Konstantin Pileckas

Existence, Uniqueness and Asymptotics of Steady Jets

We consider the three-dimensional flow through an aperture in a plane either with a prescribed flux or pressure drop condition. We discuss the existence and uniqueness of solutions for small data in weighted spaces and derive their complete asymptotic behaviour at infinity. Moreover, we show that each solution with a bounded Dirichlet integral, which has a certain weak additional decay, behaves like O(r−2) as r=¦x¦→∞ and admits a wide jet region. These investigations are based on the solvability properties of the linear Stokes system in a half space ℝ+3. To investigate the Stokes problem in ℝ+3, we apply the Mellin transform technique and reduce the Stokes problem to the determination of the spectrum of the corresponding invariant Stokes-Beltrami operator on the hemisphere.

Steady Flows of Viscoelastic Fluids in Domains with Outlets to Infinity

Abstract. The equations governing the motion of incompressible viscoelastic fluids of Rivlin—Ericksen and Oldroyd type are investigated in domains with cylindrical and paraboloidal outlets to infinity. For sufficiently small fluxes, prescribed in each outlet, existence and uniqueness of solutions are proven in weighted Hölder spaces. In domains with paraboloidal outlets the solution is obtained as a perturbation of the corresponding Navier—Stokes solution and in domains with cylindrical outlets as a perturbation of a flux carrier, constructed by joining together the exact solutions found in each outlet. These exact solutions are shown to be either rectilinear flows of Poiseuille type or flows composed of a rectilinear and of a transverse secondary component.

Asymptotic analysis of the nonsteady viscous flow with a given flow rate in a thin pipe

The nonsteady Navier–Stokes equations are considered in a thin infinite pipe with the small diameter ϵ in the case of the Reynolds number of order ϵ. The time-dependent flow rate is a given function. The complete asymptotic expansion is constructed and justified. The error estimate of order O(ϵ J ) for the difference of the exact solution and the J-th asymptotic approximation is proved for any real J.

Divergence equation in thin-tube structures

Divergence equation is considered in a thin-tube structure, connected finite union of thin finite cylinders (in the case of two dimensions, respectively, thin rectangles) where the ratio of the diameter and the heights of the cylinders is the small parameter . Various norms of the solution of divergence equation are estimated over the norms of the right-hand side. The exact dependence of constants in these estimates on the small parameter is obtained.

The Liouville Theorem for the Steady-State Navier–Stokes Problem for Axially Symmetric 3D Solutions in Absence of Swirl

We study the Navier–Stokes equations of steady motion of a viscous incompressible fluid in

On singular solutions of time-periodic and steady Stokes problems in a power cusp domain

{\mathbb{R}^{3}}

Leray’s Problem on Existence of Steady State Solutions for the Navier-Stokes Flow

R3. We prove that there are no nontrivial solution of these equations defined in the whole space