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

TL;DR

This paper establishes the existence and uniqueness of steady incompressible fluid flow solutions in the plane with prescribed velocity at infinity, revealing a directional spatial structure related to the Oseen flow using Fourier analysis.

Contribution

It introduces a novel approach treating flow direction as a time variable and employs Fourier transform techniques to analyze spatial asymptotics of solutions.

Findings

Existence of unique solutions for large prescribed velocities.

Characterization of the spatial structure of solutions.

Method applicable to other elliptic systems.

Abstract

We investigate a steady flow of incompressible fluid in the plane. The motion is governed by the Navier-Stokes equations with prescribed velocity $u_\infty$ at infinity. The main result shows the existence of unique solutions for arbitrary force, provided sufficient largeness of $u_\infty$. Furthermore a spacial 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 directino of the flow as \emph{time} direction. The analysis is done in 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 modeled by elliptic equations.