Some mathematical aspects of capillary surfaces

A. Mellet∗

April 9, 2008

Abstract

This is a review of various mathematical aspects of the study of cap-illary surfaces. A special emphasis is put on the behavior of the contactline, the three phases junction between the liquid, solid and vapor.

1 Introduction

A small drop of water lying on a flat surface offers a familiar example of capillarysurface. More generally, we call capillary surface the interface between two fluidsthat are in contact with each other without mixing. The shape of such surfacesis mainly determined by the phenomenon of surface tension, which results fromthe action of cohesive forces between liquid molecules (”molecules of liquid arehappier when surrounded by other molecules of liquid”).

The first attempts of mathematical analysis of those phenomena go backto T. Young [You05] and P. S. Laplace [Lap05] (later followed by C.F. Gauss)and the introduction of the notion of mean-curvature. The force generated bysurface tension (sometime referred to as capillary force) tends to minimize thearea of the free surface and is responsible for a pressure difference across theinterface (capillary pressure) which is proportional to the mean-curvature ofthe free surface. Equilibrium capillary surfaces thus satisfy a mean-curvatureequation, known as Young-Laplace’s equation.

Interesting behaviors occur when the capillary surface is in contact with asolid support, for example when a liquid drop is resting on a solid surface. Alongthe triple junction, where liquid, gas and solid meet, a phenomenon similar tosurface tension takes place, with three phases instead of two. The balance offorces along this contact line gives rise to a contact angle condition which playsa central role in this review.

This note contains two parts devoted respectively to statics and dynamicsmodels. In the first part, we present two aspects of the theory of equilibriumsurfaces: The capillary tube and the sessile drop. In both case, the mathe-matical analysis relies heavily on the theory of functions of bounded variation

∗Dept of Mathematics, University of British Columbia, Vancouver, BC V6T 1Z2, Canada

1

* S*LiquidVapor

S

S

Figure 1: Three phases configuration

and Caccioppoli sets. Our presentation of the capillary tube problem is mainlybased on the work of M. Emmer, L. Miranda, U. Massari, E. Giusti and R. Finn(we will follow the wonderful book of R. Finn [Fin86] for much of the presen-tation). The sessile drop problem gives rise to a free boundary problems whichhas been extensively studied in particular by L. Caffarelli and A. Friedman. Wealso describe some recent results concerning hysteresis phenomena.

Concerning the dynamical case, the main issue is to find a model that accu-rately describes the motion of the contact line. This turns out to be considerablymore difficult than in the static case, because the motion of the liquid needs tobe taken into account, typically through Navier-Stokes equations. We will focuson two models that have received the attention of the mathematics community:the quasi-static model and the lubrication approximation for thin droplets (theso-called thin film equation).

2 Equilibrium capillary surfaces

2.1 The Young-Laplace law

In this first section, we described the mathematical theory for two particulartypes of capillary surfaces: The capillary tube (narrow tube filled with liquid)and the sessile drops (drop of liquid lying on a solid support). But first, letus derive the main equations satisfied by capillary surfaces: Young-Laplace’sequations.

The energy balance Following R. Finn [Fin86], we consider a three phasesconfiguration liquid/vapor/solid (see Figure 1). We denote by S the free inter-face between the liquid and the solid and by S∗ (resp. S

∗) the contact surface

between the solid support and the liquid (resp. the vapor). Various energiescome into play in determining the shape of the free surface S:

(i) The surface tension energy: As stated in the introduction, the effect of

2

surface tension is to minimize the area of the interface. The resulting energy isthus of the form:

ES = σLV S

where σLV is the surface tension coefficient between the liquid and the vaporphases. Note that here and below we denote by S either the surface itself or itsarea (the notion of area will be made precise in the next section).

(ii) The wetting energy: Similarly, the energy resulting from the contact betweenthe fluids and the solid is of the form:

EW = σSLS∗ + σSVS

∗.

Since any modification of the liquid/solid contact area results in the oppositemodification of the vapor/solid contact area, we may write S

∗= S0 − S∗. The

wetting energy, is thus given by (up to a constant):

EW = (σSL − σSV)S∗ = −σLVβS∗

where β = − (σSL−σSV)σLV

is the relative adhesion coefficient.

(iii) The gravitational energy: The gravitational energy, or any other bodyforces, may be written as

EΓ =∫

Γρ dx

where ρ is the density of mass of the liquid, Γ is the potential energy and theintegral is evaluated over the region occupied by the liquid.

(iv) The volume constraint: Finally, one can take the volume constraint intoaccount via a Lagrange multiplier:

EV = σLVλV

where V is the volume occupied by the liquid.

The total energy associated to such a configuration is thus

ET = σ

[S − βS∗ +

1σ

∫Γρ dx + λV

](1)

with σ = σLV.

First variations: Euler-Lagrange equations. Equilibrium states corre-spond to local minimizers of the total energy (1). It is well known (see R. Finn[Fin86] for details) that area minimization leads to a mean curvature equation.

3

LiquidVapor!

Figure 2: Contact angle

More precisely, the first variation of the total energy shows that the mean cur-vature of the free surface H must satisfy

2H = λ +1σ

Γρ. (2)

Similarly, small perturbations near the contact line (the triple junction whereS, S∗ and S

∗meet) leads to:

cos γ = − (σSL − σSV)σLV

= β (3)

where γ denotes the angle with which the free surface S intersects the solidsurface (see Figure 2). This angle γ is called the contact angle. The contactangle condition (3) plays a central role in this review. In particular, we notethat for equilibrium to be reached, the relative adhesion coefficient must satisfy−1 ≤ β ≤ 1 at the contact line.

Equations (2) and (3) form what is sometime referred to as Young-Laplace’slaw.

2.2 Mathematical framework

The development of the theory of BV functions and sets of finite perimeter(or Caccioppoli sets) allowed important advances in the mathematical analysisof capillary phenomena. We present here some key results of this theory (astandard reference for BV theory is E. Giusti [Giu84]). Though the naturalsetting for our study is R3, the dimension plays no significant role, so we setour problem in Rn+1 for some n ≥ 2.

Let Ω be an open subset of Rn+1; BV (Ω) denotes the set of all functions inL1(Ω) with bounded variation:

BV (Ω) =

f ∈ L1 :∫

Ω

|Df | < +∞

where ∫Ω

|Df | = sup∫

Ω

f(x) div g(x)dx : g ∈ [C10 (Ω)]n+1, |g| ≤ 1

.

4

If E is a Borel set in Rn+1, we denote by ϕE its characteristic function. Theperimeter of E in Ω is then defined by

P (E,Ω) =∫

Ω

|DϕE |

which is thus finite as soon as ϕE ∈ BV (Ω). A Caccioppoli set is a Borel setE that has locally finite perimeter (i.e. P (E,B) < ∞ for every bounded opensubset B of Ω).

We recall the following relevant facts from the theory of BV functions (theproofs can be found in [Giu84]):

1. If E ∈ Rn+1 is smooth enough (∂E of class C2), then P (E,Ω) is equal tothe n-Hausdorff measure of ∂E. P (E,Ω) is thus a natural extension ofthe notion of area of surfaces for sets that are not smooth.

2. Sets of finite perimeter are defined only up to sets of measure 0. It ishenceforth usual to normalize E so that

0 < |E ∩B(x, ρ)| < |B(x, ρ)| for all x ∈ ∂E and all ρ > 0.

