Abstract or Additional Information
We consider the nonhomogeneous boundary value problem for the steady Navier-Stokes equations under the slip boundary conditions in a two-dimensional bounded domain with multiple boundary components. By the incompressibility condition of the fluid, the total flux of the given boundary datum through the boundary must be zero. We prove that this problem has a solution if the domain and the given data satisfy certain symmetry conditions. No smallness restriction is imposed on the fluxes through each connected component of the boundary. The existence of a solution is established with the aid of the Leray-Schauder theorem. The required a priori estimate is proved by a contradiction argument due to Leray (1933) and by an application of Bernoulli's law for a weak solution to the steady Euler equations.