Asymptotic behavior of the two‐phase flow around the planar Couette flow in three‐dimensional space

This paper is concerned with the nonlinear stability of planar Couette flow for the three‐dimensional NS‐NS equations, which are used to model the motion of two‐phase flow. Our result shows that the planar Couette flow is asymptotically stable for the initial perturbations sufficiently small in some Sobolev space if the background velocity and the Reynolds number are sufficiently small. Moreover, it is proved that the solution converges to the stationary state at an algebraic time decay rate, and we also give the decay rate of the relative velocity.