Asymptotic Stability of the Three-Dimensional Couette Flow for the Stokes-Transport Equations

Abstract

In this paper, we investigate the asymptotic stability of the three-dimensional Couette flow in a stratified fluid governed by the Stokes-transport equations. We observe that a lift-up effect similar to that in the three-dimensional Navier–Stokes equation near Couette flow destabilizes the system, while the inviscid-damping-type decay due to the Couette flow \((Y,0,0)^\top \), together with the damping structure induced by the decreasing background density \(\varrho _\textrm{s}(Y)\), stabilizes the system. More precisely, we prove that if the initial density is close to a linearly decreasing function in the Gevrey-\(\frac{1}{s}\) class with \(\frac{1}{2}< s\leqq 1\), namely, \(\Vert \varrho _\textrm{in}(X,Y,Z)-\varrho _\textrm{s}(Y)\Vert _{\mathcal {G}^{s}}\leqq \varepsilon \), then the perturbed density remains close to \(\varrho _\textrm{s}(Y)\). Moreover, the associated velocity field converges to Couette flow \((Y, 0, 0)^{\top }\) with a convergence rate of \(\frac{1}{\langle t\rangle ^3}\).