Stabilizability of 2D and 3D Navier-Stokes equations with memory around a non-constant steady state

TL;DR

This paper studies the stabilization of 2D and 3D Navier-Stokes equations with memory effects around non-constant steady states using localized control, spectral analysis, and fixed-point methods.

Contribution

It introduces a novel approach to stabilize Navier-Stokes equations with memory by analyzing the principal operator and extending results to the full nonlinear system.

Findings

Established feedback stabilization for the principal linear system.

Extended stabilization results to the full nonlinear Navier-Stokes system.

Proved stabilizability of the vorticity equation around non-constant steady states.

Abstract

In this article, we investigate the stabilizability of the two- and three-dimensional Navier-Stokes equations with memory effects around a non-constant steady state using a localized interior control. The system is first linearized around a non-constant steady state and then reformulated into a coupled system by introducing a new variable to handle the integral term. Due to the presence of variable coefficients in the linear operator, the rigorous computation of eigenvalues and eigenfunctions becomes infeasible. Therefore, we concentrate on the principal operator, and investigate its analyticity and spectral properties. We establish a feedback stabilization result for the principal system, ensuring a specific decay rate. Using the feedback operator derived from this analysis, we extend the approach to the full system, constructing a closed-loop system. By proving a suitable regularity result and applying a fixed-point argument, we ultimately demonstrate the stabilizability of the full system. We also discuss the stabilizability of the corresponding vorticity equation around a non-constant steady state.