### 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)} < c_{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 |x1|}) for some positive constant &zeta .

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