3. Bounded family of functions in BV (Ω) are precompact in L1(Ω).

4. If the boundary ∂Ω of Ω is locally Lipschitz, then each function f ∈ BV (Ω)has a trace f+ in L1(∂Ω). Moreover, if fn → f in L1(Ω) and

∫|Dfn| →∫

|Df |, then f+n → f+ in L1(∂Ω).

5. If ∂E ∩ Ω is the graph of a BV function, i.e. if A is a subset of Rn,u ∈ BV (A) and

E = (x, z) ∈ A× R ; 0 < z < u(x),

thenP (E,A× (0,∞)) =

∫A

√1 + |∇u|2 dx.

Note that for u ∈ BV (A), the integral in the right hand side can be definedby ∫

A

√1 + |∇u|2 dx

= sup

∫A

gn+1 + u

n∑i=1

Digidx : g ∈ [C10 (Ω)]n+1, |g| ≤ 1

and satisfies ∫A

|Du| ≤∫

A

√1 + |Du|2 dx ≤ |A|+

∫A

|Du|.

5

Sessile drop Capillary Tube

!

E

E

!

Figure 3: Sessile drop and capillary tube

In the sequel, we study in details two situations: The sessile drop and thecapillary tube (see Figure 3). With the notations above, the total energy for asessile drop occupying a set E ⊂ Ω and in contact with the solid surface ∂Ω,can be written as

ET = σ

[P (E,Ω)−

∫∂Ω

βϕEdσ(x) +1σ

∫Ω

ρΓϕE dx + λ

∫Ω

ϕE dx

].

When Ω = Σ×R+ with Σ open set of Rn, and if the free interface of the liquidis the graph of a function u(x) defined over Σ, as in the capillary tube case, thetotal energy can be written as

ET = σ

[∫Σ

√1 + |∇u|2 dx−

∫∂Σ

∫ u

0

β dz dσ(x)

+1σ

∫∂Σ

∫ u

0

ρΓ dz dx + λ

∫Σ

u dx

].

2.3 The capillary tube

Throughout this section, Σ is a fixed subset of Rn and we consider a liquidfilling a cylinder Ω = Σ× R+. Furthermore, we assume that β is constant andβ ∈ [−1, 1]. Finally, the gravitational potential is given by Γ = gz.

Under such assumptions, it was proved by M. Miranda [Mir64] that the freesurface projects simply on Ω. We can thus restrict ourself to sets whose freesurfaces are graphs, i.e. sets of the form:

E = (x, z) ∈ Σ× R+ ; 0 < z < u(x), u ∈ BV (Σ). (4)

The energy associated to a set E of the form (4) is then given by

J (u) =∫

Σ

√1 + |∇u|2 dx− β

∫∂Σ

u(x)dσ(x) +ρg

σ

∫Σ

u2

2dx + λ

∫Σ

u(x) dx

6

and the equilibrium shape of the free surface is determined by the followingminimization problem:

J (u) = minv∈BV (Σ)

J (v). (5)

In this framework, Young’s Laplace law readsdiv

(∇u√

1 + |∇u|2

)= λ + κu in Σ

∇u · ν√1 + |∇u|2

= β on ∂Σ

(6)

where κ = ρgσ is the capillary constant. In the sequel we denote by γ the real

number in [0, π] such thatcos γ = β.

Note that the Lagrange multiplier is uniquely determined by γ and the vol-ume constraint: Integrating (6) over Σ, we get

λ =1Σ

[∂Σ cos γ − κV ].

In particular, if κ = 0 (gravity free case), we have

λ =∂ΣΣ

cos γ (7)

which is independent of the volume constraint. When κ 6= 0 we can set v =u + λ/κ which solves (6) with λ = 0. The shape of the free surface is thus, ineither case, independent of the volume.

Surfaces of prescribed mean curvature were first studied by Laplace [Lap05].Further results have been obtained by S. Bernstein [Ber09], and the existenceof C2 surfaces with prescribed mean curvature has been studied in particular bySerrin [Ser70]. Numerous existence results for the variational problem (5) havebeen obtained with various requirements on Σ. One of the first results is that ofM. Emmer [Emm73], later followed by the work of C. Gerhardt [Ger74, Ger75,Ger76], R. Finn and C. Gerhardt [FG77], E. Giusti [Giu76, Giu78], M. Giaquinta[Gia74], P. Concus and R. Finn [CF74] and L.F. Tam [Tam86a, Tam86b]. Verygeneral results, based on the ideas of M. Miranda and E. Giusti can be foundin [Fin86].

We discuss below some those results in the particular case where κ = 0(gravity free). Results with positive gravity may be found in E. Giusti [Giu76]and R. Finn [Fin86] (and reference therein).

A necessary condition As pointed out by P. Concus and R. Finn [CF74] thecapillary tube problem may not have any solution. This is readily seen in (5),since when β ∈ (0, 1] (γ ∈ [0, π/2)) the functional J may be not be bounded

7

’2!

"

2l sin !

l"

Figure 4: Domain with corner of opening 2α

from below. Physically, this means that the liquid will rise indefinitely alongthe boundary of the tube and will never reach an equilibrium configuration.

A necessary condition for the existence of a solution in C2,α(Σ) ∩ C(Σ) canactually easily be found by integrating (6) over any subset Σ′ ⊂ Σ: We get

λΣ′ = ∂Σ ∩ Σ′ cos γ +∫

∂Σ′∩Σ

∇u · ν√1 + |∇u|2

dx

and so ∣∣∣λΣ′ − ∂Σ ∩ Σ′ cos γ∣∣∣ ≤ ∂Σ′ ∩ Σ

with strict inequality whenever Σ′ 6= Σ. Using (7), we deduce:∣∣∣∂ΣΣ

Σ′ − ∂Σ ∩ Σ′∣∣∣ cos γ < ∂Σ′ ∩ Σ for all Σ′ ⊂ Σ. (8)

This necessary condition is a geometric condition on Σ, depending on γ but noton u. One can better understand the meaning of (8) by considering a domainΣ containing a corner of opening angle 2α. Taking Σ′ as in Figure 4 and lettingl → 0, it is readily seen that condition (8) yields cos γ ≤ sinα or

γ + α ≥ π

2.

We deduce a first non-existence result, due to P. Concus and R. Finn:

Theorem 2.1 (P. Concus and R. Finn [CF74]) If Σ contains a corner withopening angle 2α such that α + γ < π/2, then (6) admits no solutions. Thiscondition is sharp (see Finn [Fin86] for details).

One could wonder if the lack of solutions to an apparently simple physical prob-lem should be taken as an indication that the mathematical model is flawed.However, it can be verified experimentally that when a liquid is contained be-tween two solid plate forming a corner, there is a critical opening angle forwhich the liquid will rise to infinity (or to the top of the container). This re-sult is thus in accord with experimental observations. Finally, note that thisphenomenon is not due to the singularity of the boundary of Σ, since one couldsmooth the corner in the previous computation and reach a similar conclusionwith ∂Σ ∈ C∞. This also implies a strong instability of capillary surfaces withrespect to boundary perturbations.

