Relaxation to equilibrium in the one-dimensional thin-film equation with partial wetting

Microscopical Justification of the Winterbottom problem for well-separated Lattices

We consider the discrete atomistic setting introduced in \cite{PiVe1} to microscopically justify the continuum model related to the \emph{Winterbottom problem}, i.e., the problem of determining the equilibrium shape of crystalline film drops resting on a substrate, and relax the rigidity assumption considered in \cite{PiVe1} to characterize the \emph{wetting} and \emph{dewetting regimes} and to perform the \emph{discrete to continuum passage}. In particular, all results of \cite{PiVe1} are extended to the setting where the distance between the reference lattices for the film and the substrate …

Invariant Manifolds for the Thin Film Equation

The large-time behavior of solutions to the thin film equation with linear mobility in the complete wetting regime on $\mathbb{R}^N$ is examined: We investigate the higher order asymptotics of solutions converging towards self-similar Smyth--Hill solutions under certain symmetry assumptions on the initial data. The analysis is based on a construction of finite-dimensional invariant manifolds that solutions approximate to an arbitrarily prescribed order.

The Cox-Voinov 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 $m(h) = h^3+λ^{3-n}h^n$, where $h \ge 0$ is the film height, $λ> 0$ 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 un…

接触线摩擦下的液滴运动：完全润湿情形下的长时间渐近行为

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

本文研究完全润湿情形下带接触线摩擦的薄液膜方程，利用准自相似解刻画液滴铺展速率与轮廓的长时间渐近行为，并结合瞬态数值解验证理论结果。

完美润湿状态下具有一般迁移率的薄膜方程的经典解

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

本文利用时间 $L^p$-估计和泛函分析方法，在完美润湿状态下对一般迁移率 $m(h)=h^n$ 的薄膜方程证明了行波解的适定性、部分正则性与渐近稳定性，扩展了现有结果中迁移率指数的适用范围。

渐近稳定性与强迫项：非牛顿薄液膜流动的分析

Asymptotic Stability and the Forcing Term: An Analysis of Non-Newtonian Thin-Film Flows

本文研究了非牛顿薄液膜流动在固态表面的"完全润湿"状态下，通过分析四阶拟线性退化抛物型方程的解的收敛速率，推导出双侧估计并验证了物理现象的一致性。

Weak solutions to a two-phase thin film model with insoluble surfactant driven by capillary effects

Of concern is the study of a system of three equations describing the motion of a viscous complete wetting two-phase thin film endowed with a layer of insoluble surfactant on the surface of the upper fluid under the effects of capillary forces. The governing equations for the film heights of the two-phase flow are degenerate, parabolic and strongly coupled fourth-order equations, which are additionally coupled to a second-order parabolic transport equation for the surfactant concentration. A result on the existence of non-negative global weak solutions is presented.

去润湿区域中弹性薄膜岛的运动

Motion of Islands of Elastic Thin Films in the Dewetting Regime

本文针对薄膜固态去润湿二维变分模型，利用$H^{-1}$梯度流结构建立带曲率正则化的表面扩散方程短时存在性，创新在于考虑移动接触线。

Gradient flow structure for domain relaxation in Langmuir films

We describe a gradient flow structure for the inviscid Langmuir layer Stokesian subfluid model introduced recently by Alexander et al [1].

Electrified thin films: Global existence of non-negative solutions

We consider an equation modeling the evolution of a viscous liquid thin film wetting a horizontal solid substrate destabilized by an electric field normal to the substrate. The effects of the electric field are modeled by a lower order non-local term. We introduce the good functional analysis framework to study this equation on a bounded domain and prove the existence of weak solutions defined globally in time for general initial data (with finite energy).