On Leray’s problem under Navier boundary conditions

Abstract

We consider the stationary motions of a viscous incompressible fluid, of kinematical viscosity ν, in domains with cylindrical outlets to infinity of section &Sigma, and axis e 1 (distorted cylinders). We require that the fluid satisfies the classical boundary Navier condition u·n=0, [&gamma u+T (u,p)·n] &tau =0 (&gamma&ge 0) and tends at infinity to the laminar flow u N in the cylinder with section &Sigma having an assigned flux through &Sigma (Leray problem). We prove that there is a computable positive constant c 0 depending only on &Sigma, such that if ||u N || L &infin (&Sigma) 0 &nu, then the problem has at least a weak solution. Moreover, for &gamma=0 we can alternatively require that u tends at infinity to an assigned constant vector &mu e 1 , &mu&ne 0. In contrast with the most part of the known existence theorems for the steady--state Navier--Stokes problem we show that a solution (u,p) of such a problem exists for every &mu and &nu, is unique for large &nu and converges to &mu e 1 at infinity according to u(x)=&mu e 1 +O(e -&zeta |x 1 | ) for some positive constant &zeta .