Some Mathematical Aspects of Capillary Surfaces

Some mathematical aspects of capillary surfaces

A. Mellet∗ April 9, 2008

Abstract This is a review of various mathematical aspects of the study of capillary surfaces. A special emphasis is put on the behavior of the contact line, 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 capillary surface. More generally, we call capillary surface the interface between two fluids that are in contact with each other without mixing. The shape of such surfaces is mainly determined by the phenomenon of surface tension, which results from the action of cohesive forces between liquid molecules (”molecules of liquid are happier when surrounded by other molecules of liquid”). The first attempts of mathematical analysis of those phenomena go back to 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 by surface tension (sometime referred to as capillary force) tends to minimize the area of the free surface and is responsible for a pressure difference across the interface (capillary pressure) which is proportional to the mean-curvature of the free surface. Equilibrium capillary surfaces thus satisfy a mean-curvature equation, known as Young-Laplace’s equation. Interesting behaviors occur when the capillary surface is in contact with a solid support, for example when a liquid drop is resting on a solid surface. Along the triple junction, where liquid, gas and solid meet, a phenomenon similar to surface tension takes place, with three phases instead of two. The balance of forces along this contact line gives rise to a contact angle condition which plays a central role in this review.

This note contains two parts devoted respectively to statics and dynamics models. In the ﬁrst part, we present two aspects of the theory of equilibrium surfaces: The capillary tube and the sessile drop. In both case, the mathematical 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

Vapor Liquid S* S*

Figure 1: Three phases conﬁguration and Caccioppoli sets. Our presentation of the capillary tube problem is mainly based 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 presentation). The sessile drop problem gives rise to a free boundary problems which has been extensively studied in particular by L. Caﬀarelli and A. Friedman. We also 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 considerably more difficult than in the static case, because the motion of the liquid needs to be taken into account, typically through Navier-Stokes equations. We will focus on two models that have received the attention of the mathematics community: the quasi-static model and the lubrication approximation for thin droplets (the so-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 particular types 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, let us derive the main equations satisfied by capillary surfaces: Young-Laplace’s equations.

The energy balance Following R. Finn [Fin86], we consider a three phases conﬁguration 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 energies come 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 is thus of the form:

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

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

∗ ∗ EW = σSLS + σSVS .

Since any modiﬁcation of the liquid/solid contact area results in the opposite ∗ ∗ modiﬁcation 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) is the relative adhesion coeﬃcient. σLV (iii) The gravitational energy: The gravitational energy, or any other body forces, may be written as Z EΓ = Γρ dx where ρ is the density of mass of the liquid, Γ is the potential energy and the integral is evaluated over the region occupied by the liquid.

(iv) The volume constraint: Finally, one can take the volume constraint into account 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

1 Z E = σ S − βS∗ + Γρ dx + λV (1) T σ

with σ = σLV.

First variations: Euler-Lagrange equations. Equilibrium states correspond 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 Vapor Liquid !

Figure 2: Contact angle

More precisely, the ﬁrst variation of the total energy shows that the mean curvature of the free surface H must satisfy 1 2H = λ + Γρ. (2) σ Similarly, small perturbations near the contact line (the triple junction where ∗ S, S∗ and S meet) leads to:

(σ − σ ) cos γ = − SL SV = β (3) σLV where γ denotes the angle with which the free surface S intersects the solid surface (see Figure 2). This angle γ is called the contact angle. The contact angle condition (3) plays a central role in this review. In particular, we note that 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’s law.

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 analysis of capillary phenomena. We present here some key results of this theory (a standard reference for BV theory is E. Giusti [Giu84]). Though the natural setting for our study is R3, the dimension plays no signiﬁcant role, so we set our 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 in L1(Ω) with bounded variation: Z BV (Ω) = f ∈ L1 : |Df| < +∞ Ω where Z Z 1 n+1 |Df| = sup f(x) div g(x)dx : g ∈ [C0 (Ω)] , |g| ≤ 1 . Ω Ω

4 n+1 If E is a Borel set in R , we denote by ϕE its characteristic function. The perimeter of E in Ω is then defined by Z P (E, Ω) = |DϕE| Ω which is thus finite as soon as ϕE ∈ BV (Ω). A Caccioppoli set is a Borel set E that has locally finite perimeter (i.e. P (E,B) < ∞ for every bounded open subset B of Ω). We recall the following relevant facts from the theory of BV functions (the proofs can be found in [Giu84]):

1. If E ∈ Rn+1 is smooth enough (∂E of class C2), then P (E, Ω) is equal to the n-Hausdorff measure of ∂E. P (E, Ω) is thus a natural extension of the 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 is henceforth 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) + 1 1 R has a trace f in L (∂Ω). Moreover, if fn → f in L (Ω) and |Dfn| → R + + 1 |Df|, then fn → f in L (∂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)}, then Z P (E,A × (0, ∞)) = p1 + |∇u|2 dx. A Note that for u ∈ BV (A), the integral in the right hand side can be deﬁned by Z p1 + |∇u|2 dx A (Z n ) X 1 n+1 = sup gn+1 + u Digidx : g ∈ [C0 (Ω)] , |g| ≤ 1 A i=1 and satisﬁes Z Z Z |Du| ≤ p1 + |Du|2 dx ≤ |A| + |Du|. A A A

5 !

