  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
$\Omega$. 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 $e^{- c_\# t}$ of the global solution. The majority is that the
derivative $f'(\phi)$ of the physical relevant energy density $f(\phi)$ will
generate an additional damping effect under the perturbation $\phi = \varphi
\pm 1$.

Analysis of PDEs