On the initial-boundary value problem of two-phase incompressible flows with variable density in smooth bounded domain

In this work, we study the so-called Allen-Cahn-Navier-Stokes equations, a diffuse-interface model for two-phase incompressible flows with different densities. We first prove the local-in-time existence and uniqueness of classical solutions with finite initial energy over the smooth bounded domain $\mathrm{\Omega }$. The key point is to transform the boundary values of the higher order spatial derivatives to that of the higher order time derivatives by employing the well-known Agmon-Douglis-Nireberg theory in [6]. We then prove global existence near the equilibrium $(0,\pm 1)$ and justify the time exponetial decay $e^{-c_{\mathrm{\#}}t}$ of the global solution. The majority is that the derivative $f^{\mathrm{\prime }}(\phi )$ of the physical relevant energy density $f(\phi )$ will generate an additional damping effect under the perturbation $\phi =\phi \pm 1$.