Musterbildung, Energielandschaften und Skalierungsgesetze
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Felix Otto
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Katja Heid
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Viscous Thin Films
Authors: Elias Esselborn, Julian Fischer, and Manuel V. Gnann
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Lubrication theory
Thin liquid films appear in a variety of situations in nature and in engineering applications. Examples include tear films in the eye, membranes in biophysics, linings in the lungs of animals, or paints. Despite the diversity of phenomena and applications, the mathematical modeling is quite similar if the film is sufficiently viscous. Excluding other contributions such as rotational forces, gravity, or vanderWaals (molecular) interactions, it is reasonable to assume that the dynamics of the film are only driven by surface tension and viscosity, indicating the gradientflow structure of the problem. In the regime of thin films, one commonly refers to this approach as lubrication approximation.
The thinfilm equation
The resulting partial differential equation is a fourthorder degenerateparabolic equation that  for a \((d+1)\)dimensional thin film on a \(d\)dimensional flat solid  reads
\(\partial_t h + \nabla \cdot \left(\left(h^3 + \lambda^{3n} h^n\right) \nabla \Delta h\right) = 0,\) (1)
where \(h = h(t,y)\) is the height of the film, \(t\) the time, \(\lambda\) the slip length, \(0 < n < 3\) the mobility exponent, and \(y \in R^d\) the lateral variable (base point). The parameters \(\lambda\) and \(n\) are determined by the underlying fluid model from which equation (1) originates: The case \(\lambda = 0\) can be derived by an asymptotic expansion from the (Navier) Stokes system as the underlying model, with the standard noslip condition at the liquidsolid interface [f,m]. This model, however, leads to the wellknown noslip paradox [ac]: A contact line (triple junction), separating the three phases (liquid, gas, and solid), can only move by inserting an infinite amount of energy into the system to overcome dissipation. This is reflected by a singularity of the solution of (1) at the boundary \(\partial \{h > 0\}\) of the droplet and in fact the underlying Stokes system is illposed. One way to remove this paradox is to introduce slippage: The noslip condition is replaced by a Navierslip condition, where a nonzero horizontal component of the fluid velocity at the liquidsolid interface is allowed. This leads to the additive term \(\lambda^{3n} h^n\) in the lubrication model (1).
The thinfilm equation as a free boundary problem
Equation (1) is a fourthorder equation with a moving (free) boundary, that is, 3 boundary conditions are needed:
 \(h = 0\) at \(\partial \{h > 0\}\). This condition merely defines the position of \(\partial \{h > 0\}\).
 \(\partial_\nu h = 0\) on \(\partial \{h > 0\}\), where \(\nu\) is the interior normal of \(\partial \{h > 0\}\). For a quasistatic evolution of the thin film, the contact angle is determined by an equilibrium of the surface tensions of the three interfaces (liquidgas, liquidsolid, solidgas). Assuming \(\partial_\nu h = 0\) at \(\partial \{h > 0\}\) implies that such a local equilibrium is never attained and the film will generically not stop spreading (complete wetting regime), in contrast to the case of partial wetting, where without loss of generality \(\left\lvert{\partial_\nu h}\right\rvert = 1\) on \(\partial \{h > 0\}\). Finally, the simplifying assumption of a quasistatic movement can be relaxed to dynamic contact angles.
 One can also view equation (1) as a continuity equation, that is, \(\partial_t h + \nabla \cdot (V h) = 0\) in \(\{h > 0\}\), where the transport velocity \(V\) is given by \(V = \left(h^2 + \lambda^{3n} h^{n1}\right) \nabla \Delta h\). By compatibility, the boundary value of \(V\) on \(\partial \{h > 0\}\) has to equal the velocity of the contact line. This condition ensures conservation of mass.