! E E

Sessile drop Capillary Tube

Figure 3: Sessile drop and capillary tube

In the sequel, we study in details two situations: The sessile drop and the capillary tube (see Figure 3). With the notations above, the total energy for a sessile drop occupying a set E ⊂ Ω and in contact with the solid surface ∂Ω, can be written as Z 1 Z Z ET = σ P (E, Ω) − βϕEdσ(x) + ρΓϕE dx + λ ϕE dx . ∂Ω σ Ω Ω When Ω = Σ × R+ with Σ open set of Rn, and if the free interface of the liquid is the graph of a function u(x) deﬁned over Σ, as in the capillary tube case, the total energy can be written as Z Z Z u p 2 ET = σ 1 + |∇u| dx − β dz dσ(x) Σ ∂Σ 0 1 Z Z u Z + ρΓ dz dx + λ u dx . σ ∂Σ 0 Σ

2.3 The capillary tube

Throughout this section, Σ is a ﬁxed subset of Rn and we consider a liquid ﬁlling a cylinder Ω = Σ × R+. Furthermore, we assume that β is constant and β ∈ [−1, 1]. Finally, the gravitational potential is given by Γ = gz. Under such assumptions, it was proved by M. Miranda [Mir64] that the free surface projects simply on Ω. We can thus restrict ourself to sets whose free surfaces 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 Z p Z ρg Z u2 Z J (u) = 1 + |∇u|2 dx − β u(x)dσ(x) + dx + λ u(x) dx Σ ∂Σ σ Σ 2 Σ

6 and the equilibrium shape of the free surface is determined by the following minimization problem: J (u) = min J (v). (5) v∈BV (Σ) In this framework, Young’s Laplace law reads  !  ∇u  div p = λ + κu in Σ  1 + |∇u|2 (6)  ∇u · ν  = β on ∂Σ  p1 + |∇u|2

ρg where κ = σ is the capillary constant. In the sequel we denote by γ the real number in [0, π] such that cos γ = β. Note that the Lagrange multiplier is uniquely determined by γ and the volume constraint: Integrating (6) over Σ, we get 1 λ = [∂Σ cos γ − κV ]. Σ In particular, if κ = 0 (gravity free case), we have ∂Σ λ = cos γ (7) Σ 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, in either 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 existence of C2 surfaces with prescribed mean curvature has been studied in particular by Serrin [Ser70]. Numerous existence results for the variational problem (5) have been obtained with various requirements on Σ. One of the ﬁrst results is that of M. 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]. Very general results, based on the ideas of M. Miranda and E. Giusti can be found in [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] the capillary 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 "

2l sin ! "’ l 2!

Figure 4: Domain with corner of opening 2α from below. Physically, this means that the liquid will rise indefinitely along the 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) can actually easily be found by integrating (6) over any subset Σ0 ⊂ Σ: We get Z ∇u · ν λΣ0 = ∂Σ ∩ Σ0 cos γ + dx p 2 ∂Σ0∩Σ 1 + |∇u| and so 0 0 0 λΣ − ∂Σ ∩ Σ cos γ ≤ ∂Σ ∩ Σ with strict inequality whenever Σ0 6= Σ. Using (7), we deduce: ∂Σ Σ0 − ∂Σ ∩ Σ0 cos γ < ∂Σ0 ∩ Σ for all Σ0 ⊂ Σ. (8) Σ This necessary condition is a geometric condition on Σ, depending on γ but not on u. One can better understand the meaning of (8) by considering a domain Σ containing a corner of opening angle 2α. Taking Σ0 as in Figure 4 and letting l → 0, it is readily seen that condition (8) yields cos γ ≤ sin α or π γ + α ≥ . 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 with opening angle 2α such that α + γ < π/2, then (6) admits no solutions. This condition is sharp (see Finn [Fin86] for details). One could wonder if the lack of solutions to an apparently simple physical problem should be taken as an indication that the mathematical model is flawed. However, it can be verified experimentally that when a liquid is contained between two solid plate forming a corner, there is a critical opening angle for which the liquid will rise to infinity (or to the top of the container). This result is thus in accord with experimental observations. Finally, note that this phenomenon is not due to the singularity of the boundary of Σ, since one could smooth the corner in the previous computation and reach a similar conclusion with ∂Σ ∈ C∞. This also implies a strong instability of capillary surfaces with respect to boundary perturbations.

