Stabilizability of 2D and 3D Navier-Stokes equations with memory around a non-constant steady state
Wasim Akram, Manika Bag, Manil T. Mohan
TL;DR
The paper addresses the stabilization of 2D/3D Navier–Stokes equations with memory around a non-constant steady state using localized interior control. It linearizes about the steady state and reformulates memory via an auxiliary variable, then analyzes a principal operator $\mathcal{A}$ that generates an analytic semigroup and has a spectrum with negative real parts; stabilizability is achieved for any decay rate $\nu$ with $0<\nu<\nu_0$, where $\nu_0=\frac{\kappa}{\eta}+\lambda$, through a Riccati-based feedback $K$. This principal-system feedback is extended to the full nonlinear system by proving regularity estimates and applying a Banach fixed-point argument, yielding exponential stabilization for small initial data in $\mathbb{V}$ and an associated pressure estimate. The vorticity formulation is treated similarly, with analogous stabilization results, and the work highlights the novelty of stabilizing systems with memory around non-constant equilibria by focusing on the analytically tractable principal operator. The methods combine analytic semigroup theory, spectral analysis, Hautus-type criteria, Riccati-based feedback, and fixed-point theory to achieve local stabilization of NSEs with memory and offer a pathway for similar evolution equations with memory 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.