On an inhomogeneous slip-inflow boundary value problem for a steady flow of a viscous compressible fluid in a cylindrical domain

We investigate a steady flow of a viscous compressible fluid with inflow boundary condition oil the density and inhomogeneous slip boundary conditions on the velocity in a cylindrical domain Omega = Omega(0) x (0 L) is an element of R(3).We show existence of a solution (nu, rho) is an element of W(p)(2)(Omega) x W(p)(1)(Omega), p > 3, where nu is the velocity of the fluid and rho is the density, that is a small perturbation of a constant flow ((nu) over bar equivalent to [1, 0, 0], (rho) over bar equivalent to 1). We also show that this solution is unique in a class of small perturbations of ((v) over bar, (rho) over bar). The term u del w in the continuity equation makes it impossible to show the existence applying directly a fixed point method. Thus in order to show existence of the solution we construct a sequence (nu(n), rho(n)) that is bounded in W(p)(2)(Omega) x W(p)(1)(Omega) and satisfies the Cauchy condition in a larger space L(infinity)(0, L, L(2) (Omega(0))) what enables us to deduce that the weak limit of a subsequence of (nu(n), rho(n)) is in fact a strong solution to our problem (C) 2009 Elsevier Inc All rights reserved