8

Existence result for γ ∈ (0, π/2). Condition (8) is thus necessary for theexistence of a solution. As it turns out, it is not far from being also a sufficientcondition, as stated in the following theorem, which follows from E. Giusti[Giu76]:

Theorem 2.2 Assume that γ > 0, Σ is a Lipschitz domain and that there existsε0 such that for all proper subset Σ′ ⊂ Σ we have:∣∣∣∂Σ

ΣΣ− ∂Σ ∩ Σ′

∣∣∣ cos γ < (1− ε0)∂Σ′ ∩ Σ. (9)

Then there exists a minimizer of J in BV (Σ). Furthermore u solves (6) andis bounded. Finally any other solution of (6) is equal to u up to an additiveconstant.

Condition (9) implies that for any v ∈ BV (Ω) we have the following inequality

cos γ

∫∂Σ

v dHn−1(x) ≤ (1− ε)∫

Σ

|Dv|+ C

∫Σ

|v| dx (10)

which is the key in proving that J is bounded below.In fact, E. Giusti [Giu76] proves that Theorem 2.2 is valid as soon as (8) and

(10) hold. Inequality (10) first appeared in M. Emmer [Emm73] who provedthat it holds with (1 − ε) =

√1 + L2 cos γ when ∂Σ is Lipshitz with Lispchitz

constant L. Note that when Σ has a corner with opening angle 2α the condition√1 + L2 cos γ < 1 is exactly equivalent to

α + γ < π/2.

We refer to R. Finn [Fin86] for a detailed discussion on necessary and sufficientconditions on the domain Σ for (10) to hold.

Finally, note that multiplying (6) by v ∈ BV (Σ) and integrating by parts,we can easily show that (10) with ε = 0 is a necessary condition for the existenceof a solution of (6). Whether it is a sufficient condition is still, to my knowledge,not completely settled.

The case γ = 0. When γ = 0, (8) becomes

λΣ′ < ∂Σ′ (11)

for all proper subset Σ′ of Σ, with equality if Σ′ = Σ. In fact this condition issufficient for the existence of u, as proved by E. Giusti [Giu78] (see also R. Finn[Fin74]):

Theorem 2.3 (E. Giusti [Giu78]) If (11) holds for all proper subset Σ′ of Σwith equality if Σ′ = Σ, then there exists a unique solution (up to a constant)of (6).

Furthermore, the behavior of u along ∂Σ can be characterized. It actuallydepends on the mean curvature Γ(x) of ∂Σ at a point x:

9

• If (n− 1)Γ < λ near x0 ∈ ∂Σ then u is bounded at x0.

• If (n− 1)Γ = λ in Γ0 ⊂ ∂Σ then limx→x0 u = +∞ for all x0 ∈ Γ0.

(Note E. Giusti [Giu78] shows that the conditions in Theorem 2.3 implies inparticular that (n− 1)Γ(x) ≤ λ for all x ∈ ∂Ω).

2.4 The sessile drop

This section is devoted to the study of sessile drops lying on a horizontal plane.We denote by Ω the upper half space:

Ω = Rn × (0,+∞),

and we denote by (x, z) an arbitrary point in Ω, with x ∈ Rn and z ∈ [0,+∞).We also denote by E (V ) the class of Caccioppoli sets in Ω with total volumeV > 0:

E (V ) =

E ⊂ Ω :∫

Ω

|DϕE | < +∞, |E| = V

,

where |E| =∫Ω

ϕE demotes the Lebesgue measure of E. Since Caccioppoli setshave a trace on ∂Ω = Rn × z = 0, we can define the following functional forevery E ∈ E (V ):

J (E) =∫ ∫

z>0

|DϕE | −∫

z=0

β(x)ϕE(x, 0)dx +∫

z>0

ρ Γ ϕE dx dz (12)

= P (E,Ω)−∫

E∩z=0β(x) dH n(x) +

∫z>0

ρ Γ ϕE dx dz.

In this framework, equilibrium liquid drops are solutions of the minimizationproblem:

J (E) = infF∈E (V )

J (F ) , E ∈ E (V ). (13)

We recall that the Euler-Lagrange equation associated to this minimizationproblem with volume constraint is given by the Young-Laplace law (2-3):

nH =Γρ

σ− λ (14)

cos γ = β, (15)

where H denotes the mean-curvature of the free surface ∂E and γ denotes theangle between the free surface of the drop ∂E and the horizontal plane z = 0along the contact line ∂(E ∩z = 0) (measured within the fluid, see Figure 5).

The shape of equilibrium drops strongly depends on the coefficient β. Whenβ ≤ −1, the cost of wetting the solid support is at least the same as that of thefree surface. Equilibrium drops will thus not touch the solid surface in more

10

!

E

E

!

"<0 ">0

Figure 5: Sessile drop and contact angle

than one point; The surface is said to be hydrophobic. As soon as β > −1,it is easy to see that absolute minimizers will have a non trivial contact areaE ∩ z = 0. The main feature of the mathematical analysis of equilibriumdrops in that case is the study of the contact line ∂(E ∩ z = 0) which is acodimension 2 free surface.

Note finally that if β > 1 then the drop will spread indefinitely and nominimizer can exist. We will thus always assume

|β| ≤ β0 < 1.

In that case, it is easy to check that

J (E) ≥ 1− β0

2

∫z>0

|DϕE |+1− β0

2

∫z=0

ϕEdx +∫

z>0

ρ Γ ϕE dx dz

for all E ∈ E (V ), so that J is bounded below in E (V ).

2.4.1 Existence of minimizers.

The lower semi-continuity of the functional (12) in the L1(Ω) topology can beeasily established (see [CM07b] for instance). Moreover, it is a classical resultthat bounded subsets of BV (Ω) are pre-compact in L1

loc(Ω). Those two factsgive the convergence of minimizing sequences to minimizers of J provided|Ω| < +∞ (The existence of minimizers for the drop problem in a boundeddomain was first proved in [MP75]). Since we are considering drops lying in theupper half space, we need to be a little bit careful and ensure that minimizingsequences do not shift to infinity in one direction.

When β = β0 is constant, the existence of a minimizer is proved by E. Gon-zalez [Gon76]. An important tool, in that case is Schwarz symmetrization: Forevery E ∈ E (V ), the set

Es = (x, z) ∈ Ω ; |x| < ρ(z), where ρ(z) =(

ω−1n

∫ϕE(x, z)dx

) 1n

(16)

11

is a Caccioppoli set with same volume as E and satisfying

J (Es) ≤ J (E)

with equality if and only if E already had axial symmetry. This implies that anyminimizer should have axial symmetry (see [Gon76] for details). If we assumefurthermore that Γ = 0 (gravity free), then it can actually be shown that theminimizers are spherical caps; that is the intersection of a ball Bρ0(0, z0) inRn+1 with the upper-half space Ω. We denote by

B+ρ0

(z0) = Bρ0(0, z0) ∩ z > 0

such a spherical cap.

When the relative adhesion coefficient β depends on x, Schwarz symmetriza-tion (16) may increase the wetting energy∫

β(x)ϕE(x, 0)dx

so minimizers do not have axial symmetry in general. It is still, however, possibleto obtain the existence of minimizers under relatively general assumptions on β(see [CF85]). We also refer to [CM07b] for a detailed study of minimizers of Jwhen β is periodic and Γ = 0 (we will get back to this assumption later).

