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...

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...

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 > 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 > 0 is the slip length, y > 0 denotes the lateral variable, and n ϵ (0, 3) is the mobility exponent parameterizing the nonlinear slip...

Non-negative Martingale Solutions to the Stochastic Thin-Film Equation with Nonlinear Gradient Noise

We prove the existence of non-negative martingale solutions to a class of stochastic degenerate-parabolic fourth-order PDEs arising in surface-tension driven thin-film flow influenced by thermal noise. The construction applies to a range of mobilites including the cubic one which occurs under the assumption of a no-slip condition at the...

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...

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...