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

We investigate a steady flow of incompressible fluid in the plane. The motion is governed by the Navier–Stokes equations with prescribed velocity u∞ at infinity. The main result shows the existence of unique solutions for arbitrary force, provided sufficient largeness of u∞. Furthermore a spatial structure of the solution is obtained in comparison with the Oseen flow. A key element of our new approach is based on a setting which treats the direction of the flow as the time direction. The analysis is done in the framework of the Fourier transform taken in one (perpendicular) direction and a special choice of function spaces which take into account the inhomogeneous character of the symbol of the Oseen system. From that point of view our technique can be used as an effective tool in examining spatial asymptotics of solutions to other systems modelled by elliptic equations