2.4.2 Regularity of minimizers.

If E is a minimizer of J , it is natural to wonder what regularity we can expectfor the free surface ∂E ∩ z > 0. This is in fact a very classical result, whichis closely related to the theory of minimal surfaces:

Theorem 2.4 (U. Massari [Mas74], E. Gonzalez et al. [GMT83])

If n ≤ 6, then ∂E is analytic in Ω.

Note the restriction on the dimension n + 1 ≤ 7 which is the same as therestriction for the regularity of minimal surfaces.

The next question is the regularity of the contact line which is a codimen-sion 2 free surface in Rn+1. The formation of singularities at the contact lineis an important problem in many physical applications. From a mathematicalpoint of view, our main goal is to be able to give a rigorous meaning to thecontact angle condition (which requires the contact line to be at least C1).

2.4.3 Regularity of the contact line

The regularity of the contact line was addressed by L. Caffarelli and A. Friedmanin [CF85] when β(x) ∈ (0, 1) (i.e. γ ∈ (0, π/2)). In that case, it can be shown(using Steiner symmetrization) that minimizers are graphs, i.e. of the form:

E = (x, z) ; x ∈ Σ, 0 < z < u(x) (17)

12

with u continuous function (this fact follows from the analyticity of ∂E ∩ z >0).

Moreover, it can be shown that for sets of the form (17), we have:

J (E) = J(u) :=∫

Rn

√1 + |∇u|2 dx +

∫Rn

(1− β)χu>0 dx +ρg

σ

∫u2

2dx + λ

∫Ω

u dx

and u minimizes J among all BV functions. To sum up, we have:

Proposition 2.5 (Caffarrelli-Friedman [CF85]) Assume that β(x) ∈ (0, 1)for all x. Then the set E solution of (13) is the graph of a nonnegative contin-uous function u ∈ BV (Rn), solution of

J(u) = minv∈BV (Rn),v≥0

J(v).

Note that, formally at least, u is solution of the following free boundary problem:div

(∇u√

1 + |∇u|2

)= λ +

ρg

σu in u > 0

∇u√1 + |∇u|2

· ν = β on ∂u > 0.(18)

This free boundary problem belongs to the wide class of problems that areobtained as Euler-Lagrange equations for the minimization of functionals of theform ∫

u>0F (x, u,∇u) dx

with F (x, z, p) convex with respect to p. Those include the well known Bernoulliproblem corresponding to F (x, z, p) = |p|2 + β: ∆u = 0 in u > 0

|∇u|2 = β on ∂u > 0.

The systematic study of such problems was initiated by H. Alt and L. Caffarelliin [AC81] for the Bernoulli problem and generalized to more general functionalsby H. Alt, L. Caffarelli and A. Friedman in [ACF84]. The case of capillary drops(18) does not quite fit in the framework of [ACF84] because of the degeneracyof the mean-curvature operator (for large ∇u). It is however possible to adaptthe method, as shown by L. Caffarelli and A. Friedman in [CF85] (the key isthe derivation of a Lipschitz estimate for u, after what the theory of [ACF84]can be used).

The main results are the Lipschitz regularity of u (which is the optimalregularity in view of the free boundary condition) and non degeneracy at thefree boundary (i.e. linear growth away from the free boundary). This in turnimplies that the free boundary ∂u > 0 as finite Hausdorff measure.

13

Further regularity of the free boundary is harder to obtain and relies on localimprovement of regularity, a method which is reminiscent of De Giorgi’s proofof the regularity of minimal surfaces. The following can be shown ([AC81] and[ACF84]):

If β ∈ Cα then ∂redu > 0 ∈ C1,α

andH n−1(∂u > 0 \ ∂redu > 0) = 0

where ∂redu > 0 denotes the reduced free boundary (see [Giu84]).Additional regularity of β implies further regularity of ∂redu > 0:

if β ∈ Ck,α then ∂redu > 0 ∈ Ck+1,α

andif β is analytic then ∂redu > 0 is analytic.

Further characterization of the singular set ∂u > 0 \ ∂redu > 0 relies onblow-up arguments and the classification of global solutions. The result dependson the dimension, and it is known that ∂u > 0 = ∂redu > 0 in dimensionn = 2 (H. Alt and L. Caffarelli [AC81]) and n = 3 (L. Caffarelli, D. Jerison andC. Kenig [CJK04]). In the case of the drops (n = 2), we thus have:

Proposition 2.6 If n = 2, β is of class Cα and satisfies 0 < β(x) < 1, thenthe contact line is C1,α and the contact angle condition cos γ = β is satisfied inthe classical sense.

Finally, let us point out that when we only assume −1 < β(x) < 1 (insteadof β(x) ∈ (0, 1)), the previous analysis breaks down, since minimizers are nolonger graphs. However, it is likely that the same regularity result will hold inthat case. We proved in [CM07b] that the contact line ∂(E∩z = 0) has finiteHausdorff measure in that case:

Proposition 2.7 (L. Caffarelli, A. Mellet [CM07b])

Assume |β(x)| ≤ β0 < 1 and let E be the minimizer of J in Ω = Rn+1+ . Assume

furthermore that E lies in a ball of radius M |E|1

n+1 for some large M . Thenthere exists C depending only on M and the dimension such that

H n−1(Br ∩ ∂(E ∩ z = 0) ≤ Crn−1

for any ball Br ⊂ Rn.

2.5 Contact angle hysteresis

The coefficient β is determined experimentally, and depends on the properties ofthe materials (solid and liquid). It is often assumed to be constant, but it is verysensitive to small perturbations in the properties of the solid plane (chemical

14

contamination or roughness of the surface). A real solid surface is extremelyhard to clean and is never ideal; it always has a small roughness or small spatialinhomogeneities.

These inhomogeneities are responsible for many interesting phenomena, themost spectacular being contact angle hysteresis (see [JdG84], [LJ92]): In ex-periments, one almost never measures the equilibrium contact angle given byYoung-Laplace’s law. Instead, the measured static angle depends on the wayin which the drop was formed on the solid. If the equilibrium was reached byadvancing the liquid (for example by spreading or condensation), the contactangle has value γa larger than the equilibrium value. If on the contrary, theliquid interface was obtained by receding the liquid (evaporation or aspirationof a drop for example), then the contact angle has value γr, smaller than theequilibrium value (see L. Hocking [Hoc95], [Hoc81]). In extreme situations (typ-ically when the liquid is not a simple liquid, but a solution), differences of theorder of 100 degrees between γr and γa have been observed (see [LJ92]). In[HM77], C. Huh and S. G. Mason solve the Young-Laplace equation for someparticular type of periodic roughness and explicitly compute the contact anglehysteresis in that case.

Contact angle hysteresis also explains some simple phenomena observed ineveryday life, such as the sticking drop on an inclined plane:

If the support plane Π is inclined at angle θ with the horizontal in the y-direction, the potential Γ can be written as:

Γ = g(z cos θ + y sin θ). (19)

When β is constant and g, θ > 0, no minimizer to (13) can exist since anytranslation in the y < 0 direction will strictly decrease the total energy. It canalso be shown (see R. Finn and M. Shinbrot [FS88]) that (14-15) has no solutionswhen β is constant and κ, θ 6= 0. This means that on a perfect surface, the dropshould always slide down, no matter how small the inclination. However, awater drop resting on a plane that we slowly inclined will first change shapewithout sliding, and will only start sliding when the inclination angle reachesa critical value: in the lower parts of the drop, the liquid has a tendency toadvance and the contact angle increases until it reaches the advancing contactangle, in the upper parts of the drop, the liquid has a tendency to recede and thecontact angle decreases until it reaches the receding contact angle (see [DC83]).

