The mathematics of spreading droplets

Wetting phenomena at small scales are an area where chemistry,
physics, mathematics, and engineering intersect. In recent years,
driven also by molecular dynamics, new concepts have been introduced
to describe the statics and dynamics of wetting, allowing new
insights into the old problems of surface forces. Among these
problems, two prominent ones are an appropriate mathematical modeling
of the moving contact line where liquid, solid, and surrounding vapor
meet, and how such models influence the macroscopic properties of the
flow. After a general framing -- the classical setting of droplets'
statics and dynamics, diffuse and sharp interface models, classical
and new descriptions of the contact line -- I shall review the PDE
theory for one of such models -- the so-called thin-film equation --, mainly focusing on the two aforementioned problems and on some of the most interesting current challenges.