Very Weak Solutions and Asymptotic Behavior of Leray Solutions to the Stationary Navier-Stokes Equations

Let be a Leray solution to the Navier-Stokes boundary-value problem in an exterior domain, vanishing at infinity and satisfying the generalized energy inequality. We show that if there exist R>0 and s 23 q, q>6, such that the L^ s-norm of on the spherical surface of radius R divided by R is less than a constant depending only on s and q, then (x) must decay as |x|^-1 for |x|. This result is proved with an approach based on a new theory of very weak solutions in exterior domains which, as such, is of independent interest.