Local well-posedness of axisymmetric flows in a bounded cylinder with the moving contact lines

The aim of this paper is to propose a new formulation of free boundary problems of the Navier--Stokes equations in the three-dimensional Euclidean space with moving contact line in terms of classical balance laws and boundary conditions. The new system does not require any additional boundary conditions on the moving contact line, where a contact angle between a free interface and a rigid surface is oscillating in time. It should be emphasized that the total available energy of the system is conserved along with smooth solutions and the negative total available energy is a strict Lyapunov functional if the velocity field and the free interface are axisymmetric. Furthermore, we show local well-posedness of the formulated system provided that the initial data are axisymmetric. Of crucial importance for the analysis is the property of maximal $L^p - L^q$-regularity for the corresponding linearized problem.