Often one assumes that the film height \(h\) is small compared to the slip length \(\lambda\). Thus, we are lead to the study of the free boundary problem
\(\partial_t h + \nabla \cdot \left(h^n \nabla \Delta h\right)\)  \(= 0\) for \(t > 0\) and \(y \in \{h > 0\}\),  (2a)  
\(h = \partial_y h\)  \(= 0\) for \(t > 0\) and \(y \in \partial \{h > 0\}\),  (2b)  
\(\lim_{\{h > 0\} \owns y \to \partial \{h > 0\}} h^{n1} \nabla \Delta h\)  \(= U(t,y)\) for \(t > 0\) and \(\in \partial\{h > 0\}\),  (2c) 
where \(U(t,y)\) denotes the velocity of the free boundary \(\partial\{h > 0\}\).
Weak solutions
The mathematical treatment of the thinfilm equation started with the work of Bernis and Friedman [h] establishing existence of weak solutions in \(d = 1\). This approach mainly relies on the dissipation of (surface) energy \(\frac{d}{d t} \int_{R^d} \left\lvert {\nabla h} \right\rvert^2 d y =  \int_{R^d} h^n \left\lvert{\nabla \Delta h}\right\rvert^2 \, d y \le 0\), conservation of mass \(\frac{d}{d t} \int_{R^d} h \, d y = 0\), and a compactness argument. Further using dissipation of the "entropy" \(\frac{d}{d t} \int_{R^d} \eta_n(h) \, d y =  \int_{R^d} (\Delta h)^2 \, d y \le 0\) (where \(\eta_1(h) \sim h \log h\)), the notion of global "strong" or "entropy"
Qualitative behavior of the free boundary
Rigorously analyzing the qualitative behavior of the contact line in solutions to the thinfilm equation (2) poses significant mathematical challenges: For the porous medium equation
\(\partial_t h=\Delta h^m\) (3)
(with \(m>1\))  the secondorder analogue of the thinfilm equation  , estimates on free boundary propagation for weak solutions may be established by the comparison principle or Harnack inequalities. However, the thinfilm equation is a degenerate parabolic equation of fourth order; thus, no comparison principle is available anymore and no Harnack inequalities are known either. For this reason, for solutions to the thinfilm equation it was not even known until recently whether the free boundary would ever move near a given point on the initial free boundary. The situation was no different for other higherorder degenerate parabolic equations: No lower bounds on free boundary propagation for higherorder degenerate parabolic PDEs had been available at all. In the recent papers [8,13], we have devised a technique for the derivation of lower bounds on free boundary propagation for higherorder parabolic equations. The key ingredient of our approach are new monotonicity formulas for weak solutions to the thinfilm equation of the form
\(\partial_t\int_{\mathbb{R}^d} h^{1+\alpha} yy_0^\gamma ~dy \geq c \int_{\mathbb{R}^d} h^{1+\alpha+n} yy_0^{\gamma4} ~dy\)
(for certain \(\alpha\in (1,0)\) and \(\gamma<0\)); these formulas are valid as as long as the support of the solution \(h\) does not touch the singularity of the weight at \(y_0\). Combining these formulas with a differential inequality argument due to Chipot and Sideris [e], estimates from below on support propagation may be derived. By this method, we have obtained sufficient criteria for instantaneous forward motion of the free boundary, upper bounds on socalled waiting times, as well as lower bounds on asymptotic free boundary propagation rates. For \(2<n<3\), we have succeeded in proving that the free boundary starts moving forward instantaneously near some point \(y_0\in \partial \{h_0>0\}\) if the initial data \(h_0\) grow steeper than \(yy_0^{4/n}\) near \(y_0\). This is optimal since (with a grain of salt) in [s] it has been shown that growth of at most \(yy_0^{4/n}\) entails a socalled waiting time phenomenon: The free boundary does not move beyond its initial location before some waiting time has passed. In such a case, our method yields upper bounds on the waiting time which are optimal up to a constant factor. In the borderline case \(n=2\), we obtain upper bounds on waiting times and sufficient conditions for immediate forward motion of the interface