8 Existence result for γ ∈ (0, π/2). Condition (8) is thus necessary for the existence of a solution. As it turns out, it is not far from being also a suﬃcient condition, 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 exists 0 ε0 such that for all proper subset Σ ⊂ Σ we have: ∂Σ Σ − ∂Σ ∩ Σ0 cos γ < (1 − ε )∂Σ0 ∩ Σ. (9) Σ 0 Then there exists a minimizer of J in BV (Σ). Furthermore u solves (6) and is bounded. Finally any other solution of (6) is equal to u up to an additive constant. Condition (9) implies that for any v ∈ BV (Ω) we have the following inequality Z Z Z cos γ v dHn−1(x) ≤ (1 − ε) |Dv| + C |v| dx (10) ∂Σ Σ Σ 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 proved that 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 suﬃcient conditions 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 existence of 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

λΣ0 < ∂Σ0 (11) for all proper subset Σ0 of Σ, with equality if Σ0 = Σ. In fact this condition is suﬃcient 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 Σ0 of Σ with equality if Σ0 = Σ, then there exists a unique solution (up to a constant) of (6). Furthermore, the behavior of u along ∂Σ can be characterized. It actually depends 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 in particular 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:

n Ω = R × (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 volume V > 0: Z E (V ) = E ⊂ Ω: |DϕE| < +∞, |E| = V , Ω R where |E| = Ω ϕE demotes the Lebesgue measure of E. Since Caccioppoli sets have a trace on ∂Ω = Rn × {z = 0}, we can deﬁne the following functional for every E ∈ E (V ): ZZ Z Z J (E) = |DϕE| − β(x)ϕE(x, 0)dx + ρ Γ ϕE dx dz (12) z>0 z=0 z>0 Z Z n = P (E, Ω) − β(x) dH (x) + ρ Γ ϕE dx dz. E∩{z=0} z>0 In this framework, equilibrium liquid drops are solutions of the minimization problem: J (E) = inf J (F ) ,E ∈ E (V ). (13) F ∈E (V ) We recall that the Euler-Lagrange equation associated to this minimization problem with volume constraint is given by the Young-Laplace law (2-3): Γρ nH = − λ (14) σ cos γ = β, (15) where H denotes the mean-curvature of the free surface ∂E and γ denotes the angle between the free surface of the drop ∂E and the horizontal plane {z = 0} along the contact line ∂(E ∩ {z = 0}) (measured within the ﬂuid, see Figure 5).

The shape of equilibrium drops strongly depends on the coeﬃcient β. When β ≤ −1, the cost of wetting the solid support is at least the same as that of the free 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 area E ∩ {z = 0}. The main feature of the mathematical analysis of equilibrium drops in that case is the study of the contact line ∂(E ∩ {z = 0}) which is a codimension 2 free surface. Note ﬁnally that if β > 1 then the drop will spread indeﬁnitely and no minimizer can exist. We will thus always assume

|β| ≤ β0 < 1.

In that case, it is easy to check that Z Z Z 1 − β0 1 − β0 J (E) ≥ |DϕE| + ϕEdx + ρ Γ ϕE dx dz 2 z>0 2 z=0 z>0 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 be easily established (see [CM07b] for instance). Moreover, it is a classical result 1 that bounded subsets of BV (Ω) are pre-compact in Lloc(Ω). Those two facts give the convergence of minimizing sequences to minimizers of J provided |Ω| < +∞ (The existence of minimizers for the drop problem in a bounded domain was ﬁrst proved in [MP75]). Since we are considering drops lying in the upper half space, we need to be a little bit careful and ensure that minimizing sequences 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: For every E ∈ E (V ), the set

1 Z n s −1 E = {(x, z) ∈ Ω; |x| < ρ(z)}, where ρ(z) = ωn ϕE(x, z)dx (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 any minimizer should have axial symmetry (see [Gon76] for details). If we assume furthermore that Γ = 0 (gravity free), then it can actually be shown that the minimizers are spherical caps; that is the intersection of a ball Bρ0 (0, z0) in Rn+1 with the upper-half space Ω. We denote by B+ (z ) = B (0, z ) ∩ {z > 0} ρ0 0 ρ0 0 such a spherical cap.

When the relative adhesion coeﬃcient β depends on x, Schwarz symmetrization (16) may increase the wetting energy Z β(x)ϕE(x, 0)dx so minimizers do not have axial symmetry in general. It is still, however, possible to 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 J when 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 expect for the free surface ∂E ∩ {z > 0}. This is in fact a very classical result, which is 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 the restriction for the regularity of minimal surfaces.

The next question is the regularity of the contact line which is a codimension 2 free surface in Rn+1. The formation of singularities at the contact line is an important problem in many physical applications. From a mathematical point of view, our main goal is to be able to give a rigorous meaning to the contact 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. Friedman in [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:

Z Z Z 2 Z p 2 ρg u J (E) = J(u) := 1 + |∇u| dx + (1 − β)χu>0 dx + dx + λ u dx n n σ 2 R R Ω 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 continuous function u ∈ BV (Rn), solution of J(u) = min J(v). n v∈BV (R ),v≥0 Note that, formally at least, u is solution of the following free boundary problem:  !  ∇u ρg  div = λ + u in {u > 0}  p1 + |∇u|2 σ (18)  ∇u  · ν = β on ∂{u > 0}.  p1 + |∇u|2

This free boundary problem belongs to the wide class of problems that are obtained as Euler-Lagrange equations for the minimization of functionals of the form Z F (x, u, ∇u) dx {u>0} with F (x, z, p) convex with respect to p. Those include the well known Bernoulli problem 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. Caffarelli in [AC81] for the Bernoulli problem and generalized to more general functionals by 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 degeneracy of the mean-curvature operator (for large ∇u). It is however possible to adapt the method, as shown by L. Caffarelli and A. Friedman in [CF85] (the key is the 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 optimal regularity in view of the free boundary condition) and non degeneracy at the free boundary (i.e. linear growth away from the free boundary). This in turn implies that the free boundary ∂{u > 0} as finite Hausdorff measure.

