The stochastic thin-film equation: existence of nonnegative martingale solutions

We consider the stochastic thin-film equation with colored Gaussian Stratonovich noise in one space dimension and establish the existence of nonnegative weak (martingale) solutions. The construction is based on a Trotter-Kato-type decomposition into a deterministic and a stochastic evolution, which yields an easy to implement numerical algorithm. Compared to previous work, no interface potential has to be included, the initial data and the solution can have de-wetted regions of positive measure, and the Trotter-Kato scheme allows for a simpler proof of existence than in case of It\^o noise.Comment: 38 pages, revised version, nonnegativity proof changed, details to time regularity and interpolation of operators adde

The Navier-slip thin-film equation for 3D fluid films: existence and uniqueness

We consider the thin-film equation $\partial_t h + \nabla \cdot \left(h^2 \nabla \Delta h\right) = 0$ in physical space dimensions (i.e., one dimension in time $t$ and two lateral dimensions with $h$ denoting the height of the film in the third spatial dimension), which corresponds to the lubrication approximation of the Navier-Stokes equations of a three-dimensional viscous thin fluid film with Navier-slip at the substrate. This equation can have a free boundary (the contact line), moving with finite speed, at which we assume a zero contact angle condition (complete-wetting regime). Previous results have focused on the $1+1$-dimensional version, where it has been found that solutions are not smooth as a function of the distance to the free boundary. In particular, a well-posedness and regularity theory is more intricate than for the second-order counterpart, the porous-medium equation, or the thin-film equation with linear mobility (corresponding to Darcy dynamics in the Hele-Shaw cell). Here, we prove existence and uniqueness of classical solutions that are perturbations of an asymptotically stable traveling-wave profile. This leads to control on the free boundary and in particular its velocity.Comment: 86 pages, 2 figures, revised version; norms, embeddings, and nonlinear estimates correcte

Well-posedness and self-similar asymptotics for a thin-film equation

We investigate compactly supported solutions for a thin-film equation with linear mobility in the regime of perfect wetting. This problem has already been addressed by Carrillo and Toscani, proving that the source-type self-similar profile is a global attractor of entropy solutions with compactly supported initial data. Here we study small perturbations of source-type self-similar solutions for the corresponding classical free boundary problem and set up a global existence and uniqueness theory within weighted L2-spaces under minimal assumptions. Furthermore, we derive asymptotics for the evolution of the solution, the free boundary, and the center of mass. As spatial translations are scaled out in our reference frame, the rate of convergence is higher than the one obtained by Carrillo and Toscani.Funding of this work was provided by the International Max Planck Research School (IMPRS) of the Max Planck Institute for Mathematics in the Sciences (MPI MIS) in Leipzig and the Fields Institute in Toronto. The author’s research was partially supported by the National Science Foundation under grant NSF DMS-1054115.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/117366/1/Gnann_2015.pdfDescription of Gnann_2015.pdf : Main article (published version

Classical solutions to the thin-film equation with general mobility in the perfect-wetting regime

We prove well-posedness, partial regularity, and stability of a thin-film equation $h_t + (m(h) h_{zzz})_z = 0$ with general mobility $m(h) = h^n$ and mobility exponent $n\in (1,\tfrac{3}{2})\cup (\tfrac{3}{2},3)$ in the regime of perfect wetting (zero contact angle). After a suitable coordinate transformation to fix the free boundary (the contact line where liquid, air, and solid coalesce), the thin-film equation is rewritten as an abstract Cauchy problem and we obtain maximal $L^{p}_t$-regularity for the linearized evolution. Partial regularity close to the free boundary is obtained by studying the elliptic regularity of the spatial part of the linearization. This yields solutions that are non-smooth in the distance to the free boundary, in line with previous findings for source-type self-similar solutions. In a scaling-wise quasi-minimal norm for the initial data, we obtain a well-posedness and asymptotic stability result for perturbations of traveling waves. The novelty of this work lies in the usage of $L^{p}$ estimates in time, where $1 < p < \infty$, while the existing literature mostly deals with $p = 2$ at least for nonlinear mobilities. This turns out to be essential to obtain for the first time a well-posedness result in the perfect-wetting regime for all physical nonlinear slip conditions except for a strongly degenerate case at $n = \tfrac 3 2$ and the well-understood perfect-slip case $n = 1$. Furthermore, compared to [J. Differential Equations, 257(1):15-81, 2014] by Giacomelli, the first author of this paper, Kn\"upfer, and Otto, where a PDE-approach yields $L^2_t$-estimates, well-posedness, and stability for $1.8384 \approx \tfrac{3}{17}(15-\sqrt{21}) < n < \tfrac{3}{11}(7+\sqrt{5}) \approx 2.5189$, our functional-analytic approach is significantly shorter while at the same time giving a more general result.Comment: 40 pages, 4 figure

