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 Itô noise.</p

The Cox-Voinov law for traveling waves in the partial wetting regime

We consider the thin-film equation ∂th+∂ym(h)∂y3h=0 in {h &gt; 0} with partial-wetting boundary conditions and inhomogeneous mobility of the form m(h) = h 3 + λ 3-n h n , where h ∼ 0 is the film height, λ &gt; 0 is the slip length, y &gt; 0 denotes the lateral variable, and n ϵ (0, 3) is the mobility exponent parameterizing the nonlinear slip condition. The partial-wetting regime implies the boundary condition ∂ y h = const. &gt; 0 at the triple junction ∂{h &gt; 0} (nonzero microscopic contact angle). Existence and uniqueness of traveling-wave solutions to this problem under the constraint ∂y2h→0 as h → ∞ have been proved in previous work by Chiricotto and Giacomelli (2011 Commun. Appl. Ind. Math. 2 e-388, 16). We are interested in the asymptotics as h ↓ 0 and h → ∞. By reformulating the problem as h ↓ 0 as a dynamical system for the difference 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 ↓ 0 depending on n. Together with the asymptotics as h → ∞ characterizing the Cox-Voinov law for the velocity-dependent macroscopic contact angle as found by Giacomelli, the first author of this work, and Otto (2016 Nonlinearity 29 2497-536), the rigorous asymptotics of traveling-wave solutions 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 the Cox-Voinov law for the macroscopic contact angle depends continuously differentiably on the microscopic contact angle. Analysi

Multiscale analysis for traveling-pulse solutions to the stochastic fitzhugh–nagumo equations

We investigate the stability of traveling-pulse solutions to the stochastic FitzHugh–Nagumo equations with additive noise. Special attention is given to the effect of small noise on the classical deterministically stable fast traveling pulse. Our method is based on adapting the velocity of the traveling wave by solving a scalar stochastic ordinary differential equation (SODE) and tracking perturbations to the wave meeting a system of a scalar stochastic partial differential equation (SPDE) coupled to a scalar ordinary differential equation (ODE). This approach has been recently employed by Krüger and Stannat (Nonlinear Anal. 162 (2017) 197–223) for scalar stochastic bistable reaction–diffusion equations such as the Nagumo equation. A main difference in our situation of an SPDE coupled to an ODE is that the linearization has essential spectrum parallel to the imaginary axis and thus only generates a strongly continuous semigroup. Furthermore, the linearization around the traveling wave is not self-adjoint anymore, so that fluctuations around the wave cannot be expected to be orthogonal in a corresponding inner product. We demonstrate that this problem can be overcome by making use of Riesz instead of orthogonal spectral projections as recently employed in a series of papers by Hamster and Hupkes in case of analytic semigroups. We expect that our approach can also be applied to traveling waves and other patterns in more general situations such as systems of SPDEs with linearizations only generating a strongly continuous semigroup. This provides a relevant generalization as these systems are prevalent in many applications.Green Open Access added to TU Delft Institutional Repository ‘You share, we take care!’ – Taverne project https://www.openaccess.nl/en/you-share-we-take-care Otherwise as indicated in the copyright section: the publisher is the copyright holder of this work and the author uses the Dutch legislation to make this work public.Analysi

Droplet motion with contact-line friction: long-time asymptotics in complete wetting

We consider the thin-film equation for a class of free boundary conditions modelling friction at the contact line, as introduced by E and Ren. Our analysis focuses on formal long-time asymptotics of solutions in the perfect wetting regime. In particular, through the analysis of quasi-self-similar solutions, we characterize the profile and the spreading rate of solutions depending on the strength of friction at the contact line, as well as their (global or local) corrections, which are due to the dynamical nature of the free boundary conditions. These results are complemented with full transient numerical solutions of the free boundary problem. Green Open Access added to TU Delft Institutional Repository ‘You share, we take care!’ – Taverne project https://www.openaccess.nl/en/you-share-we-take-care Otherwise as indicated in the copyright section: the publisher is the copyright holder of this work and the author uses the Dutch legislation to make this work public.Mathematical Physic

Stability of receding traveling waves for a fourth order degenerate parabolic free boundary problem

Consider the thin-film equation h t +(hh yyy ) y =0 with a zero contact angle at the free boundary, that is, at the triple junction where liquid, gas, and solid meet. Previous results on stability and well-posedness of this equation have focused on perturbations of equilibrium-stationary or self-similar profiles, the latter eventually wetting the whole surface. These solutions have their counterparts for the second-order porous-medium equation h t −(h m ) yy =0, where m&gt;1 is a free parameter. Both porous-medium and thin-film equation degenerate as h↘0, but the porous-medium equation additionally fulfills a comparison principle while the thin-film equation does not. In this note, we consider traveling waves h=[Formula presented]x 3 +νx 2 for x≥0, where x=y−Vt and V,ν≥0 are free parameters. These traveling waves are receding and therefore describe de-wetting, a phenomenon genuinely linked to the fourth-order nature of the thin-film equation and not encountered in the porous-medium case as it violates the comparison principle. The linear stability analysis leads to a linear fourth-order degenerate-parabolic operator for which we prove maximal-regularity estimates to arbitrary orders of the expansion in x in a right-neighborhood of the contact line x=0. This leads to a well-posedness and stability result for the corresponding nonlinear equation. As the linearized evolution has different scaling as x↘0 and x→∞ the analysis is more intricate than in related previous works. We anticipate that our approach is a natural step towards investigating other situations in which the comparison principle is violated, such as droplet rupture. Accepted author manuscriptAnalysi

Computer simulations of structure, dynamics, and phase behavior of colloidal fluids in confined geometry and under shear

Using computer simulations, colloidal systems in different external fields are investigated. Colloid-polymer mixtures, described in terms of the Asakura-Oosawa (AO) model, are considered under strong confinement. Both in cylindrical and spherical confinement, the demixing transition of the three-dimensional AO model is rounded and, using Monte Carlo simulations, we analyze in detail the consequences of this rounding (occurrence of multi-domain states in cylindrical geometry, non-equivalence of conjugate ensembles due to different finite-size corrections in spherical geometry etc.). For the case of the AO model confined between two parallel walls, spinodal decomposition is studied using a combination of molecular dynamics simulation and the multiparticle collision dynamics method. This allows us to investigate the influence of hydrodynamic interactions on the domain growth during spinodal decomposition. For a binary glass-forming Yukawa mixture, non-linear active micro-rheology is considered, i.e. a single particle is pulled through a deeply supercooled liquid. The diffusion dynamics of the pulled particle is analyzed in terms of the van Hove correlation function. Finally, the Yukawa mixture in the glass state, confined between walls, is studied under the imposition of a uniform shear stress. Below and around the yield stress, persistent creep in the form of shear-banded structures is observed