13 Further regularity of the free boundary is harder to obtain and relies on local improvement of regularity, a method which is reminiscent of De Giorgi’s proof of the regularity of minimal surfaces. The following can be shown ([AC81] and [ACF84]): α 1,α If β ∈ C then ∂red{u > 0} ∈ C and n−1 H (∂{u > 0}\ ∂red{u > 0}) = 0 where ∂red{u > 0} denotes the reduced free boundary (see [Giu84]). Additional regularity of β implies further regularity of ∂red{u > 0}:

k,α k+1,α if β ∈ C then ∂red{u > 0} ∈ C and if β is analytic then ∂red{u > 0} is analytic.

Further characterization of the singular set ∂{u > 0}\ ∂red{u > 0} relies on blow-up arguments and the classification of global solutions. The result depends on the dimension, and it is known that ∂{u > 0} = ∂red{u > 0} in dimension n = 2 (H. Alt and L. Caffarelli [AC81]) and n = 3 (L. Caffarelli, D. Jerison and C. 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, then the contact line is C1,α and the contact angle condition cos γ = β is satisfied in the classical sense. Finally, let us point out that when we only assume −1 < β(x) < 1 (instead of β(x) ∈ (0, 1)), the previous analysis breaks down, since minimizers are no longer graphs. However, it is likely that the same regularity result will hold in that case. We proved in [CM07b] that the contact line ∂(E ∩ {z = 0}) has finite Hausdorff measure in that case: Proposition 2.7 (L. Caffarelli, A. Mellet [CM07b]) n+1 Assume |β(x)| ≤ β0 < 1 and let E be the minimizer of J in Ω = R+ . Assume 1 furthermore that E lies in a ball of radius M|E| n+1 for some large M. Then there exists C depending only on M and the dimension such that

n−1 n−1 H (Br ∩ ∂(E ∩ {z = 0}) ≤ Cr

n for any ball Br ⊂ R .

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

14 contamination or roughness of the surface). A real solid surface is extremely hard to clean and is never ideal; it always has a small roughness or small spatial inhomogeneities. These inhomogeneities are responsible for many interesting phenomena, the most spectacular being contact angle hysteresis (see [JdG84], [LJ92]): In experiments, one almost never measures the equilibrium contact angle given by Young-Laplace’s law. Instead, the measured static angle depends on the way in which the drop was formed on the solid. If the equilibrium was reached by advancing the liquid (for example by spreading or condensation), the contact angle has value γa larger than the equilibrium value. If on the contrary, the liquid interface was obtained by receding the liquid (evaporation or aspiration of a drop for example), then the contact angle has value γr, smaller than the equilibrium value (see L. Hocking [Hoc95], [Hoc81]). In extreme situations (typically when the liquid is not a simple liquid, but a solution), diﬀerences of the order 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 some particular type of periodic roughness and explicitly compute the contact angle hysteresis in that case. Contact angle hysteresis also explains some simple phenomena observed in everyday 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 any translation in the y < 0 direction will strictly decrease the total energy. It can also be shown (see R. Finn and M. Shinbrot [FS88]) that (14-15) has no solutions when β is constant and κ, θ 6= 0. This means that on a perfect surface, the drop should always slide down, no matter how small the inclination. However, a water drop resting on a plane that we slowly inclined will first change shape without sliding, and will only start sliding when the inclination angle reaches a critical value: in the lower parts of the drop, the liquid has a tendency to advance and the contact angle increases until it reaches the advancing contact angle, in the upper parts of the drop, the liquid has a tendency to recede and the contact angle decreases until it reaches the receding contact angle (see [DC83]). In [CM07b] and [CM07a], we rigorously investigate the approach suggested by C. Huh and S. G. Mason [HM77]: Roughness of the surface is taken into account via small scale oscillations of the coefficient β. In order to understand the effects of those microscopic oscillations on the large (or macroscopic) scale shape of the drop, we use a classical mathematical artifact: 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 ZZ Z Z Jε(E) = |DϕE| − β(x/ε)ϕE(x, 0)dx + ρ Γ ϕE dx dz (20) z>0 z=0 z>0 as ε → 0 will describe the eﬀects that the small scale inhomogeneities have on the overall shape of the drops (which is the main goal of the theory of homogenization). For the sake of simplicity, we assume Γ = 0 throughout the rest of this section.

