Well-posedness of Navier-Stokes/Cahn-Hilliard equations modeling the dynamics of contact line in a channel

Abstract: In this paper, we study the contact line problem in a channel. Precisely, we consider the incompressible Navier-Stokes/Cahn-Hilliard system with eneralized Navier boundary condition and relaxation boundary condition in a channel, which is the phase field model for the moving contact line problem in fluid mechanics. We establish the existence and uniqueness of local-in-time strong solution to this initial boundary value problem in 2D. To our knowledge, this is the first result to give the local-in-time well-posedness of Navier-Stokes/Cahn-Hilliard system with generalized Navier boundary condition and relaxation boundary condition. This result provides a rigorous mathematical analysis to confirm that the physical and numerical results by Qian-Wang-Sheng [Phys. Rev. E 68 (2003), 016306, 1-15; J. Fluid Mech. 564 (2006), 333-360] are well-posed and reasonable.