Vanishing Capillarity-Viscosity Limit For The 3D Incompressible Inhomogeneous Fluid Models Of Korteweg Type With Slip Boundary Conditions

In this paper, we investigated the solvability, regularity and vanishing capillarity-viscosity limit for the 3D incompressible inhomogeneous fluid models of Korteweg type with a slip boundary condition. In order to solve the problem of vanishing capillarity-viscosity limits, we added the boundary condition for initial density,which is ??0 · n = 0 on the boundary of the domain. Then we showed that when the viscous coefficient and capillarity coefficient vanishing on the boundary, the solutions of incompressible inhomogeneous fluid models of Korteweg type can con-verge to the corresponding ideal incompressible inhomogeneous Euler equations,and obtained a convergence result.