2.5.1 Global minimizers We denote by hβi the average of β: Z hβi = β(x) dx, [0,1]n and by B+ (z ) = B (0, z ) ∩ {z > 0} the minimizer of the functional with ρ0 0 ρ0 0 constant adhesion coeﬃcient: ZZ Z J0(E) = |DϕE| − hβiϕE(x, 0)dx. z>0 z=0 We recall that the axial symmetry of minimizers when β is constant follows from Schwarz symmetrization and the isoperimetric inequality. When β depends on (x, y) the method fails, since the rearrangement (16) could increase the wetting energy Z β(x/ε, y/ε)ϕE(x, y, 0) dx dy. (21)

We remark however, that when β is periodic, the contribution to the wetting energy of any cell of size ε which is included in E ∩ {z = 0} is equal to hβiεn. Thus, the diﬀerence in the wetting energy (21) of a set E and that of its Schwarz n symmetrization Es is at most hβiε times the number of cells that intersect the contact line ∂(E∩{z = 0}) (see Figure 6). That number can readily be estimated if we control the (n − 1)-Hausorﬀ measure of the contact line. Proposition 2.7 thus yields: Jε(Es) ≤ Jε(E) + Cε In other words, we do not have equality in the isoperimetric inequality but almost equality. With the help of Bonnesen type inequalities (quantitative isoperimetric inequalities) we deduce that the minimizer E must have almost axial symmetry. One can then show that E almost coincides with a spherical cap. 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

1 1 E ⊂ {|x| ≤ R0V n+1 , z ∈ [0,T0V 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+ ⊂ E ⊂ B+ . (1−η)ρ0 (1+η)ρ0 In other words, the free surface ∂E ∩ {z > 0} lies between ∂B+ and (1+η)ρ0 ∂B+ for ε small enough. (1−η)ρ0

This result gives the existence of “sphere-like” minimizers of Jε, which, when β ∈ (0, 1), correspond to “sphere-like” viscosity solutions of the free boundary problem:  !  Du  div = κ in {u > 0}  p1 + |Du|2 (22)  Du  · ν = β(x/ε) on ∂{u > 0}.  p1 + |Du|2

In particular, Theorem 2.8 implies that the global minimizers of Jε converge uniformly to minimizers of J0 (which are all spherical caps with contact angle condition cos γ = hβi). We also deduce that there exists a solution of (22) which converges, as ε goes to zero, to a solution of  !  Du  div = κ in {u > 0}  p1 + |Du|2

 Du  · ν = hβi on ∂{u > 0}.  p1 + |Du|2

17 1 " f(x/ ! ) min

"max x 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 the energy, 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 be captured 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, ∞),   2 |ux| = 2f(x/ε) on ∂{u > 0}, (23)   u(0) = 1. which is the Euler-Lagrange equation for the minimization of Z ∞ 1 2 Jε(u) = |ux| + f(x/ε)χ{u>0} dx. (24) 0 2 Classical solutions of (23) are straight lines with slope γ intersecting the x-axis at a point x0 such that

2 γ = 2f(x0/ε).

We thus have one solution for every x0 such that (see Figure 7) x 1 f = ε 2x2

When ε → 0 we obtain a family of lines with slope γ ∈ [γmin, γmax] where p p γmin = 2 inf f and γmax = 2 sup f.

2 The homogenization of the free boundary condition |ux| = 2f(x/ε) thus gives |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 from simple in general. The main thing is that the free boundary is free to avoid the bad 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 a periodic metric (see [Mor24]), and for minimal surfaces (see L. Caﬀarelli, R. de la Llave [CdlL01]) In [CLM06], we generalized the result above to higher dimension, proving, in particular that there exists some solutions of the elliptic free boundary problem

( ∆u = 0 in {u > 0} (25) |∇u|2 = f(x/ε, y/ε) on ∂{u > 0}, 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 propagation in periodic media. Finally, note that if we look at the global minimizer of (24) (which is a solution of (23)), it is not very diﬃcult to show that it converges, as ε goes to zero, to the unique minimizer of Z ∞ 1 2 Jε(u) = |ux| + hfiχ{u>0} dx, 0 2 which satisﬁes |∇u|2 = 2hfi along its free boundary. This is similar to what we observed with the global minimizer of the sessile drop problem.

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

min max β(x, y) < hβi. (26) y x To construct local minimizers we recall that the wetting surface corresponding to the asymptotic minimizer B+ is a disk of radius ρo

p 2 Ro = ρo 1 − hβi .

So we introduce 2 Σt = {(x, y) ∈ R ; 0 ≤ y ≤ 2Ro − t}, and look for minimizers of Jε whose wetting area stays within the region Σt. Clearly, for t > 0, E (which is arbitrarily close to B+ for small ε) is not a ε ρo 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, then Jε has a local minimizer Eeε of volume V not equal to the global minimizer Eε. Furthermore, when ε goes to zero, Eeε converges to some set Ee ∈ E (V ) satisfying the following equation:

cos γ ≤ hβi, 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−1hβi in some directions. This result is purely qualitative: The proof does not give any estimate on the size of the interval of admissible contact angles. It is fairly easy to construct barriers that yield hysteresis phenomena in any ﬁnite number of directions (for appropriate β). To get inﬁnitely many directions, however, one would probably have to consider random inhomogeneities.

The solution given by Theorem 2.9 plays the role of barrier during the formation of a liquid drop by slow spreading or condensation. This explains the so-called stick-jump phenomenon: As the volume of the drop increases, the contact line remains unchanged at ﬁrst, while the contact angle increases. Only when the contact angle reaches a critical value does the contact line jump to the next equilibrium position (see [HM77]). When the surface of the drop is a graph z = u(x, y), Theorem 2.9 gives the existence of a solution of (22) which converges as ε → 0, to a function u satisfying Du · ν ≤ hβi on ∂{u > 0} p1 + |Du|2 with strict inequality on part of the free boundary, which is reminiscent of the results obtained for (25) in [CLM06]. However solutions of (25) were constructed by studying viscosity solution of a singular nonlinear equation rather than by minimizing a functional.

