Stationary solutions of the Navier-Stokes equations in periodic tubes

Let be a tubular domain in Rⁿ, n=2,3, with a Lipschitz boundary , invariant with respect to a translation by the vector. It is proven that, for any prescribed real number _o, there exists at least one solution of the nonhomogeneous boundary-value problem for a stationary Navier-Stokes system with a periodic and pressure , having the drop _o over the period. (The exterior forces and the boundary values of the velocity field are assumed to be periodic.) In addition, one proves the existence of a critical nonnegative number ^*, depending only on the geometry of the domain , the viscosity coefficient, the exterior forces and the boundary values of, such that for ¦₀¦>^* the fluid flows along the direction of the decrease of the pressure.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 115, pp. 104–113, 1982.The author is grateful to O. A. Ladyzhenskaya for her interest in the paper.