Stabilizing the Long-time Behavior of the Navier-Stokes Equations and Damped Euler Systems by Fast Oscillating Forces

TL;DR

This paper demonstrates that fast oscillating external forces can stabilize solutions of Navier-Stokes and damped Euler systems, causing convergence to periodic flows in 2D and similar effects in 3D weak solutions.

Contribution

It introduces a novel stabilization method using fast oscillating forces, applicable even with stationary forces having large momentum, for both 2D and 3D fluid systems.

Findings

2D solutions converge to periodic flows under oscillating forces

Stabilization occurs even with stationary forces with large momentum

Analogous stabilization results for 3D weak solutions

Abstract

The paper studies the issue of stability of solutions to the Navier-Stokes and damped Euler systems in periodic boxes. We show that under action of fast oscillating-in- time external forces all two dimensional regular solutions converge to a time periodic flow. Unexpectedly, effects of stabilization can be also obtained for systems with stationary forces with large total momentum (average of the velocity). Thanks to the Galilean transformation and space boundary conditions, the stationary force changes into one with time oscillations. In the three dimensional case we show an analogical result for weak solutions to the Navier- Stokes equations.