Directional approach to spatial structure of solutions to the Navier-Stokes equations in the plane

We investigate the steady state Navier-Stokes equations considered in the full space $\mathbb{R}^2$. We suplement the system with a condition at infinity which requires the solution (the velocity) to tend to a prescribed constant vector field. This problem is strictly connected with an open problem of a flow past an obstacle on the plane. The main difficulty there is to assure the convergence of a solution to a prescribed velocity at infinity. We propose a new method to deal with this problem. The class of functions, where we look for a solution, is different from standard Sobolev spaces. This is due to the fact that our analysis is carried through in a Fourier space only in one direction. In these spaces we show existence of solutions together with their basic asymptotic structure.