Remark 2.10 Throughout this section, we have considered drops with complete contact with the underlying surface (known as Wenzel drops in the literature). However, another effect of roughness is to allow for small pockets of vapor to form underneath the drop (Cassie-Baxter drops) thus affecting the wetting energy. The effect of partial wetting on contact hysteresis has been investigated in 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 motion of the contact line when a droplet is spreading on a flat surface or sliding down an inclined plane. This problem is considerably more difficult to model than the static case: On top of the capillary effects at the drop surface and the wetting dynamic at the contact line, one must also account for the motion of the liquid inside the drop. We discuss two models: The quasi-static approximation, which essentially allows us to neglect the dynamic of the fluid inside the drop, and the lubrication approximation which is valid for small, very viscous drops. The effects of gravity are neglected throughout this section.

3.1 Quasi-static approximation Maybe the simplest model for contact line dynamics is the quasi-static model which assumes the slowness of the contact line motion in comparison to the time for capillary relaxation (see K. Glasner [Gla05] for a detailed discussion). We can then consider that the ﬂuid pressure inside the drop is constant at all time and that the free surface has constant mean-curvature. The inbalance of surface forces along the contact line is responsible for the motion of the contact line. This is usually described by a constitutive law linking the velocity of the contact 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 are led to the following system of equations:  ! Dh  div = −λ(t, h) in {h > 0},  p 2  1 + |Dh|  Z (27) h(x, t) dx = V for all t,     v = F (|∇h|, x) on ∂{h > 0} where v denotes the velocity of the contact line ∂{h > 0} in the direction normal ht to ∂{h > 0}. Note that we have v = |∇h| , so the free boundary condition also reads ht = |∇h|F (|∇h|, x) on ∂{h > 0}. Alternatively, one can approximate the mean curvature by the laplacian of h 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 determined at each time by the volume constraint. In particular we cannot expect the comparison principle to hold. The determination of the constitutive law (the function F ) is a major dif- 3 3 ficulty of the study of contact line motion. Empirical laws give v = θ − θe or

21 v = κ(θ)(θ −θe) where θ denotes the measured contact angle and θe the equilibrium 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 law is always satisfied at the molecular level, but that there is a difference between the microscopic contact angle θe and the apparent contact angle θ which is responsible 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 difficult to study mathematically (due to the lack of comparison principle, as noted above). Classical solutions do not exist globally in time: Numerical computa- tions point out that singularity may form along the contact line. Even initially smooth and convex drops may develop corners. Also topological singularities may 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 down in such events (each connex component must be treated separately with a different coefficient λ).

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

3.2 Lubrication approximation and the thin film equation The lubrication approximation consists of a depth-averaged equation of mass conservation and a simplified form of the Navier-Stokes equations that is appropriate for very thin drops (vertical length scale much smaller than the horizontal length scale) of very viscous liquid. In this framework, the evolution of capillary surfaces can be described by a fourth order degenerate diffusion equation, known as the thin film equation. We refer to H. Greenspan [Gre78] for details about the validity of the lubrication approximation, and we point out the review of A. Bertozzi [Ber98] in which one can find a complete introduction to the mathematical 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 by h(x, t) the height of the drop (for x ∈ R2, t ∈ (0, ∞)) and by v(x, t) the vertical average of the horizontal component of the velocity ﬁeld of the ﬂuid (uH(x, z, t), where z is the vertical coordinate): 1 Z h v(x, t) = uH(x, z, t) dz. h 0

22 The depth-averaged conservation of mass yields ∂h + div (hv) = 0, ∂t x while, under the lubrication approximation, the momentum equation reduces to ∂2u µ H = ∇ p(x, t). ∂z2 x Along the free surface z = h, we assume that the horizontal shear stress vanishes, while we take no-slip condition at z = 0: ∂u H | = 0, u| = 0. ∂z z=h z=0 Integrating the momentum equation then gives 1 1 u = ∇ p z2 − hz H µ x 2 and so h2 v = − ∇ p. 3µ x

