  The moving contact line paradox discussed in the famous paper by Huh and
Scriven has lead to an extensive scientific discussion about singularities in
continuum mechanical models of dynamic wetting in the framework of the
two-phase Navier-Stokes equations. Since the no-slip condition introduces a
non-integrable and therefore unphysical singularity into the model, various
models to relax the singularity have been proposed. Many of the relaxation
mechanisms still retain a weak (integrable) singularity, while other approaches
look for completely regular solutions with finite curvature and pressure at the
moving contact line. In particular, the model introduced recently in (Lukyanov,
Pryer, Langmuir 2017) aims for regular solutions through modified boundary
conditions. The present work applies the mathematical tool of compatibility
analysis to continuum models of dynamic wetting. The basic idea is that the
boundary conditions have to be compatible at the contact line in order to allow
for regular solutions. Remarkably, the method allows to compute explicit
expressions for the pressure and the curvature locally at the moving contact
line for regular solutions to the model of Lukyanov and Pryer. It is found that
solutions may still be singular for the latter model.
