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

TL;DR

This work addresses the large-distance behavior of Leray solutions to the stationary Navier–Stokes equations in exterior domains by developing a theory of very weak solutions. It first builds existence, uniqueness, and decay estimates for very weak solutions to the linear Stokes problem with low-regularity data, then extends to the nonlinear Navier–Stokes problem via a contraction mapping in a small-data regime. Under a smallness condition on the boundary trace of the velocity on a large sphere, the authors prove that Leray solutions decay like $|x|^{-1}$ at infinity, with higher derivatives decaying as $|x|^{-1-|oldsymboleta|}$ and $|x|^{-2-|oldsymboleta|}$ for the pressure. The approach connects exterior-domain very weak solutions to the asymptotic behavior of Leray solutions through weak-strong uniqueness, yielding a sharp decay rate and a framework that may inform Liouville-type questions and Cauchy-problem behavior for Navier–Stokes in exterior domains.

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 ${\sf s}\ge \frac23 q$, $q>6$, such that the $L^{\sf s}-$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.

Theorems & Definitions (11)

• Definition 2.1
• Theorem 2.1
• Theorem 2.2
• Definition 3.1
• Theorem 3.1
• Theorem 3.2
• Definition 4.1
• Proposition 4.1
• Lemma 4.1
• Theorem 4.1