In [CM07b] and [CM07a], we rigorously investigate the approach suggestedby C. Huh and S. G. Mason [HM77]: Roughness of the surface is taken intoaccount via small scale oscillations of the coefficient β.

In order to understand the effects of those microscopic oscillations on thelarge (or macroscopic) scale shape of the drop, we use a classical mathematicalartifact: Denoting by ε 1 the scale of those oscillations, we set β = β(x/ε)and we consider periodic inhomogeneities. We thus assume

β = β(x/ε), with y 7→ β(y) Zn-periodic.

15

The investigation of the behavior of the minimizers of

Jε(E) =∫ ∫

z>0

|DϕE | −∫

z=0

β(x/ε)ϕE(x, 0)dx +∫

z>0

ρ Γ ϕE dx dz (20)

as ε → 0 will describe the effects that the small scale inhomogeneities haveon the overall shape of the drops (which is the main goal of the theory ofhomogenization).

For the sake of simplicity, we assume Γ = 0 throughout the rest of thissection.

2.5.1 Global minimizers

We denote by 〈β〉 the average of β:

〈β〉 =∫

[0,1]nβ(x) dx,

and by B+ρ0

(z0) = Bρ0(0, z0) ∩ z > 0 the minimizer of the functional withconstant adhesion coefficient:

J0(E) =∫ ∫

z>0

|DϕE | −∫

z=0

〈β〉ϕE(x, 0)dx.

We recall that the axial symmetry of minimizers when β is constant follows fromSchwarz symmetrization and the isoperimetric inequality. When β depends on(x, y) the method fails, since the rearrangement (16) could increase the wettingenergy ∫

β(x/ε, y/ε)ϕE(x, y, 0) dx dy. (21)

We remark however, that when β is periodic, the contribution to the wettingenergy of any cell of size ε which is included in E ∩ z = 0 is equal to 〈β〉εn.Thus, the difference in the wetting energy (21) of a set E and that of its Schwarzsymmetrization Es is at most 〈β〉εn times the number of cells that intersect thecontact line ∂(E∩z = 0) (see Figure 6). That number can readily be estimatedif we control the (n − 1)-Hausorff measure of the contact line. Proposition 2.7thus yields:

Jε(Es) ≤ Jε(E) + Cε

In other words, we do not have equality in the isoperimetric inequality butalmost equality. With the help of Bonnesen type inequalities (quantitativeisoperimetric inequalities) we deduce that the minimizer E must have almostaxial symmetry. One can then show that E almost coincides with a sphericalcap. More precisely, we proved the following result in [CM07b]:

Theorem 2.8

16

!

Contact line

Figure 6: Wetting surface of E and its Schwarz symmetrization

(i) For all V and ε > 0, there exists E minimizer of Jε in E (V ). Moreover,up to a translation, we can always assume that

E ⊂ |x| ≤ R0V1

n+1 , z ∈ [0, T0V1

n+1 ]

with R0 and T0 universal constants.

(ii) The contact line ∂(E∩z = 0) has finite (n−1) Hausdorff measure in Rn.

(iii) There exists a constant C(V ) such that if ε ≤ Cη(n+1)/α, then

B+(1−η)ρ0

⊂ E ⊂ B+(1+η)ρ0

.

In other words, the free surface ∂E ∩ z > 0 lies between ∂B+(1+η)ρ0

and∂B+

(1−η)ρ0for ε small enough.

This result gives the existence of “sphere-like” minimizers of Jε, which,when β ∈ (0, 1), correspond to “sphere-like” viscosity solutions of the freeboundary problem:

div

(Du√

1 + |Du|2

)= κ in u > 0

Du√1 + |Du|2

· ν = β(x/ε) on ∂u > 0.(22)

In particular, Theorem 2.8 implies that the global minimizers of Jε convergeuniformly to minimizers of J0 (which are all spherical caps with contact anglecondition cos γ = 〈β〉). We also deduce that there exists a solution of (22) whichconverges, as ε goes to zero, to a solution of

div

(Du√

1 + |Du|2

)= κ in u > 0

Du√1 + |Du|2

· ν = 〈β〉 on ∂u > 0.

17

x

!

1

"min

"max

x

f(x/ )

Figure 7: Solutions of f (x/ε) = 1/(2x2) and the corresponding solutions of (23)

This behavior was expected from solutions obtained as global minimizers of theenergy, and it seems to exclude contact angle hysteresis phenomena. However,when a drop is formed, its shape does not necessarily achieve an absolute min-imum of energy. In fact, contact angle hysteresis phenomenon may only becaptured by looking for local minimizers of the energy functional.

2.5.2 Local minimizers

A simple (but illustrative) example. We first consider a simpler problem:uxx = 0 in u > 0 ∩ (0,∞),

|ux|2 = 2f(x/ε) on ∂u > 0,

u(0) = 1.

(23)

which is the Euler-Lagrange equation for the minimization of

Jε(u) =∫ ∞

0

12|ux|2 + f(x/ε)χu>0 dx. (24)

Classical solutions of (23) are straight lines with slope γ intersecting thex-axis at a point x0 such that

γ2 = 2f(x0/ε).

We thus have one solution for every x0 such that (see Figure 7)

f(x

ε

)=

12x2

When ε → 0 we obtain a family of lines with slope γ ∈ [γmin, γmax] where

γmin =√

2 inf f and γmax =√

2 sup f.

The homogenization of the free boundary condition |ux|2 = 2f(x/ε) thusgives

|ux| ∈ [γmin, γmax] on ∂u > 0

18

which is exactly the type of condition that we expect for hysteresis phenomena.This result shows that the homogenization of free boundary problems is far fromsimple in general. The main thing is that the free boundary is free to avoid thebad area of the medium and will only “see” certain values of the coefficients.This type of behavior is, for instance, well known for geodesics associated to aperiodic metric (see [Mor24]), and for minimal surfaces (see L. Caffarelli, R. dela Llave [CdlL01])

In [CLM06], we generalized the result above to higher dimension, proving, inparticular that there exists some solutions of the elliptic free boundary problem

∆u = 0 in u > 0|∇u|2 = f(x/ε, y/ε) on ∂u > 0,

(25)

which converge to functions of the form u(x, y) = γ max(xe1 + ye2, 0) with

γ ∈ [γmin, γmax].

We refer to [CLM06] for some applications of this result to fronts propagationin periodic media.

Finally, note that if we look at the global minimizer of (24) (which is asolution of (23)), it is not very difficult to show that it converges, as ε goes tozero, to the unique minimizer of

Jε(u) =∫ ∞

0

12|ux|2 + 〈f〉χu>0 dx,

which satisfies |∇u|2 = 2〈f〉 along its free boundary. This is similar to what weobserved with the global minimizer of the sessile drop problem.

Back to the drop problem. We now consider the 3-dimensional problemand denote by (x, y) ∈ R2 the horizontal components. From the discussionabove, it is clear that in order to get some interesting phenomena, we need tomake sure that β is non constant. One possible assumption is the following:

miny

maxx

β(x, y) < 〈β〉. (26)

To construct local minimizers we recall that the wetting surface correspond-ing to the asymptotic minimizer B+

ρois a disk of radius

Ro = ρo

√1− 〈β〉2.

