Global existence and uniqueness of the density-dependent incompressible Navier-Stokes-Korteweg system with variable capillarity and viscosity coefficients

We consider the global well-posedness of the inhomogeneous incompressible Navier-Stokes-Korteweg system with a general capillary term. Based on the maximal regularity property, we obtain the global existence and uniqueness of solutions to the incompressible Navier-Stokes-Korteweg system with variable viscosity and capillary terms. By assuming the initial density ${\rho }_0$ is close to a positive constant, additionally, the initial velocity $u_0$ and the initial density $\mathrm{\nabla }{\rho }_0$ are small in critical space ${\dot{B}}_{p,1}^{-1+d\mathrm{/}p}({\mathbb{R}}^d)$ $(1<p<d)\mathrm{.}$ This work relies on the maximal regularity property of the heat equation, of the Stokes equation, and of the Lamé equation.