  In this paper, we are concerned with the well-posedness and large time
behavior of Cauchy problem for 3D incompressible Navier-Stokes-Cahn-Hilliard
equations. First, using Banach fixed point theorem, we establish the local
well-posedness of solutions. Second, assuming
$\|(u_0,\nabla\phi_0)\|_{\dot{H}^\frac12}$ is sufficiently small, we obtain the
global well-posedness of solutions. Moreover, the optimal decay rates of the
higher-order spatial derivatives of the solution are also obtained.