which are optimal up to a logarithmic correction term. Interestingly, for \(n<2\) the shorttime behavior of free boundaries is more complex. Again, for initial data \(h_0\) growing at most like \(yy_0^{4/n}\), in [s] it has been shown that a waiting time phenomenon must occur. However, already in one spatial dimension the stationary solution \((yy_0)_+^2\) shows that growth of the initial data \(h_0\) steeper than \(yy_0^{4/n}\) does not necessarily entail instantaneous forward motion of the interface. Nevertheless, for \(1<n<2\) we have been able to construct initial data \(h_0\) which grow just a bit steeper than \((yy_0)_+^{4/n}\) and for which instantaneous forward motion of the interface occurs [11]. As these initial data are bounded from above by the steady state \((yy_0)_+^2\), this is a drastic example of a violation of any comparison principle and highlights an important difference to the case of secondorder degenerate parabolic equations: In the case of the secondorder porous medium equation (3) the initial behaviour of the free boundary is dictated by the growth of the initial data at the free boundary. For growth steeper than \((yy_0)_+^{2/(m1)}\), instantaneous forward motion happens, while a waiting time phenomenon occurs otherwise. In contrast, in the case of the thinfilm equation with \(1<n<2\), the initial behaviour of the interface is not determined just by the growth of the initial data at the free boundary. Regarding asymptotic propagation of the free boundary, for \(\frac{3}{2}<n<3\) we are able to show that for large times the support of any solution to the thinfilm equation must spread at about the same speed as the corresponding selfsimilar solution. More precisely, for any \(t>0\) and any \(y_s\in \operatorname{supp} u_0\) the inclusion
\(B_{R(t)}(y_s)\subset \operatorname{supp} u(.,t)\)
holds with
\(R(t):=c(d,n)u_0_{L^1}^{n/(4+n d)} t^{1/(4+nd)}\operatorname{diam}(\operatorname{supp} u_0)\).
Our method for the derivation of lower bounds on free boundary propagation is not limited to the thinfilm equation, but is flexible enough to be applied to other higherorder nonnegativitypreserving parabolic equations: For example, in the case of the socalled quantum driftdiffusion equation an adaption of our ansatz can be used to prove infinite speed of propagation [12].
Gradient Flow structure of the thinfilm equation
It is wellknown since [j] that the thinfilm equation can be seen as the gradient flow of the surface energy with respect to a Wassersteintype metric for all mobilities \(n\). Considering the space of functions
\(\mathcal{N} = \left\{h:\mathbb{R} \rightarrow [0,\infty[ \middle \int_{R} h \,dy = 1 \right\}\),
we can think of its tangent space as
\(T_{h}\mathcal{N} = \left\{\delta h:\mathbb{R} \rightarrow \mathbb{R} \middle \int_{R} \delta h \,dy = 0 \right\}\).
Identifying a tangent vector \(\delta h \in T_{h}\mathcal{N}\) with a solution \(v\) of the equation
\(\delta h + \partial_y(v h^n) = 0\),
we define a metric tensor by
\(\left\langle \delta h, \delta h \right\rangle_{h,n} := \int_{\mathbb{R}} v^2 h^n \,dy\).
One observes that the gradient flow with respect to this metric and the free energy
\(E_{\alpha}(h) := \frac{1}{2}\int_{\{h>0\}} \left(\partial_y h\right)^2 \, dy + \frac{\alpha}{2} \, \{h > 0\}\), (4)
for \(\alpha \in \{0,1\}\) leads to the thinfilm equation with complete / partial wetting boundary conditions, i.e.
\(\partial_y h(y) = \alpha \text{ for } y = Y(t)\).
This insight was used to obtain a first existence result for weak solutions in the partial wetting regime [1], where an approximative timediscrete solution for timestep size \(\tau\) was constructed via the minimizing movement scheme
\( h_{\tau}^{(k)}\) is minimizer of \(h \mapsto \frac{d^2(h_{\tau}^{(k1)},h)}{2\tau} + E(h)\), (5)