Finally, we approximate the capillary pressure by p|z=h = −σ∆h (replacing the mean-curvature operator by the Laplacian). We obtain the so-called thin film equation ∂h σ + div(h3∇∆h) = 0. (28) ∂t 3µ 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 null flux condition hv = 0: h3∇∆h = 0 which guarantees conservation of mass. With only two conditions at the free boundary, the fourth order equation (28) is not well posed, which suggests that we impose a contact angle condition |∇h| = θ0 with θ0 = 0 (complete wetting, appropriate for wetting on a moist surface) or θ0 > 0 (partial wetting). Another possibility, suggested for instance by Greenspan [Gre78] and Hocking [Hoc81], is to assume that the velocity of the contact line and the contact angle are linked by a constitutive law: ht = |∇h|F (∇h). There is still some debates about the appropriate boundary condition (from a physical point of view). Mathematically, it is not known which condition would make the problem well posed. As we will see in the sequel, fairly general 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 contact lines. One way to see this, is to notice that the motion of the contact line would lead to a multivalued velocity field at the contact line. More interestingly, it was observed by Huh and Scriven [HS71] that, independently of the contact angle condition, the dissipation energy has a logarithmic singularity at the contact 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 solutions and source type solutions exist for (28). However, liquid do spread, so some of the assumptions leading to (28) must be inappropriate, and more physics needs to be taken into account in the model near the contact line. Several ways have been suggested to remove this singularity. One possibility is to include microscopic scale forces, in the form of long-range forces of Van Der Waals interactions between the solid and the liquid (leading to a hyperdiffusive term, see A. Bertozzi [Ber98]). Another approach is to assume the presence of a precursor film of very thin height (b 1), which amounts to look for positive solutions satisfying h = b > 0 at the boundary of the domain. A third possibility is to relax the no-slip boundary condition at z = 0 and use the Navier slip condition instead. This condition reads ∂u u = Γ(h) at z = 0. ∂z

Λ The slip coeﬃcient is given by Γ(h) = 3h2−s with typically s = 1 (singular slip) or s = 2 (constant slip). Equation (28) then becomes:

∂h σ + div((h3 + Λhs)∇∆h) = 0. (29) ∂t 3µ Note that we have not changed the degenerate character of the equation, but we have removed the dissipation energy singularity at the contact line.

Lubrication approximation in a Hele-Shaw cell. Interestingly an equation similar to (28) arises when studying thin film in Hele-Shaw cells under the lubrication 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 lubrication approximation can be used (i.e. if we are considering the very slow flow of a viscous fluid), the vertically-averaged velocity field is given by Dracy’s Law:

b2 v(x, t) = − ∇ p(x, t) 12µ x

2 ∂ uH (obtained by integration of Stokes law µ ∂z2 = ∇p together with no-slip conditions at both plates z = 0 and z = b). The incompressibility condition div u = 0 thus yields ∆p = 0 in Ωt (30)

24 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 the boundary of Ωt, we have a capillary pressure condition p = σH (31)

(where H denotes the mean curvature of ∂Ωt), together with the kinematic condition ∂tp + v · ∇p = 0 which, using Darcy’s law, leads to b2 ∂ p = |∇p|2 (32) t 12µ (this last condition also says that the contact line ∂{p > 0} is moving with speed b2 12µ |∇p| in the direction normal to ∂{p > 0}). Equations (30-32) form a free boundary problem which has been intensively studied, often under the condition that σ = 0 (neglecting the capillary eﬀects). In that case, and assuming that the pressure is constant and equal to zero outside the region occupied by the liquid, we obtain the classical Hele-Shaw free boundary problem:  ∆p = 0 in {p > 0}  b2 ∂ p = |∇p|2 on ∂{p > 0}.  t 12µ 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 are focusing on the eﬀects of capillary forces.

Equations (30-32) constitute a rather diﬃcult free boundary problem. It turns out however that two interesting situations lead to an equation similar to the thin ﬁlm equation: The thin neck of liquid and the thin layer at the edge of the Hele-Shaw cell (see Figure 8). In either case, if we denote by h(x, t) the height of liquid (in the direction parallel to the plates), using Darcy’s law and approximating the mean curvature of ∂Ωt by hxx, we obtain the following fourth order degenerate equation: b2 ∂ h + ∂ (h ∂ h) = 0 (33) t 12µ x xxx

25 together with a contact angle condition at h = 0 (assuming no-slip boundary condition at the edge). Note that the rigorous derivation of (33) is performed by L. Giacomelli and F. Otto [GO03] with zero contact angle condition (complete wetting).

Self similar solutions and Tanner’s Law. We now consider the general equation ∂h + div(hn∇∆h) = 0, (34) ∂t which include the ill-posed thin ﬁlm equation (n = 3) and the Hele-Shaw lubrication approximation (n = 1). A particular role is played by compactly supported self similar solutions of (34). Such solutions are of the form 1 h(x, t) = tdδH(|x| tδ), δ = − 4 + dn with H deﬁned on (0, 1) satisfying

HnH000 = αxH and H0(0) = 0,H(1) = 0,H0(1) = 0 (the last condition is the zero contact angle condition) and where d = 1 or 2 for planar 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 solutions as they satisfy h(x, 0) = c δ(x). It is also proved in [BPW92] that no such solution exists for n = 3. Finally, we note that for n < 3, there exist other self similar solutions with compact support and satisfying a non zero contact angle condition.

For n = 3 (thin film equation), the scaling of the self similar solutions suggests that the support of spherical drops expand like t1/10 (despite the fact that such solutions do not exist). This is known as Tanner’s law and it has been verified experimentally with remarkable precision. It was suggested by P.- G. DeGennes that the modification of the model near the contact line (via the Navier slip condition or Van der Waals forces) should only have a weak effect on 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 planar film: While the scaling of self-similar solutions predicts that the support will 1 spread like t 7 , L. Giacomelli and F. Otto show that solutions of (29) with s = 2 (constant slip) satisfy

1 t 7 meas(h > Λ) ∼ . log(1/t)

26 Properties of the thin film equation. Equation (34) is reminiscent of the porous media equation γ ∂tu − div (u ∇u) = 0, for which existence and uniqueness of weak solutions is well known (see for instance A. Friedman [Fri88] and references therein). Other properties of the porous media equation include the existence of self similar source type solutions and traveling wave solutions, the finite speed expansion of the support (if the initial data has compact support, then the solution has compact support for all time) and the fact that the support is always expanding (in fact strictly expanding, 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 known that non-negative initial data may generate changing sign solutions of the fourth order equation ∂th + ∂xxxxh = 0. Such solutions would not make sense in the framework of thin film. It is however a remarkable feature of (34) that the degeneracy of the diffusion coefficient permits the existence of non-negative solutions.

