On 2D incompressible and density-dependent Navier–Stokes equations: global stabilization under dynamic Couette flow

This paper studies the long-time dynamics of classical solutions to an initial-boundary value problem for the 2D Navier–Stokes equations of an incompressible and inhomogeneous fluid flow: \(\begin{aligned} \begin{aligned} \rho _t+\textbf{u}\cdot \nabla \rho&=0,&\textbf{x}&\in \Omega ,\ t>0,\\ \rho \textbf{u}_t+\rho \textbf{u}\cdot \nabla \textbf{u}+ \nabla P&=\mu \Delta \textbf{u}+\rho \nabla \psi ,&\textbf{x}&\in \Omega ,\ t>0,\\ \nabla \cdot \textbf{u}&=0,&\textbf{x}&\in \Omega ,\ t>0, \end{aligned} \end{aligned}\) $$\begin{array}{c}\hfill \begin{array}{cccc}\hfill {\rho }_t+\mathbf{u}·\mathrm{\nabla }\rho & =0,\hfill & \hfill \mathbf{x}& \in \mathrm{\Omega },t>0,\hfill \\ \hfill \rho {\mathbf{u}}_t+\rho \mathbf{u}·\mathrm{\nabla }\mathbf{u}+\mathrm{\nabla }P& =\mu \mathrm{\Delta }\mathbf{u}+\rho \mathrm{\nabla }\psi ,\hfill & \hfill \mathbf{x}& \in \mathrm{\Omega },t>0,\hfill \\ \hfill \mathrm{\nabla }·\mathbf{u}& =0,\hfill & \hfill \mathbf{x}& \in \mathrm{\Omega },t>0,\hfill \end{array}\end{array}$$ where \(\Omega = (0,L)\times (0,H)\) $\mathrm{\Omega }=(0,L)\times (0,H)$ is a rectangle. The system of equations is supplemented with time-dependent boundary condition of Couette type for the velocity field: \(\textbf{u}|_{\partial \Omega }=(\alpha (t)x_2,0)^\textrm{T} \equiv \widehat{\textbf{u}}(t)\) ${\mathbf{u}|}_{\partial \mathrm{\Omega }}={(\alpha (t)x_2,0)}^{\text{T}}\equiv \widehat{\mathbf{u}}(t)$ , where \(\alpha \) $\alpha$ is a smooth function of t, and periodic boundary condition for \(\rho \) $\rho$ and \(\psi \) $\psi$ on two vertical sides of \(\Omega \) $\mathrm{\Omega }$ . Under certain regularity conditions for \(\psi \) $\psi$ and \(\alpha \) $\alpha$ , it is shown that for given generic initial data in \(H^3(\Omega \) $H^3(\mathrm{\Omega }$ ), there exists a unique solution to the initial-boundary value problem and the perturbation \((\textbf{u}-\widehat{\textbf{u}})(\textbf{x},t)\) $(\mathbf{u}-\widehat{\mathbf{u}})(\mathbf{x},t)$ converges to zero in certain topology as time goes to infinity.