which is formally equivalent to the timediscrete gradient flow equation. Here \(d\) denotes the Riemannian distance induced by \(\left\langle \cdot, \cdot \right\rangle_{h,n}\), which in the case \(n=1\) is known to be the wellstudied Wasserstein distance. The gradient flow structure is also crucial in understanding how (2) with \(n=1\) arises as the lubrication approximation of the Darcy flow in a HeleShaw cell. In a first work [2] it was shown that the scheme (5) can be seen as the \(\Gamma\)limit of the suitably rescaled corresponding discrete schemes of the HeleShaw flows. The lubrication approximation for the full equation was then made rigorous in the complete wetting case in [4], using one of the main insights from [2] that the contact angle is a consequence of an instantaneous energy relaxation at the triple point rather than an imposed contraint. The energy landscape described by \(E\) and \(\left\langle \cdot, \cdot \right\rangle_{h,n}\) is globally nonconvex. Nevertheless, in the partial wetting case with linear mobility it is convex in a region close to the stationary solution \((x)_+\), an observation which leads to natural relaxation rates of perturbations of the stationary solution [16]. Including intermolecular forces in the energy (4) in the form of a potential \(\mathcal{U}\) leads to a phenomenon where a configuration of droplets coarsens. This means that the number of droplets decreases, while the average size of single droplets increases. The rates by which this happens are investigated in [5,7], for a more detailed discussion, please see the related page on coarsening.
Wellposedness and regularity for the thinfilm freeboundary problem
It appears natural to ask whether the introduction of slippage indeed removes the singular behavior at the contact line. This leads to the mathematical question of regularity of the solution at the free boundary. In fact, developing a regularity theory for degenerateparabolic fourthorder equations is a relatively new field. For the thinfilm problem (2) we refer to the works of Giacomelli, Knüpfer, and two of the group members [6,10,v,y,z], in particular addressing the case of \(n = 1\) in the complete wetting regime. Here the solution is in fact smooth up to the contact line. However, such qualitative behavior cannot be expected for other mobility exponents as first noticed by Knüpfer in the partial wetting case and \(d = 1\) [w,x]. One of the ongoing projects of our group is to understand the case of complete wetting, where a moving contact line is the generic situation.
Apart from the applied point of view, there is also a theoretical interest in the questions detailed above, since before the analysis starting with [6], uniqueness results have not been available for (2). The existence results for weak solutions [h,i,k,o,q] always relied on a compactness argument since the control of the solutions at the free boundary was not strong enough to apply the contraction mapping theorem. Again it is the detailed understanding of the regularity at the free boundary that enables us to prove existence and uniqueness of solutions for short times or for initial data close to generic solutions (stationary, traveling waves, or selfsimilar solutions, see below).
There is another theoretical motivation for our analysis, coming from the porous medium equation (3). Here a welldeveloped existence and uniqueness theory is available [g,p,r], which, however, at least partially relies on the use of a maximum (or comparison) principle. Therefore it seems of interest to study which of the analysis does or does not transfer from the secondorder to the fourthorder case. Although the works for the particular case \(n = 1\) [6,10,v] support the claim that the analysis does transfer, this is not true for all other mobility exponents. Only for \(n = 1\) the partial differential operator of a suitable linearization turns out to be the square of the wellunderstood linearized porousmedium operator.
