The Navier-Stokes equation with time quasi-periodic external force: existence and stability of quasi-periodic solutions

TL;DR

This paper proves the existence and stability of small amplitude, time-quasi-periodic solutions for the incompressible Navier-Stokes equation on a torus under a quasi-periodic external force, demonstrating their asymptotic stability and exponential convergence.

Contribution

It establishes the existence and orbital stability of quasi-periodic solutions for Navier-Stokes with quasi-periodic forcing, extending understanding of long-term behavior.

Findings

Existence of small amplitude quasi-periodic solutions.

Orbital and asymptotic stability in high Sobolev spaces.

Exponential convergence to invariant tori.

Abstract

We prove the existence of small amplitude, time-quasi-periodic solutions (invariant tori) for the incompressible Navier-Stokes equation on the $d$-dimensional torus $\T^d$, with a small, quasi-periodic in time external force. We also show that they are orbitally and asymptotically stable in $H^s$ (for $s$ large enough). More precisely, for any initial datum which is close to the invariant torus, there exists a unique global in time solution which stays close to the invariant torus for all times. Moreover, the solution converges asymptotically to the invariant torus for $t \to + \infty$, with an exponential rate of convergence $O( e^{- \alpha t })$ for any arbitrary $\alpha \in (0, 1)$.