Annihilation of slowly-decaying terms of Navier-Stokes flows by external forcing

TL;DR

This paper presents an algorithm to construct localized external forces that induce faster energy decay in Navier-Stokes flows, effectively eliminating slowly-decaying terms and surpassing traditional decay rates.

Contribution

It introduces a method to explicitly design localized forces that accelerate energy dissipation in Navier-Stokes solutions, surpassing standard decay rates.

Findings

Constructs localized external forces with prescribed profiles.

Achieves energy decay faster than the usual optimal rate.

Force can be supported in finite space-time regions.

Abstract

The goal of this paper is to provide an algorithm that, for any sufficiently localised, divergence-free small initial data, explicitly constructs a localised external force leading to a rapidly dissipative solutions of the Navier-Stokes equations $\mathbb{R}^n$: namely, the energy decay rate of the flow will be forced to satisfy $\|u(t)\|_2^2 = o(t^{-(n+2)/2})$ as $t \to \infty$, which is beyond the usual optimal rate. An important feature of our construction is that this force can always be taken compactly supported in space-time, and its profile arbitrarily prescribed up to a spatial rescaling. Since the forcing term vanishes after a finite time interval, our result suggests that nontrivial interactions between the linear and nonlinear parts occur, annihilating all the slowly decaying terms contained in Miyakawa and Schonbek's asymptotic profiles.