Mathematical study of multi‐phase flow under shear through order parameter formulation

In this paper we study the coupling of the Navier–Stokes equations and the Cahn–Hilliard equation which stands for a model of a multi‐phase fluid under shear. We first study existence and uniqueness of solutions of the system in dimension $2$ and $3$ even if the diffusion coefficient is allowed to degenerate. In the last part, an asymptotic stability result is shown.