So we introduceΣt = (x, y) ∈ R2 ; 0 ≤ y ≤ 2Ro − t,

and look for minimizers of Jε whose wetting area stays within the region Σt.Clearly, for t > 0, Eε (which is arbitrarily close to B+

ρofor small ε) is not a

candidate anymore. In fact, we prove the following:

19

Theorem 2.9 ([CM07a]) Assume that the relative adhesion coefficient β(x, y)satisfies (26). Then, for any volume V , there exists εo such that if ε < εo, thenJε has a local minimizer Eε of volume V not equal to the global minimizer Eε.

Furthermore, when ε goes to zero, Eε converges to some set E ∈ E (V )satisfying the following equation:

cos γ ≤ 〈β〉, along the contact line,

with a strict inequality on parts of the contact line.

The last part of Theorem 2.9 says that the apparent contact angle (or ho-mogenized contact angle) is indeed larger than cos−1〈β〉 in some directions.This result is purely qualitative: The proof does not give any estimate on thesize of the interval of admissible contact angles. It is fairly easy to constructbarriers that yield hysteresis phenomena in any finite number of directions (forappropriate β). To get infinitely many directions, however, one would probablyhave to consider random inhomogeneities.

The solution given by Theorem 2.9 plays the role of barrier during the for-mation of a liquid drop by slow spreading or condensation. This explains theso-called stick-jump phenomenon: As the volume of the drop increases, the con-tact line remains unchanged at first, while the contact angle increases. Onlywhen the contact angle reaches a critical value does the contact line jump tothe next equilibrium position (see [HM77]).

When the surface of the drop is a graph z = u(x, y), Theorem 2.9 givesthe existence of a solution of (22) which converges as ε → 0, to a function usatisfying

Du√1 + |Du|2

· ν ≤ 〈β〉 on ∂u > 0

with strict inequality on part of the free boundary, which is reminiscent of theresults obtained for (25) in [CLM06]. However solutions of (25) were constructedby studying viscosity solution of a singular nonlinear equation rather than byminimizing a functional.

Remark 2.10 Throughout this section, we have considered drops with completecontact with the underlying surface (known as Wenzel drops in the literature).However, another effect of roughness is to allow for small pockets of vapor toform underneath the drop (Cassie-Baxter drops) thus affecting the wetting en-ergy. The effect of partial wetting on contact hysteresis has been investigatedin particular, by G. Alberti and A. DeSimone [AD05] and A. DeSimone, N.Grunewald and F. Otto [DGO07].

20

3 The Motion of the contact line

In the second part of this review, we discuss some models describing the motionof the contact line when a droplet is spreading on a flat surface or sliding downan inclined plane. This problem is considerably more difficult to model than thestatic case: On top of the capillary effects at the drop surface and the wettingdynamic at the contact line, one must also account for the motion of the liquidinside the drop. We discuss two models: The quasi-static approximation, whichessentially allows us to neglect the dynamic of the fluid inside the drop, andthe lubrication approximation which is valid for small, very viscous drops. Theeffects of gravity are neglected throughout this section.

3.1 Quasi-static approximation

Maybe the simplest model for contact line dynamics is the quasi-static modelwhich assumes the slowness of the contact line motion in comparison to the timefor capillary relaxation (see K. Glasner [Gla05] for a detailed discussion).

We can then consider that the fluid pressure inside the drop is constant at alltime and that the free surface has constant mean-curvature. The inbalance ofsurface forces along the contact line is responsible for the motion of the contactline. This is usually described by a constitutive law linking the velocity of thecontact line and the contact angle.

If we denote by h(x, t) the height of the drop at a point x and time t, we areled to the following system of equations:

div

(Dh√

1 + |Dh|2

)= −λ(t, h) in h > 0,

∫h(x, t) dx = V for all t,

v = F (|∇h|, x) on ∂h > 0

(27)

where v denotes the velocity of the contact line ∂h > 0 in the direction normalto ∂h > 0. Note that we have v = ht

|∇h| , so the free boundary condition alsoreads

ht = |∇h|F (|∇h|, x) on ∂h > 0.

Alternatively, one can approximate the mean curvature by the laplacian ofh and thus replace the first equation by

∆h = −λ(t, h).

The main difficulty lies in the fact that the coefficient λ(t, h) is determinedat each time by the volume constraint. In particular we cannot expect thecomparison principle to hold.

The determination of the constitutive law (the function F ) is a major dif-ficulty of the study of contact line motion. Empirical laws give v = θ3 − θ3

e or

21

v = κ(θ)(θ−θe) where θ denotes the measured contact angle and θe the equilib-rium contact angle predicted by Young-Laplace’s law. Such laws are valid when|θ − θe| θe. One understanding of such relations is that Young-Laplace’s lawis always satisfied at the molecular level, but that there is a difference betweenthe microscopic contact angle θe and the apparent contact angle θ which isresponsible for the contact line motion.

We can thus for example take:

F (|∇h|) = |∇h|3 − 1

(note that |∇h| = sin(θ)).Such models are widely used in numerical experiments, but are very diffi-

cult to study mathematically (due to the lack of comparison principle, as notedabove). Classical solutions do not exist globally in time: Numerical computa-tions point out that singularity may form along the contact line. Even initiallysmooth and convex drops may develop corners. Also topological singularitiesmay arise when drops split or reconnect.

Recently, a viscosity solution theory was developed by K. Glasner and I. Kim[GK07] that can handle the formation of corner type singularities. However,topological singularities cannot be taken into account as the model breaks downin such events (each connex component must be treated separately with a dif-ferent coefficient λ).

This model is rather nice, but experiments suggest that F should also dependon the droplet size and external flow field. A global theory is therefore neededthat takes into account the motion of the fluid inside the drop. This is theobject of the next section.

3.2 Lubrication approximation and the thin film equation

The lubrication approximation consists of a depth-averaged equation of massconservation and a simplified form of the Navier-Stokes equations that is appro-priate for very thin drops (vertical length scale much smaller than the horizontallength scale) of very viscous liquid. In this framework, the evolution of capil-lary surfaces can be described by a fourth order degenerate diffusion equation,known as the thin film equation. We refer to H. Greenspan [Gre78] for detailsabout the validity of the lubrication approximation, and we point out the re-view of A. Bertozzi [Ber98] in which one can find a complete introduction to themathematical theory of thin films which we attempt to briefly describe below.

We consider a thin drop of liquid on a solid surface (R2). We denote byh(x, t) the height of the drop (for x ∈ R2, t ∈ (0,∞)) and by v(x, t) the verticalaverage of the horizontal component of the velocity field of the fluid (uH(x, z, t),where z is the vertical coordinate):

v(x, t) =1h

∫ h

0

uH(x, z, t) dz.

22

The depth-averaged conservation of mass yields

∂h

∂t+ divx(hv) = 0,

while, under the lubrication approximation, the momentum equation reduces to

µ∂2uH

∂z2= ∇xp(x, t).

Along the free surface z = h, we assume that the horizontal shear stress vanishes,while we take no-slip condition at z = 0:

∂uH

∂z|z=h = 0, u|z=0 = 0.

Integrating the momentum equation then gives

uH =1µ∇xp

(12z2 − hz

)and so

v = −h2

3µ∇xp.

