The Navier-slip thin-film equation for 3D fluid films: existence and uniqueness

Abstract

We consider the thin-film equation ${\partial }_{t}h+\nabla \cdot \left(h^2\nabla \Delta h\right)=0$ in physical space dimensions (i.e., one dimension in time $t$ and two lateral dimensions with $h$ denoting the height of the film in the third spatial dimension), which corresponds to the lubrication approximation of the Navier-Stokes equations of a three-dimensional viscous thin fluid film with Navier-slip at the substrate. This equation can have a free boundary (the contact line), moving with finite speed, at which we assume a zero contact angle condition (complete-wetting regime). Previous results have focused on the $1+1$-dimensional version, where it has been found that solutions are not smooth as a function of the distance to the free boundary. In particular, a well-posedness and regularity theory is more intricate than for the second-order counterpart, the porous-medium equation, or the thin-film equation with linear mobility (corresponding to Darcy dynamics in the Hele-Shaw cell). Here, we prove existence and uniqueness of classical solutions that are perturbations of an asymptotically stable traveling-wave profile. This leads to control on the free boundary and in particular its velocity.

86 pages, 2 figures, revised version; norms, embeddings, and nonlinear estimates corrected

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• Spreading equilibria under mildly singular potentials: pancakes versus droplets
• Thin-film equations with singular potentials: an alternative solution to the contact-line paradox
• Classical solutions to the thin-film equation with general mobility in the perfect-wetting regime
• Relaxation to equilibrium in the one-dimensional thin-film equation with partial wetting
• Droplet motion with contact-line friction: long-time asymptotics in complete wetting
• Stochastic elastohydrodynamics of contact and coarsening during membrane adhesion
• A Class of Functional Inequalities and their Applications to Fourth-Order Nonlinear Parabolic Equations