Z. Angew. Math. Phys. (2018) 69:89
2018 Springer International Publishing AG,
part of Springer Nature
Zeitschrift f¨ur angewandte
Mathematik und Physik ZAMP
On dry spot and droplet solutions for thin ﬁlms on the plane
Abstract. Steady states including axisymmetric droplet and dry spot solutions of the thin ﬁlm-type equation are studied.
Results on the existence and energy stability of these solutions are obtained.
Mathematics Subject Classiﬁcation. Primary 76A20; Secondary 35B35, 35K55.
Keywords. Thin ﬁlm-type equation, Steady state, Droplet with zero contact angle, Dry spot, Energy stability.
Consider the thin ﬁlm-type equation
+ ∇·(a(h)∇h + Bb(h)∇h)=0,
in a planar domain Ω. Letting f satisfy f
= Bb/a, this equation can be written alternatively as
+ ∇·(a(h)∇(h + f(h))) = 0. (1.1)
Here, a(z) is a positive continuous function for z>0 and vanishes at z = 0, and the solution h(x, t)
is assumed to be nonnegative. When a(z)=z
, this equation describes the motion of a thin liquid ﬁlm
whose height is given by h. The derivation of (1.1) from the Navier–Stokes equations with free boundaries
is based on the lubrication approximation. The fourth-order term comes from the surface tension of the
ﬁlm, and the lower-order term ∇·(b∇h) represents other physical eﬀects such as gravity, capillarity, van
der Waals molecular forces. The constant B could be positive or negative. Here, we will take it to be
positive so that surface tension and other eﬀects are in a competing position. Its magnitude measures
the importance of these eﬀects as compared to the surface tension. The thin ﬁlm-type equation embraces
many interesting models concerning ﬂuids and has been studied by many people. One may consult the
surveys Oron et al.  and Myers  for many known experimental and numerical results.
Equation (1.1) will be studied under the “no ﬂux” conditions
∇h · ν = a(h)∇(h + f(h)) · ν =0 on ∂Ω, (1.2)
where ν is the outer unit normal on the boundary of Ω. A classical solution of (1.1) and (1.2) satisﬁes
two fundamental identities. First, we have the conservation of mass
h(x, t)dx =
is the initial function. Second, we have the energy dissipation relation
a(h)|∇(h + f(h)|
dxdt = E(h