arXiv subjects
Publications and source records attributed to F. L. Bakharev.
We describe the spectrum structure for the restricted Dirichlet fractional Laplacian in multi-tubes, i.e. domains with cylindrical outlets to infinity. Some new effects in comparison with the local case are discovered. In this version, Theorem 4 is essentially improved.
The paper presents a stochastic analysis of the growth rate of viscous fingers in miscible displacement in a heterogeneous porous medium. The statistical parameters characterizing the permeability distribution of a reservoir vary over a wide range. The formation of fingers is provided by the mixing of different-viscosity fluids -- water and polymer solution. The distribution functions of the growth rate of viscous fingers are numerically determined and visualized. Careful data processing reveals the non-monotonic nature of the dependence of the front end of the mixing zone on the correlation length of the permeability (describing the medium graininess) of the reservoir formation. It is demonstrated that an increase in graininess up to a certain value causes an expansion of the distribution shape and a shift of the distribution maximum to the region of higher velocities. In addition, an increase in the standard deviation of permeability leads to a slight change in the shape and characteristics of the density distribution of the growth rates of viscous fingers. The theoretical predictions within the framework of the transverse flow equilibrium approximation and the Koval model are contrasted with the numerically computed velocity distributions.
We describe the Dirichlet spectrum structure for the Fichera layers and crosses in any dimension $n\ge3$. Also the application of the obtained results to the classical Brownian exit times problem in these domains.
We give an example of a scalar second order differential operator in the plane with double periodic coefficients and describe its modification, which causes an additional spectral band in the essential spectrum. The modified operator is obtained by applying to the coefficients a mirror reflection with respect to a vertical or horizontal line. This change gives rise to Rayleigh type waves localized near the line. The results are proven using asymptotic analysis, and they are based on high contrast of the coefficient functions.
Spatial cruciform quantum waveguides (the Dirichlet problem for Laplace operator) are constructed such that the total multiplicity of the discrete spectrum exceeds any preassigned number.
We study the spectral linear elasticity problem in an unbounded periodic waveguide, which consists of a sequence of identical bounded cells connected by thin ligaments of diameter of order $ h >0$. The essential spectrum of the problem is known to have band-gap structure. We derive asymptotic formulas for the position of the spectral bands and gaps, as $h \to 0$.
Sufficient conditions for a discrete spectrum of the biharmonic equation in a two-dimensional peak-shaped domain are established. Different boundary conditions from Kirchhoff's plate theory are imposed on the boundary and the results depend both on the type of boundary conditions and the sharpness exponent of the peak.
We will study the spectral problem related to the Laplace operator in a singularly perturbed periodic waveguide. The waveguide is a quasi-cylinder with contains periodic arrangement of inclusions. On the boundary of the waveguide we consider both Neumann and Dirichlet conditions. We will prove that provided the diameter of the inclusion is small enough in the spectrum of Laplacian opens spectral gaps, i.e. frequencies that does not propagate through the waveguide. The existence of the band gaps will verified using the asymptotic analysis of elliptic operators.