Parabolic-scalings on large-time behavior of the incompressible Navier--Stokes flow

TL;DR

This paper investigates the large-time behavior of incompressible Navier--Stokes flow using asymptotic expansion and parabolic scalings, revealing higher-order decay and logarithmic evolutions.

Contribution

It introduces a higher-order asymptotic expansion with 2n-th order and uncovers logarithmic evolutions in the flow's decay behavior.

Findings

Asymptotic expansion with 2n-th order derived

Logarithmic evolutions in higher-order decay discovered

Parabolic scalings ensure uniqueness of the expansion

Abstract

Through asymptotic expansion, the large-time behavior of incompressible Navier--Stokes flow in $n$-dimensional whole space is depicted. Especially, from their parabolic scalings, large-time behaviors of any terms on the expansion are clarified. The parabolic scalings also guarantee the uniqueness of the expansion. In the preceding work, the expansion with the $n$th order has already been derived. They also predicted that the flow has some logarithmic evolutions in higher-order decay. In this paper, an asymptotic expansion with $2n$th order is presented. Furthermore, logarithmic evolutions are discovered.