A convective Cahn–Hilliard model with dynamic boundary conditions

We consider a general class of convective bulk-surface Cahn–Hilliard systems
with singular potentials. In contrast to classical Neumann boundary conditions,
the dynamic boundary conditions of Cahn–Hilliard type allow for dynamic
changes of the contact angle between the diffuse interface and the boundary, a
convection-induced motion of the contact line as well as absorption of material
by the boundary. The coupling conditions for bulk and surface quantities involve
parameters $K,L\in [0,\infty]$, whose choice declares whether these conditions
are of Dirichlet, Robin or Neumann type.
In this talk, I present some recent results on the well-posedness of this system.
After briefly recalling the results for regular potentials, we focus on singular
potentials. Here, we make use of the Yosida approximation to regularise these
potentials, which allows us to apply the results for regular potentials and eventually
pass to the limit in this approximation scheme to obtain a global-in-time
weak solution. Afterwards, under additional assumptions on the mobility functions,
we prove higher regularity estimates for two different classes of velocity
fields, and in particular, for those having Leray-type regularity. Finally, exploiting
these higher regularity estimates, we can establish separation properties of
the phase-fields.
This is based on joint work with Andrea Giorgini and Patrik Knopf.