Spatial decay/asymptotics in the Navier-Stokes equation

TL;DR

This paper investigates the spatial asymptotic behavior of solutions to the Navier-Stokes equation on Euclidean space, establishing well-posedness in weighted spaces and analyzing how solutions develop non-trivial asymptotics at infinity.

Contribution

It proves local well-posedness of Navier-Stokes in weighted Sobolev and asymptotic spaces and characterizes the analytic dependence and asymptotic development of solutions.

Findings

Solutions depend analytically on initial data and time.

Solutions develop non-trivial asymptotic terms at infinity.

Solutions exhibit spatial smoothing depending on asymptotic order.

Abstract

We discuss the appearance of spatial asymptotic expansions of solutions of the Navier-Stokes equation on $\mathbb{R}^n$. In particular, we prove that the Navier-Stokes equation is locally well-posed in a class of weighted Sobolev and asymptotic spaces. The solutions depend analytically on the initial data and time and (generically) develop non-trivial asymptotic terms as $|x|\to\infty$. In addition, the solutions have a spatial smoothing property that depends on the order of the asymptotic expansion.