Flux problem for a certain class of two-dimensional domains

We consider the steady Navier-Stokes equations in a two-dimensional bounded domain with multiply connected boundary with nonhomogeneous slip boundary conditions. Under a geometrical constraint, restricting the shape of the domain, we construct an a priori bound of solutions for arbitrarily large data, including fluxes through each component of the boundary. Our result is an exception to the general theory of the flux problem.