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

Abstract

Let $\bfu$ 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 $\mathsf{s}\ge \frac{2}{3}q$, $q>6$, such that the $L^{\mathsf{s}}-$norm of $\bfu$ on the spherical surface of radius $R$ divided by $R$ is less than a constant depending only on {\sf s} and $q$, then $\bfu(x)$ must decay as $|x{|}^{-1}$ for $|x|\to \infty$. 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.