Solutions of the Navier-Stokes equations with forced rapid space-time decay

TL;DR

This paper demonstrates how to construct external forces and solutions for the Navier-Stokes equations that achieve faster decay rates in space and time than typical solutions, with a fixed spatial profile of the forcing.

Contribution

It introduces a method to control the decay properties of Navier-Stokes solutions by designing external forces with fixed spatial profiles and adaptable temporal profiles.

Findings

Constructed solutions with decay rates surpassing standard limits.

External forcing with fixed spatial profile can be independent of initial data.

Temporal profile of forcing depends on initial conditions.

Abstract

We study the pointwise decay properties of solutions to the incompressible Navier-Stokes equations, both in the space and time variables. It is well known that generic global solutions on $\mathbb{R}^n$ do not decay faster at infinity than $|x|^{-(n+1)}$ and $t^{-(n+1)/2}$ in the pointwise sense. In this paper, we address the control problem of constructing an external forcing and a solution to the Navier-Stokes equations whose space-time decay properties go beyond these limiting rates. A distinctive feature of the forcing term is that its spatial profile can be fixed once and for all, independently of the initial data of the problem, and localized in an arbitrarily small region of $\mathbb{R}^n$. Only the temporal profile of the external force displays a dependency on the initial datum.