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 Lˢ-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.