Related papers: Self-similar lifting and persistent touch-down poi…
We consider asymptotically self-similar blow-up profiles of the thin film equation consisting of a stabilising fourth order and destabilising second order term. It has previously been shown that blow up is only possible when the exponent in…
It is shown that a Riemann-type problem with discontinuous data of sign-type for the thin film equation, which degenerates at +1 and -1, admits a self-similar solution. Both FBP and the Cauchy problem (oscillatory solutions near interfaces)…
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…
In the present work, we study well-posedness and regularity of the multidimensional thin film equation with linear mobility in a neighborhood of the self-similar Smyth--Hill solutions. To be more specific, we perform a von Mises change of…
We study a higher-dimensional thin film equation that incorporates competitive effects between aggregation and repulsion, where repulsion is modeled by fourth-order diffusion and aggregation by backward second-order degenerate diffusion,…
We are interested in a complete characterization of the contact-line singularity of thin-film flows for zero and nonzero contact angles. By treating the model problem of source-type self-similar solutions, we demonstrate that this…
The large time behavior of non-negative weak solutions to a thin film approximation of the two-phase Muskat problem is studied. A classification of self-similar solutions is first provided: there is always a unique even self-similar…
We consider the thin-film equation with linear mobility and a stabilizing second-order porous-medium type term modeling gravity. The model admits self-similar solutions, and our goal is to analyze their stability. We reformulate the problem…
Continuous branches of similarity solutions of the fourth-order stable thin film equation are studied.
We investigate stationary solutions of a thin-film model for liquid two-layer flows in an energetic formulation that is motivated by its gradient flow structure. The goal is to achieve a rigorous understanding of the contact-angle…
The surface structure of converging thin fluid films displays self-similar behavior, as was shown in the work by Diez et al [Q. Appl. Math 210, 155, 1990]. Extracting the related similarity scaling exponents from either numerical or…
In this paper, self-similar solutions for a fractional thin film equation governing hydraulic fractures are constructed. One of the boundary conditions, which accounts for the energy required to break the rock, involves the toughness…
We study the flow of two immiscible fluids located on a solid bottom, where the lower fluid is Newtonian and the upper fluid is a non-Newtonian Ellis fluid. Neglecting gravitational effects, we consider the formal asymptotic limit of small…
We compare the capillary levelling of a random surface perturbation on a thin polystyrene film with a theoretical study on the two-dimensional capillary-driven thin film equation. Using atomic force microscopy, we follow the time evolution…
The Geometric Thin-Film equation is a mathematical model of droplet spreading in the long-wave limit, which includes a regularization of the contact-line singularity. We show that the weak formulation of the problem, given initial Radon…
We consider a third order non-autonomous ODE that arises as a model of fluid accumulation in a two dimensional thin-film flow driven by surface tension and gravity. With the appropriate matching conditions, the equation describes the inner…
A thin liquid film coating a planar horizontal substrate may be unstable to perturbations in the film thickness due to unfavourable intermolecular interactions between the liquid and the substrate, which may lead to finite-time rupture. The…
The gravitational interaction is scale-free in both Newtonian gravity and general theory of relativity. The concept of self-similarity arises from this nature. Self-similar solutions reproduce themselves as the scale changes. This property…
In this paper we consider a system of equations that describes a class of mass-conserving aggregation phenomena, including gravitational collapse and bacterial chemotaxis. In spatial dimensions strictly larger than two, and under the…
The surface of a thin liquid film with nonconstant curvature is unstable, as the Laplace pressure drives a flow mediated by viscosity. We present the results of experiments on one of the simplest variable curvature surfaces: a stepped…