Global Well-Posedness of Contact Lines: 2D Navier-Stokes Flow

Based on the global a priori estimates in [Guo-Tice, J. Eur. Math. Soc. (2024)], we establish the well-posedness of a viscous fluid model satisfying the dynamic law for the contact line {equation*} W(_tζ(±,t))=[\![γ]\!]∓σ{_1ζ}{(1+|_1ζ|^2)1/2}(±,t) {equation*} in 2D domain, where ζ(x₁,t) is a free surface with two contact points ζ(±,t), [\![γ]\!] and σ are constants characterizing the solid-fluid-gas free energy, and the increasing W is the contact point velocity response function. Motivated by the energy-dissipation structure, our construction relies on the construction of a pressureless weak solution for the coupled velocity and free interface for the linearized problems, via a Galerkin approximation with a time-dependent basis and an artificial regularization for the capillary operator.