Formally at least, solutions of (34) satisfy the conservation of mass: d Z h(x, t) dx = 0 dt and the dissipation of surface tension energy: d Z 1 Z |∇ h|2 dx = − hn(∇∆h)2 dx ≤ 0. dt 2 x Furthermore, they satisfy an entropy-like inequality: d Z Z G(h) dx = − |∆h|2 dx ≤ 0 dt where G00(s) = s−n. This last inequality proves particularly useful to show the existence of non negative solutions. As a matter of fact, we notice that for n = 2, we have G(h) = − log h so the entropy can be used to keep the solution away from zero.

Particular solutions. Besides the self similar source type solutions already mentioned, the thin ﬁlm equation also has traveling-wave solutions of the form u(x, t) = H(x − ct) (see [BKO93]). Note however that (34) has no advancing fronts solution for n ≥ 3. Finally, there are exact steady solutions with compact support for all n: √ A − Bx2, |x| < A/ B h(x, t) = 0, otherwise.

27 Some existence results and properties of the solutions. In this last section, we give an overview of the existence and regularity theory for the thin ﬁlm equation. We restrict ourself to the 1-dimensional case (i.e. we consider planar ﬁlm). We set Q = (−a, a) × (0, +∞) and consider an initial data h0(x) such that 1 h0 ∈ H (−a, a), h0 ≥ 0. We are thus looking for h(x, t) solution of the following fourth order degenerate parabolic equation:

n ∂th + ∂x(h ∂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)

n hx = |h| hxxx = 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. Further results were later obtained, by similar technics, by E. Beretta, M. Bertsch and R. Dal Passo [BBDP95] and A. Bertozzi and M. Pugh [BP96]. The particular case 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 diffusion coefficient and the initial data. For instance, we can define hε to be the classical solution of

ε ε n ε ∂th + ∂x((|h | + ε)∂xxxh ) = 0 t > 0 , x ∈ (−a, a) with initial condition ε h (x, 0) = h0(x) + ε. Then the function h(x, t) = lim hε(x, t) (38) ε→0 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 1 , 1 h(x, t) ≥ 0, h ∈ C 2 8 (Q) and ZZ ZZ n h φt dx dt + h hxxxφx dx dt = 0 Q P for all φ ∈ Lip(Q) with compact support in [−a, a] × (0, ∞). Here, P denotes the positivity set of h:

P := {(x, t) ∈ Q ; h(x, t) > 0}.

28 Furthermore, h satisfies Z a Z a h(x, t) dx = h0(x) dx ∀t > 0. −a −a This result was proved in [BF90] for n > 1 and extended, with some minor modifications, to n > 0 in [BBDP95]. Note that these solutions satisfy (35) is a very weak sense, since the flux is integrated only over the positivity set of h. In [BP96], it is proven that for 3 8 < 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 below), so the solutions given by Theorem 3.1 may strongly depend on the chosen regularization 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 all time and is unique.

Expansion of the support: Finite speed propagation of the support (as for the porous media equation) was proved by F. Bernis [Ber96a, Ber96b] for 0 < n < 3. Thus, a given compactly supported drop will stay compactly supported (until it reaches the boundary of the domain). Further characterization of the behavior of the support of h is also relevant in applications: Strictly expanding support means 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 the evolution of the support also implies that drops will never break up into smaller droplets. The following is known (see [BBDP95]): • If n ≥ 3/2, then the support of the solution given by Theorem 3.1 is increasing 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 is numerical evidence (R. Almgren, A. Bertozzi and M. Brenner [ABB96]) that break up should occur for n = 1 (Hele-Shaw cell). The critical value for 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, it satisﬁes a zero contact angle condition along its free boundary. Sharper regularity results are also derived in [BP96]. Furthermore, h becomes strictly positive after ﬁnite time and converges to its mean value as t → ∞ (and the convergence is exponential in time, see [BP96]).

29 Non uniqueness results [BBDP95]: As noted earlier, no uniqueness results have been proved. In fact, if 0 < n < 3, it is known that there exists another solution to (35-37) with non-expanding spatial support (and thus diﬀerent from the one given by Theorem 3.1). This solution is also approximated by classical positive solutions of non-degenerate problems. It is not known what additional condition (contact angle condition?) would guarantee uniqueness.

Non-zero contact angle solutions: The solution given by Theorem 2.8 satisﬁes a zero-contact angle condition. This fact follows from the approximation method rather 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 case of non zero contact angle. F. Otto, in [Ott98], proved the existence of such solutions 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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