Finally, we approximate the capillary pressure by p|z=h = −σ∆h (replacing themean-curvature operator by the Laplacian). We obtain the so-called thin filmequation

∂h

∂t+

σ

3µdiv(h3∇∆h) = 0. (28)

Equation (28) is satisfied by h for x ∈ h > 0. Along the free boundary∂h > 0, it is natural to assume, besides the natural condition h = 0, a nullflux condition hv = 0:

h3∇∆h = 0

which guarantees conservation of mass. With only two conditions at the freeboundary, the fourth order equation (28) is not well posed, which suggests thatwe impose a contact angle condition |∇h| = θ0 with θ0 = 0 (complete wetting,appropriate for wetting on a moist surface) or θ0 > 0 (partial wetting). Anotherpossibility, suggested for instance by Greenspan [Gre78] and Hocking [Hoc81], isto assume that the velocity of the contact line and the contact angle are linkedby a constitutive law:

ht = |∇h|F (∇h).

There is still some debates about the appropriate boundary condition (froma physical point of view). Mathematically, it is not known which conditionwould make the problem well posed. As we will see in the sequel, fairly gen-eral existence results have been obtained with the zero contact angle condition(|∇h| = 0), and some results are available with non-zero contact angle condition(|∇h| = 1). But to our knowledge, no uniqueness result is known in either cases.

23

No-slip paradox. Equation (28) actually fails to describe the motion of con-tact lines. One way to see this, is to notice that the motion of the contact linewould lead to a multivalued velocity field at the contact line. More interestingly,it was observed by Huh and Scriven [HS71] that, independently of the contactangle condition, the dissipation energy has a logarithmic singularity at the con-tact line. The motion of the contact line would thus require an infinite energy,leading to the conclusion that ”... even Herackles could not sink a solid.” (see[HS71]). Finally, we will see later that no advancing travelling waves solutionsand source type solutions exist for (28).

However, liquid do spread, so some of the assumptions leading to (28) mustbe inappropriate, and more physics needs to be taken into account in the modelnear the contact line. Several ways have been suggested to remove this sin-gularity. One possibility is to include microscopic scale forces, in the form oflong-range forces of Van Der Waals interactions between the solid and the liquid(leading to a hyperdiffusive term, see A. Bertozzi [Ber98]). Another approachis to assume the presence of a precursor film of very thin height (b 1), whichamounts to look for positive solutions satisfying h = b > 0 at the boundary ofthe domain. A third possibility is to relax the no-slip boundary condition atz = 0 and use the Navier slip condition instead. This condition reads

u = Γ(h)∂u

∂zat z = 0.

The slip coefficient is given by Γ(h) = Λ3h2−s with typically s = 1 (singular slip)

or s = 2 (constant slip). Equation (28) then becomes:

∂h

∂t+

σ

3µdiv((h3 + Λhs)∇∆h) = 0. (29)

Note that we have not changed the degenerate character of the equation, butwe have removed the dissipation energy singularity at the contact line.

Lubrication approximation in a Hele-Shaw cell. Interestingly an equa-tion similar to (28) arises when studying thin film in Hele-Shaw cells under thelubrication approximation. A Hele-Shaw cell consists of two parallel plates sep-arated by a very thin gap (of size b 1). If we assume that the lubricationapproximation can be used (i.e. if we are considering the very slow flow of aviscous fluid), the vertically-averaged velocity field is given by Dracy’s Law:

v(x, t) = − b2

12µ∇xp(x, t)

(obtained by integration of Stokes law µ∂2uH∂z2 = ∇p together with no-slip condi-

tions at both plates z = 0 and z = b). The incompressibility condition div u = 0thus yields

∆p = 0 in Ωt (30)

24

h

h

Thin neck of fluid Thin film of fluid at the edge

Figure 8: Lubrication approximation in Hele-Shaw cell

where Ωt denotes the domain occupied by the liquid at time t. Along theboundary of Ωt, we have a capillary pressure condition

p = σH (31)

(where H denotes the mean curvature of ∂Ωt), together with the kinematiccondition ∂tp + v · ∇p = 0 which, using Darcy’s law, leads to

∂tp =b2

12µ|∇p|2 (32)

(this last condition also says that the contact line ∂p > 0 is moving with speedb2

12µ |∇p| in the direction normal to ∂p > 0). Equations (30-32) form a freeboundary problem which has been intensively studied, often under the conditionthat σ = 0 (neglecting the capillary effects). In that case, and assuming thatthe pressure is constant and equal to zero outside the region occupied by theliquid, we obtain the classical Hele-Shaw free boundary problem:

∆p = 0 in p > 0

∂tp =b2

12µ|∇p|2 on ∂p > 0.

We refer to [EJ81] and [Gus85] and the references therein for further discus-sions about this problem, which is beyond the scope of this review since we arefocusing on the effects of capillary forces.

Equations (30-32) constitute a rather difficult free boundary problem. Itturns out however that two interesting situations lead to an equation similar tothe thin film equation: The thin neck of liquid and the thin layer at the edge ofthe Hele-Shaw cell (see Figure 8).

In either case, if we denote by h(x, t) the height of liquid (in the directionparallel to the plates), using Darcy’s law and approximating the mean curvatureof ∂Ωt by hxx, we obtain the following fourth order degenerate equation:

∂th +b2

12µ∂x(h ∂xxxh) = 0 (33)

25

together with a contact angle condition at h = 0 (assuming no-slip boundarycondition at the edge).

Note that the rigorous derivation of (33) is performed by L. Giacomelli andF. Otto [GO03] with zero contact angle condition (complete wetting).

Self similar solutions and Tanner’s Law. We now consider the generalequation

∂h

∂t+ div(hn∇∆h) = 0, (34)

which include the ill-posed thin film equation (n = 3) and the Hele-Shaw lu-brication approximation (n = 1). A particular role is played by compactlysupported self similar solutions of (34). Such solutions are of the form

h(x, t) = tdδH(|x| tδ), δ = − 14 + dn

with H defined on (0, 1) satisfying

HnH ′′′ = αxH

andH ′(0) = 0, H(1) = 0, H ′(1) = 0

(the last condition is the zero contact angle condition) and where d = 1 or 2 forplanar or radial symmetric film.

The existence of self similar solutions for n < 3 was proved by F. Bernis, L.Peletier and S. Williams [BPW92]. These solutions are source type solutionsas they satisfy h(x, 0) = c δ(x). It is also proved in [BPW92] that no suchsolution exists for n = 3. Finally, we note that for n < 3, there exist other selfsimilar solutions with compact support and satisfying a non zero contact anglecondition.

For n = 3 (thin film equation), the scaling of the self similar solutionssuggests that the support of spherical drops expand like t1/10 (despite the factthat such solutions do not exist). This is known as Tanner’s law and it hasbeen verified experimentally with remarkable precision. It was suggested by P.-G. DeGennes that the modification of the model near the contact line (via theNavier slip condition or Van der Waals forces) should only have a weak effecton the macroscopic behavior of the drops, and in particular on Tanner’s law.

This fact was proved by L. Giacomelli and F. Otto in [GO02] for a planarfilm: While the scaling of self-similar solutions predicts that the support willspread like t

17 , L. Giacomelli and F. Otto show that solutions of (29) with s = 2

(constant slip) satisfy

meas(h > Λ) ∼(

t

log(1/t)

) 17

.