Tanner's law for traveling waves in the partial wetting regime

We consider the thin-film equation $\partial_t h + \partial_y \left(m(h) \partial_y^3 h\right) = 0$ in $\{h > 0\}$ with partial-wetting boundary conditions and inhomogeneous mobility of the form $m(h) = h^3+\lambda^{3-n}h^n$, where $h \ge 0$ is the film height, $\lambda > 0$ is the slip length, $y > 0$ denotes the lateral variable, and $n \in (0,3)$ is the mobility exponent parameterizing the nonlinear slip condition. The partial-wetting regime implies the boundary condition $\partial_y h = \mathrm{const.} > 0$ at the triple junction $\partial\{h > 0\}$ (nonzero microscopic contact angle). Existence and uniqueness of traveling-wave solutions to this problem under the constraint $\partial_y^2 h \to 0$ as $h \to \infty$ have been proved in previous work by Chiricotto and Giacomelli in [Commun. Appl. Ind. Math., 2(2):e-388, 16, 2011]. We are interested in the asymptotics as $h \downarrow 0$ and $h \to \infty$. By reformulating the problem as $h \downarrow 0$ as a dynamical system for the error between the solution and the microscopic contact angle, values for $n$ are found for which linear as well as nonlinear resonances occur. These resonances lead to a different asymptotic behavior of the solution as $h\downarrow0$ depending on $n$. Together with the asymptotics as $h\to\infty$ characterizing Tanner's law for the velocity-dependent macroscopic contact angle as found by Giacomelli, the first author of this work, and Otto in [Nonlinearity, 29(9):2497-2536, 2016], the rigorous asymptotics of the traveling-wave solution to the thin-film equation in partial wetting can be characterized. Furthermore, our approach enables us to analyze the relation between the microscopic and macroscopic contact angle. It is found that Tanner's law for the macroscopic contact angle depends continuously differentiably on the microscopic contact angle.Comment: 23 pages, 3 figure

Well-Posedness of a Stochastic Parametrically-Forced Nonlinear Schr\"odinger Equation

We prove well-posedness of a parametrically-forced nonlinear Schr\"odinger equation (PFNLS) with multiplicative Stratonovitch noise in $L^2(\mathbb{R})$ and $H^1(\mathbb{R})$. The noise is white in time and correlated in space. We construct mild solutions via a fixed-point argument and blow-up criterion. Proving non-blow-up requires analysis on the evolution of spatial norms of the local solutions, as the forcing terms break the conservation laws of the NLS equation. We rely on It\^o's lemma to characterize this evolution, and carry out a regularization procedure that justifies the use of It\^o's formula on the mild solutions.Comment: 38 page

Rigorous asymptotics of traveling-wave solutions to the thin-film equation and Tanner's law

We are interested in traveling-wave solutions to the thin-film equation with zero microscopic contact angle (in the sense of complete wetting without precursor) and inhomogeneous mobility with slippage exponent n ∈ (3/2,7/3). Existence and uniqueness of these solutions have been established by Maria Chiricotto and the first of the authors in previous work under the assumption of subquadratic growth as h → ∞. In the present work we investigate the asymptotics of solutions as h → 0 (the contact-line region) and h → ∞. As h → 0 we observe, to leading order, the same asymptotics as for traveling waves or source-type self-similar solutions to the thin-film equation  with homogeneous mobility and we additionally characterize corrections to this law. Moreover, as h → ∞ we identify, to leading order, the logarithmic Tanner profile, i.e. the solution to the corresponding homogeneous problem that determines the apparent macroscopic contact angle. Besides higher-order terms, corrections turn out to affect the asymptotic law as h → ∞ only by setting the length scale in the logarithmic Tanner profile. Moreover, we prove that both the correction and the length scale depend smoothly on n. Hence, in line with the common philosophy, the precise modeling of liquid–solid interactions (within our model, the mobility exponent) does not affect the qualitative macroscopic properties of the film