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

Abstract

The time-dependent Navier-Stokes system is studied in a two-dimensional domain with strip-like outlets to infinity in weighted Sobolev function spaces. It is proved that under natural compatibility conditions there exists a unique solution with prescribed fluxes over crosssections of outlets to infinity which tends in each outlet to the corresponding time-dependent Poiseuille flow. The obtained results are proved for arbitrary large norms of the data (in particular, for arbitrary fluxes) and globally in time. (2000). 35Q30, 76D03, 76D05. Mathematics Subject Classification

Function spaces on singular manifolds

“…Zaj ' aczkowski [52]- [55], A. Kubica and W.M. Zaj ' aczkowski [28], [29], and K. Pileckas [36]- [38] (see the references in these papers for earlier work) on Stokes and Navier-Stokes equations. In all these papers, except the ones of Pileckas, ρ is the distance to the singularity set, where in Zaj ' aczkowski's publications M is obtained from a smooth subdomain of R m by eliminating a line segment.…”

Stokes equations with interface condition in an unbounded domain

“…According to (15), the bilinear form a m is coercive, then by the Lax-Milgram lemma, Problem (12) admits a unique solution u m 2 V m . Since (12) is equivalent to (3), (4), (11), there exists p m 2 L 2 (Ã m ) such that (u m , p m ) is a solution of (3), (4), (11).…”

“…The asymptotic behaviour at infinity of solutions has been studied in the case of Laplace, Stokes and Navier-Stokes equations, elasticity system, with different geometries and boundary conditions by many authors, see for instance [3][4][5][6][7][8][9][10][11][12][13][14][15]. Let us particularly mention [6] where the authors study stationary Stokes and NavierStokes equations in a junction of semi-infinite pipes (fluid outlets).…”

Time-periodic Poiseuille-type solution with minimally regular flow rate

“…It is well known that Fredholm integral equations of the second kind satisfy the Fredholm alternative (e.g., [27]). So, it is enough to prove the uniqueness of the solution to (20). Let F (t) = 0.…”