On the Cauchy problem of 3D incompressible Navier-Stokes-Cahn-Hilliard system

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 $\mathrm{\parallel }(u_0,\mathrm{\nabla }{\varphi }_0){\mathrm{\parallel }}_{{\dot{H}}^{\frac{1}{2}}}$ 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.