Refined asymptotics of the steady Navier Stokes equation around small Landau solutions

In this paper we study the large distance asymptotics of small steady solutions of the 3d Navier Stokes equation in exterior domains. It was proved by Korolev and the second author \cite{SverakKorolev} that the leading term is given by the Landau solution, and it was conjectured that the next order term should be $O(1\mathrm{/}\mathrm{\mid }x{\mathrm{\mid }}^2)$ as $x\to \mathrm{\infty }$. We confirm that this is indeed the case and we compute the next order asymptotics in terms of eigenvalues of a suitably constructed linearized operator around the Landau solution on the unit sphere. While the decay of some of the terms is precisely $O(1\mathrm{/}\mathrm{\mid }x{\mathrm{\mid }}^2)$, the the decay of other terms is slightly accelerated.