Asymptotic Analysis

We consider the flow of a viscous Newtonian fluid in a bifurcation of thin pipes with a diameter-to-length ratio of order O(ε). The model is based on the stationary Navier–Stokes equations with pressure conditions on the outflow boundaries. Existence and local uniqueness is established under the assumption of small data represented by a Reynolds number Re_ε of order O(ε). We construct an asymptotic expansion in powers of ε and Re_ε for the solution consisting of Stokes flow in the junction part of the bifurcation and Poiseuille flow in the pipes. We introduce a correction to Kirchhoff's law of the balancing fluxes in O(ε) which allows to establish error estimates for the gradient of velocity. These estimates result from the analysis of the decay properties of the flow in the layer near the bifurcation.

We give an asymptotic formula in time of solutions to the linear fourth-order Schrödinger type equation in one space dimension without the mean zero condition on the initial data. Then we apply this formula to study the long time behavior of solution to the fourth-order Schrödinger type equation with the cubic non-linearity.

Here we consider one of the basic models for many-body problems under an external field: the molecule ion H₂⁺ under the effect of an external Stark-type potential. If we consider the vibrational energy levels of the first two electronic states of the molecule ion H₂⁺ then, in the semiclassical limit and by means of a suitable modified Born–Oppenheimer method, we can prove that they switch to sharp resonances localized in the same interval of energy of the vibrational levels when an external Stark-type field, with the same direction of the nuclear axis, occurs.

We consider quasilinear and linear boundary-value problems for the second-order elliptic differential operator with rapidly oscillating coefficients in a domain Ω_ε that is ε-periodically perforated by small holes of order 𝒪(ε). The holes are divided into three ε-periodical sets depending on boundary conditions on their surfaces. On the boundaries of holes from one set we have the homogeneous Dirichlet conditions. On the boundaries of holes from the other sets, different inhomogeneous Neumann and nonlinear Robin boundary conditions involving additional perturbation parameters are imposed. For the solution to the quasilinear problem we find the leading terms of the asymptotics and prove the corresponding asymptotic estimates that show influence of the perturbation parameters. In the linear case we construct and justify the complete asymptotic expansion for the solution using two-scale asymptotic expansion method.

It is shown that the anelastic Oberbeck–Boussinesq system is a small Mach, small Péclet and small Froude number limit of the complete Navier–Stokes–Fourier system for gases with large specific heat at constant volume. This result is obtained on an arbitrary large time interval. The proof allows an intrinsic view into the process of separation of fast oscillating acoustic waves, governed by a Lighthill-type equation, from the equations describing the slow fluid flows. This is a very useful information for numerical analysts.