We consider the Navier-Stokes equations in the layer  over  with finite  Using the standard fundamental solutions of the Laplace operator and the heat operator, we reduce the Navier-Stokes equations to a nonlinear Fredholm equation of the form  where K is a compact continuous operator in anisotropic normed Hölder spaces weighted at the point at infinity with respect to the space variables. Actually, the weight function is included to provide a finite energy estimate for solutions to the Navier-Stokes equations for all  On using the particular properties of the de Rham complex we conclude that the Fréchet derivative  is continuously invertible at each point of the Banach space under consideration and the map  is open and injective in the space. In this way, the Navier-Stokes equations prove to induce an open one-to-one mapping in the scale of Hölder spaces.

Navier-Stokes equations, weighted Hölder spaces, integral representation method.