Optimal asymptotic behavior of the vorticity of a viscous flow past a two-dimensional body

Abstract

The asymptotic behavior of the vorticity for the steady incompressible Navier–Stokes equations in a two-dimensional exterior domain is described in the case where the velocity at infinity $\boldsymbol{u} _\infty$ is nonzero. It is well known that the asymptotic behavior of the velocity field is given by the fundamental solution of the Oseen system which is the linearization of the Navier–Stokes equation around $\boldsymbol{u} _\infty$. Concerning the vorticity, the previous asymptotic expansions were relevant only inside a parabolic region called the wake region. Here we present an asymptotic expansion for the vorticity relevant everywhere. Surprisingly, the found asymptotic behavior is not given by the Oseen linearization and has a power of decay that depends on the data. This strange behavior is specific to the two-dimensional problem and is not present in three dimensions.