It is instructive to further simplify the problem (2) to the case in which the behavior of solutions is selfsimilar. This is in fact the generic largetime behavior of solutions with compact support [15,t]. Here, the PDE problem (2) reduces to the study of a boundaryvalue problem for a thirdorder nonlinear ODE. Jointly with Lorenzo Giacomelli, we are able to show that the solution is generically not smooth, even if the leadingorder traveling wave is factored off [9]. Instead we are able to prove analyticity in two variables: Factoring off the travelingwave profile, the solution is an analytic function in \((x,x^\beta)\), where \(x\) denotes the distance to the boundary and \(\beta\) is an in general irrational number. \(1\) and \(\beta\) are in fact the eigenvalues at a hyperbolic stationary point (corresponding to the contact line) of a suitably chosen dynamical system and characterize the invariant manifold on which the solution lies. For the porous medium equation, in comparison, this invariant manifold is just onedimensional with the trivial eigenvalue \(1\). Essentially, it is a coincidence that for \(n = 1\) the two eigenvalues in the thinfilm case coincide.
Given the understanding for the sourcetype solution, we can also treat the general PDE problem (2) for the physical relevant case of quadratic mobility (\(n = 2\)) [14]. This work is joint with Lorenzo Giacomelli and Hans Knüpfer. Here we are able to prove wellposedness of the problem for initial data that are close to the generic solution, a traveling wave \(\sim x^{3/2}\). Our method relies on maximal regularity estimates in weighted \(L^2\)spaces and a suitable subtraction of the leadingorder singular expansion of the solution at the free boundary. While our method yields a wellposedness result, the question of higher regularity is of ongoing interest and will be adressed in future work. Furthermore, we are also interested in generalizing our result (which is in fact valid for an interval of \(n\) around \(n = 2\)) to the full range of mobility exponents \(n \in (0,3)\).
Currently our main interest lies in a deeper understanding of the full system including slippage, for which sourcetype selfsimilar solutions do not exist. Thus even simplified ODE models turn out to be mathematically subtle. In particular the travelingwave solution has no explicit characterization and exhibits two asymptotic regimes. Close to the contact line the solution has a similar asymptotic expansion as the sourcetype selfsimilar solution in the scalinginvariant case (2), whereas in the interior of the droplet we observe another asymptotic regime known as Tanner's law [d,l]. Here we are able to show that Tanner's solution (which was found in the case of noslip, i.e. \(\lambda = 0\)) is affected by the microscopic physics only in higher order corrections [17]. Furthermore, continuous (smooth) variations of the microscopic model (by e.g. varying the mobility exponent \(n\)) lead to continuous (smooth) variations of these corrections. This work is closely related to an earlier work of Giacomelli and one of the group members [3], where it is shown that the effect of slippage affects the spreading rate of the droplet only by a logarithmic correction. The proof relies on the gradient flow structure of the thinfilm equation by monitoring three physical integrals: the free energy, the dissipation, and the length of the apparent support.
In future work we would like to investigate the lubrication approximation starting from solutions of the (Navier) Stokes system with slippage. Such a rigorous lubrication approximation was indeed carried out by Giacomelli and one of the group members in earlier work for the particular case \(n = 1\) using weak solutions [2,4]. Here problem (2) can be understood as the lubrication approximation of the Darcy flow in the HeleShaw cell (see also [y,z] for the partial wetting case and classical solutions). We expect that this problem is mathematically challenging and would like to understand it first at a simplified level, that is, by studying travelingwave solutions for the Stokes system and proving the lubrication limit in the steadystate case.
Selected group publications