26

Properties of the thin film equation. Equation (34) is reminiscent of theporous media equation

∂tu− div (uγ∇u) = 0,

for which existence and uniqueness of weak solutions is well known (see forinstance A. Friedman [Fri88] and references therein). Other properties of theporous media equation include the existence of self similar source type solutionsand traveling wave solutions, the finite speed expansion of the support (if theinitial data has compact support, then the solution has compact support forall time) and the fact that the support is always expanding (in fact strictlyexpanding, except maybe for a waiting initial time).

The main difference between (34) and the porous media equation, however,is the lack of maximum principle for the former one. In fact, it is well knownthat non-negative initial data may generate changing sign solutions of the fourthorder equation ∂th + ∂xxxxh = 0. Such solutions would not make sense in theframework of thin film. It is however a remarkable feature of (34) that thedegeneracy of the diffusion coefficient permits the existence of non-negativesolutions.

Formally at least, solutions of (34) satisfy the conservation of mass:

d

dt

∫h(x, t) dx = 0

and the dissipation of surface tension energy:

d

dt

∫12|∇xh|2 dx = −

∫hn(∇∆h)2 dx ≤ 0.

Furthermore, they satisfy an entropy-like inequality:

d

dt

∫G(h) dx = −

∫|∆h|2 dx ≤ 0

where G′′(s) = s−n. This last inequality proves particularly useful to show theexistence of non negative solutions. As a matter of fact, we notice that forn = 2, we have G(h) = − log h so the entropy can be used to keep the solutionaway from zero.

Particular solutions. Besides the self similar source type solutions alreadymentioned, the thin film equation also has traveling-wave solutions of the formu(x, t) = H(x − ct) (see [BKO93]). Note however that (34) has no advancingfronts solution for n ≥ 3. Finally, there are exact steady solutions with compactsupport for all n:

h(x, t) =

A−Bx2, |x| < A/√

B0, otherwise.

27

Some existence results and properties of the solutions. In this lastsection, we give an overview of the existence and regularity theory for the thinfilm equation. We restrict ourself to the 1-dimensional case (i.e. we considerplanar film). We set Q = (−a, a) × (0,+∞) and consider an initial data h0(x)such that

h0 ∈ H1(−a, a), h0 ≥ 0.

We are thus looking for h(x, t) solution of the following fourth order degenerateparabolic equation:

∂th + ∂x(hn∂xxxh) = 0 t > 0 , x ∈ (−a, a) (35)

and satisfying the initial and boundary conditions

h(x, 0) = h0(x) for x ∈ (−a, a), (36)

hx = |h|nhxxx = 0 for x = ±a. (37)

Existence of solutions: The existence of non-negative weak solutions of (35-37)was first addressed by F. Bernis and A. Friedman [BF90] for n > 1. Furtherresults were later obtained, by similar technics, by E. Beretta, M. Bertsch andR. Dal Passo [BBDP95] and A. Bertozzi and M. Pugh [BP96]. The particularcase n = 1 (Hele-Shaw flow) has also been studied by F. Otto et al. [Ott98,GO01, GO03] via a completely different approach (gradient flows).

The method introduced in [BF90] relies on the regularization of the diffusioncoefficient and the initial data. For instance, we can define hε to be the classicalsolution of

∂thε + ∂x((|hε|n + ε)∂xxxhε) = 0 t > 0 , x ∈ (−a, a)

with initial conditionhε(x, 0) = h0(x) + ε.

Then the functionh(x, t) = lim

ε→0hε(x, t) (38)

is a weak solution of (35). More precisely:

Theorem 3.1 For n > 0, the function h(x, t) given by (38) satisfies (36)-(37)and is such that

h(x, t) ≥ 0, h ∈ C 12 , 1

8 (Q)

and ∫∫Q

h φt dx dt +∫∫

P

hnhxxxφx dx dt = 0

for all φ ∈ Lip(Q) with compact support in [−a, a] × (0,∞). Here, P denotesthe positivity set of h:

P := (x, t) ∈ Q ; h(x, t) > 0.

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Furthermore, h satisfies∫ a

−a

h(x, t) dx =∫ a

−a

h0(x) dx ∀t > 0.

This result was proved in [BF90] for n > 1 and extended, with some minormodifications, to n > 0 in [BBDP95].

Note that these solutions satisfy (35) is a very weak sense, since the fluxis integrated only over the positivity set of h. In [BP96], it is proven that for38 < n < 3, h(x, t) is also solution of (35) in the usual sense of distribution (i.e.with the flux term being integrated over all Q rather than just P ).

There is no uniqueness of the weak solutions of (35-37) in general (see be-low), so the solutions given by Theorem 3.1 may strongly depend on the chosenregularization method. Note however that when n ≥ 4 and h0 > 0 in [−a, a],it is proven in [BF90] that the solution given by Theorem 3.1 is positive for alltime and is unique.

Expansion of the support: Finite speed propagation of the support (as for theporous media equation) was proved by F. Bernis [Ber96a, Ber96b] for 0 < n < 3.Thus, a given compactly supported drop will stay compactly supported (until itreaches the boundary of the domain). Further characterization of the behaviorof the support of h is also relevant in applications: Strictly expanding supportmeans moving contact line. In particular, we recall that Huh and Scriven [HS71]predicted that the contact line would not move for n ≥ 3. Monotonicity in theevolution of the support also implies that drops will never break up into smallerdroplets.

The following is known (see [BBDP95]):

• If n ≥ 3/2, then the support of the solution given by Theorem 3.1 isincreasing in time, while if n ≥ 4, the support remains constant.

It is still an open problem to prove that this last fact holds also for n ≥ 3.

• If n ≥ 7/2 no break up can occur:

If h(x0, t0) > 0 then h(x0, t) > 0 for all t > t0.

On the other hand, break up can occur for n ∈ (0, 1/2) and there isnumerical evidence (R. Almgren, A. Bertozzi and M. Brenner [ABB96])that break up should occur for n = 1 (Hele-Shaw cell). The critical valuefor which break up may happen is not known.

Regularity and long time behavior: For 0 < n < 3, the solution given by The-orem 3.1 is C1([−a, a]) for almost all t > 0 (see [BBDP95]). In particular, itsatisfies a zero contact angle condition along its free boundary. Sharper regu-larity results are also derived in [BP96].

Furthermore, h becomes strictly positive after finite time and converges to itsmean value as t →∞ (and the convergence is exponential in time, see [BP96]).

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Non uniqueness results [BBDP95]: As noted earlier, no uniqueness results havebeen proved. In fact, if 0 < n < 3, it is known that there exists another solutionto (35-37) with non-expanding spatial support (and thus different from the onegiven by Theorem 3.1). This solution is also approximated by classical positivesolutions of non-degenerate problems.

It is not known what additional condition (contact angle condition?) wouldguarantee uniqueness.

Non-zero contact angle solutions: The solution given by Theorem 2.8 satisfies azero-contact angle condition. This fact follows from the approximation methodrather than from a conscious choice of a free boundary condition in the problem.Except for particular initial data, there is very few results addressing the caseof non zero contact angle. F. Otto, in [Ott98], proved the existence of suchsolutions for the Hele-Shaw flow free boundary problem:

∂th + ∂x(h ∂xxxh) = 0 in h > 0,

|∂xh| = 1 on ∂h > 0.

Existence of such solutions for 0 < n < 3 is still open.

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