Existence of Solutions with the Prescribed Flux of the Navier–Stokes System in an Infinite Cylinder

Abstract

The existence and uniqueness of a solution to the nonstationary Navier-Stokes system having a prescribed flux in an infinite cylinder is proved. We assume that the initial data and the external forces do not depend on x 3 and find the solution (u, p) having the following formwhere x ′ = (x 1 , x 2 ). Such solution generalize the nonstationary Poiseuille solutions. (2000). 35Q30, 76D03, 76D05. Mathematics Subject Classification

Global Solvability in W 2 2,1 -Weighted Spaces of the Two-Dimensional Navier–Stokes Problem in Domains with Strip-Like Outlets to Infinity

“…Then in each infinite channel Π j there exists a time-dependent unidirectional Poiseuille flow (see [6]):…”

“…In Section 2 we define the function spaces used in the paper, recall necessary multiplicative inequalities and prove some auxiliary results. In Section 3 we present results proved in [6] concerning the existence of the time-dependent Poiseuille flow in a two-dimensional infinite strip and we construct a flux carrier V(x, t) coinciding in each outlet to infinity Ω j with the corresponding to this outlet time-dependent Poiseuille flow. Finally, we reduce problem (1.1) to a problem for the perturbation of the constructed flux carrier (see (3.17)).…”

On the unsteady Poiseuille flow in a pipe

“…In its general form, it can be formulated as follows, see [14]. Given u 0 (x), f (x, t) and F (t) to find a pair of functions (u(x, t), q(t)) solving the following initial-boundary value problem on the cross-section σ:…”

“…In all these instances, u(x, t) becomes, at each time t, the spatial asymptotic of the relevant velocity field; see [2], [10] . Existence, uniqueness and asymptotic in time behavior of solutions to (1.1) in appropriate function classes were first obtained in [12], [13], [14]. In particular, in [13], by using a modified Fourier method, the unique solvability of problem (1.1) was studied in Hölder spaces, while the behavior in time of corresponding solutions was analyzed in [12].…”

Existence and exponential stability in Lr‐spaces of stationary Navier–Stokes flows with prescribed flux in infinite cylindrical domains

“…In the case of our cylindrical domain the existence of a weak solution to (SNS) was shown first in Reference [14] under a smallness condition on the flux, see also Reference [12], Chapter XI. Recently, in Reference [15] the instationary Navier-Stokes system in with time-dependent prescribed flux has been considered in Hilbert spaces using Galerkin approximation. For cylindrical domains with several exits to infinity and with bounded varying cross sections the stationary Stokes system is considered in Reference [16].…”