Felix Otto: Lubrication approximation with prescribed nonzero contact angleIn: Communications in partial differential equations, 23 (1998) 11/12, p. 20772164DOI: 10.1080/03605309808821411

Lorenzo Giacomelli and Felix Otto: Variational formulation for the lubrication approximation of the HeleShaw flowIn: Calculus of variations and partial differential equations, 13 (2001) 3, p. 377403MISPreprint: 32/2000 DOI: 10.1007/s005260000077

Lorenzo Giacomelli and Felix Otto: Droplet spreading : intermediate scaling law by PDE methodsIn: Communications on pure and applied mathematics, 55 (2002) 2, p. 217254MISPreprint: 74/2000 DOI: 10.1002/cpa.10017

Lorenzo Giacomelli and Felix Otto: Rigorous lubrication approximationIn: Interfaces and free boundaries, 5 (2003) 4, p. 483529DOI: 10.4171/IFB/88

Felix Otto ; Tobias Rump and Dejan Slepčev: Coarsening rates for a droplet model : rigorous upper boundsIn: SIAM journal on mathematical analysis, 38 (2006) 2, p. 503529DOI: 10.1137/050630192

Lorenzo Giacomelli ; Hans Knüpfer and Felix Otto: Smooth zerocontactangle solutions to a thinfilm equation around the steady stateIn: Journal of differential equations, 245 (2008) 6, p. 14541506DOI: 10.1016/j.jde.2008.06.005

Karl Glasner ; Felix Otto ; Tobias Rump and Dejan Slepčev: Ostwald ripening of droplets : the role of migrationIn: European journal of applied mathematics, 20 (2009) 1, p. 167DOI: 10.1017/S0956792508007559

Julian Fischer: Optimal lower bounds on asymptotic support propagation rates for the thinfilm equationIn: Journal of differential equations, 255 (2013) 10, p. 31273149DOI: 10.1016/j.jde.2013.07.028

Lorenzo Giacomelli ; Manuel V. Gnann and Felix Otto: Regularity of sourcetype solutions to the thinfilm equation with zero contact angle and mobility exponent between 3/2 and 3In: European journal of applied mathematics, 24 (2013) 5, p. 735760MISPreprint: 23/2012 DOI: 10.1017/S0956792513000156

Dominik John: On uniqueness of weak solutions for the thinfilm equationIn: Journal of differential equations, Vol. not yet known, pp. not yet knownARXIV: http://arxiv.org/abs/1310.6222

Julian Fischer: Behaviour of free boundaries in thinfilm flow : the regime of strong slippage and the regime of very weak slippageIn: Annales de l'Institut Henri Poincaré / C, Vol. not yet known, pp. not yet knownDOI: 10.1016/j.anihpc.2015.05.001

Julian Fischer: Infinite speed of support propagation for the DerridaLebowitzSpeerSpohn equation and quantum driftdiffusion modelsIn: Nonlinear differential equations and applications, 21 (2014) 1, p. 2750DOI: 10.1007/s0003001302350

Julian Fischer: Upper bounds on waiting times for the thinfilm equation : the Case of weak slippageIn: Archive for rational mechanics and analysis, 211 (2014) 3, p. 771818DOI: 10.1007/s0020501306900

Lorenzo Giacomelli ; Manuel V. Gnann ; Hans Knüpfer and Felix Otto: Wellposedness for the NavierSlip thinfilm equation in the case of complete wettingIn: Journal of differential equations, 257 (2014) 1, p. 1581MISPreprint: 1/2014 DOI: 10.1016/j.jde.2014.03.010

Manuel V. Gnann: Wellposedness and selfsimilar asymptotics for a thinfilm equationIn: SIAM journal on mathematical analysis, 47 (2015) 4, p. 28682902MISPreprint: 77/2014 DOI: 10.1137/14099190X

Elias Esselborn: Relaxation rates for a perturbation of a stationary solution to the thinfilm equationIn: SIAM journal on mathematical analysis, 48 (2016) 1, p. 349396MISPreprint: 27/2015 DOI: 10.1137/15M1017697

Lorenzo Giacomelli ; Manuel V. Gnann and Felix Otto: Rigorous asymptotics of travelingwave solutions to the thinfilm equation and Tanner's lawIn: Nonlinearity, 29 (2016) 9, p. 24972536MISPreprint: 45/2015 DOI: 10.1088/09517715/29/9/2497
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Presentations
 Logarithmic correction to droplet spreading rate because of Navier slip regularization (see PS, 222 Kbyte)
 Shorttime existence theory of smooth solutions based on linear theory (see PDF, 1.1 Mbyte)
 Towards a regularity theory for the moving contact line (see PDF, 289 Kbyte)