# Time evolution of the Navier-Stokes flow in far-field

## TL;DR

This paper establishes asymptotic expansions for the Navier-Stokes flow in the far-field, clarifying large-time behaviors and velocity evolution using renormalization and Biot-Savard law.

## Contribution

It introduces a high-order asymptotic expansion for the velocity in far-field Navier-Stokes flow under specific initial conditions.

## Key findings

- Derived high-order asymptotic expansion for velocity.
- Clarified scalings and large-time behavior of solutions.
- Provided asymptotic behavior as time approaches infinity.

## Abstract

Asymptotic expansion in far-field for the incompressive Navier-Stokes flow are established. Under moment conditions on the initial vorticity, technique of renormalization together with Biot-Savard law derives an asymptotic expansion for the velocity with high-order. Especially scalings and large-time behaviors of the expansions are clarified. By employing them, time evolution of velocity in far-field is drawn. As an appendix, asymptotic behavior of solutions as time variable tends to infinity is given.