Large-time behavior and far field asymptotics of solutions to the Navier-Stokes equations

TL;DR

This paper investigates the long-term behavior and far field asymptotics of solutions to the incompressible Navier-Stokes equations, providing high-order asymptotic expansions without initial velocity moment conditions.

Contribution

It introduces a method to derive high-order asymptotic expansions and far field behavior for Navier-Stokes solutions without requiring moment conditions on initial data.

Findings

Established high-order asymptotic expansions for large-time solutions.

Derived far field asymptotics of solutions.

Utilized Biot-Savard law and vorticity renormalization techniques.

Abstract

Asymptotic expansions of global solutions to the incompressible Navier-Stokes equation as $t$ tends to infinity with high-order is studied and large-time behavior of the expansion is clarified. Furthermore, far field asymptotics also is derived. Those expansions are provided without moment conditions on the initial velocity. The Biot-Savard law together with the renormalization for the vorticity equations yields those expansions.