The Navier-Stokes equations and the maximum principle

We consider the steady Navier-Stokes equations in a two-dimensional infinite channel with slip boundary conditions. Provided with a geometrical constraint on the domain, we are able to show existence of solutions for arbitrary large velocity data, including the flux of flow. The crucial point of the proof is the maximum principle for a reformulation of the problem. This technique gives a new type of estimates in the Hölder space for the solutions with the infinite Dirichlet integral.