Free boundary problem of steady incompressible flow with contact angle

We consider a fluid in a three-dimensional container with free upper surface, which satisfies stationary Navier–Stokes equations. The container has a smooth wall and the free upper surface is a graph of smooth plane region Σ. From the physical point of view, the fluid satisfies Dirichlet boundary condition on the wall of the container. On the other hand, the fluid satisfies a mixed boundary condition on the free upper surface, which is formulated by the balances of forces on the free upper surfaces. When a fluid is flowing in the container, singularity may be developed along the contact line because of different boundary conditions. We show the existence of flow filling a domain Ω with regular free surface in a Sobolev space , 3