses

We show that the same physical mechanism is fundamental for two seemingly different phenomena such as the formation of two-level systems in glasses and the Boson peak in the reduced density of low-frequency vibrational states g(w)/w^2. This mechanism is the vibrational instability of weakly interacting harmonic modes. Below some frequency w_c << w_0 (where w_0 is of the order of Debye frequency) the instability, controlled by the anharmonicity, creates a new stable universal spectrum of harmonic vibrations with a Boson peak feature as well as double-well potentials with a wide distribution of barrier heights. Both are determined by the strength of the interaction I ~ w_c between the oscillators. Our theory predicts in a natural way a small value for the important dimensionless parameter C ~ 10^{-4} for two-level systems in glasses. We show that C ~ I^{-3} and decreases with increasing of the interaction strength I. We show that the number of active two-level systems is very small, less than one per ten million of oscillators, in a good agreement with experiment. Within the unified approach developed in the present paper the density of the tunneling states and the density of vibrational states at the Boson peak frequency are interrelated.Comment: 28 pages, 3 figure

Anharmonic vs. relaxational sound damping in glasses: II. Vitreous silica

The temperature dependence of the frequency dispersion in the sound velocity and damping of vitreous silica is reanalyzed. Thermally activated relaxation accounts for the sound attenuation observed above 10 K at sonic and ultrasonic frequencies. Its extrapolation to the hypersonic regime reveals that the anharmonic coupling to the thermal bath becomes important in Brillouin-scattering measurements. At 35 GHz and room temperature, the damping due to this anharmonicity is found to be nearly twice that produced by thermally activated relaxation. The analysis also reveals a sizeable velocity increase with temperature which is not related with sound dispersion. This suggests that silica experiences a gradual structural change that already starts well below room temperature.Comment: 13 pages with 8 figure

Effects of Electron-Electron and Electron-Phonon Interactions in Weakly Disordered Conductors and Heterostuctures

We investigate quantum corrections to the conductivity due to the interference of electron-electron (electron-phonon) scattering and elastic electron scattering in weakly disordered conductors. The electron-electron interaction results in a negative $T^2 \ln T$-correction in a 3D conductor. In a quasi-two-dimensional conductor, $d < v_F/T$ ($d$ is the thickness, $v_F$ is the Fermi velocity), with 3D electron spectrum this correction is linear in temperature and differs from that for 2D electrons (G. Zala et. al., Phys. Rev.B {\bf 64}, 214204 (2001)) by a numerical factor. In a quasi-one-dimensional conductor, temperature-dependent correction is proportional to $T^2$. The electron interaction via exchange of virtual phonons also gives $T^2$-correction. The contribution of thermal phonons interacting with electrons via the screened deformation potential results in $T^4$-term and via unscreened deformation potential results in $T^2$-term. The interference contributions dominate over pure electron-phonon scattering in a wide temperature range, which extends with increasing disorder.Comment: 6 pages, 2figure

Generation of small-scale structures in the developed turbulence

The Navier-Stokes equation for incompressible liquid is considered in the limit of infinitely large Reynolds number. It is assumed that the flow instability leads to generation of steady-state large-scale pulsations. The excitation and evolution of the small-scale turbulence is investigated. It is shown that the developed small-scale pulsations are intermittent. The maximal amplitude of the vorticity fluctuations is reached along the vortex filaments. Basing on the obtained solution, the pair correlation function in the limit $r\to 0$ is calculated. It is shown that the function obeys the Kolmogorov law $r^{2/3}$.Comment: 18 page