Vanishing capillarity–viscosity limit of the incompressible Navier–Stokes–Korteweg equations with slip boundary condition

In this paper, we investigate the vanishing capillarity–viscosity limit of the incompressible Navier–Stokes–Korteweg (NSK) equations in a three-dimensional horizontally periodic strip domain, in which the velocity of the fluid is supplemented with slip boundary condition and the gradient of density with Dirichlet boundary condition on the boundary. We prove that there exists an positive constant $T_0$ independent on the capillarity and viscosity coefficients, such that the incompressible NSK equations have a unique strong solution on $[0,T_0]$ and the solution is uniformly bounded in $H^3$. Based on the uniform estimates, we further give the convergence rate in $H^1$ from the solutions of the incompressible NSK equations to the solution of the inhomogeneous incompressible Euler equations as the capillarity and viscosity coefficients go to zero simultaneously.