Related papers: The thin film equation with backwards second order…
We analyze the Fick's diffusion of a gas inside porous nanomaterials through the one-dimensional diffusion equation in nanopores for various cases of boundary conditions for homogeneous and non-homogeneous problems. We study the diffusion…
A fully coupled system of two second-order parabolic degenerate equations arising as a thin film approximation to the Muskat problem is interpreted as a gradient flow for the 2-Wasserstein distance in the space of probability measures with…
Finite speed of propagation is established for non-negative weak solutions to a thin film approximation of the two-phase Muskat problem. The temporal expansion rate of the support matches the scale invariance of the system. Moreover, we…
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…
Thin liquid films are ubiquitous in natural phenomena and technological applications. They have been extensively studied via deterministic hydrodynamic equations, but thermal fluctuations often play a crucial role that needs to be…
In this paper we present a mathematical analysis of thin film flow and heat transfer to a laminar liquid film from a horizontal stretching sheet. The flow of thin liquid film and subsequent heat transfer from the stretching surface is…
Fundamental global similarity solutions of the tenth-order thin film equation u_{t} = \nabla \cdot(|u|^{n} \n \D^4 u) in R^N \times R_+, where n>0 are studied. The main approach consists in passing to the limit n \to 0^+ by using Hermitian…
This article starts over the backwards diffusion problem by replacing the \emph{noncausal} diffusion equation, the direct problem, by the \emph{causal} diffusion model developed in \cite{Kow11} for the case of constant diffusion speed. For…
We present an analytical treatment of a three-dimensional variational model of a system that exhibits a second-order phase transition in the presence of dipolar interactions. Within the framework of Ginzburg-Landau theory, we concentrate on…
Motivated by lubrication problems, we consider a micropolar uid ow in a 2D domain with a rough and free boundary. We assume that the thickness and the roughness are both of order 0 < " << 1. We prove the existence and uniqueness of a…
In this contribution we prove the existence of weak solutions to degenerate parabolic systems arising from the coupled moisture movement, transport of dissolved species and heat transfer through partially saturated porous materials.…
The degenerate parabolic equation u_t + [u^3(u_xxx + u_x - sin x)]_x=0 models the evolution of a thin liquid film on a stationary horizontal cylinder. It is shown here that for each given mass there is a unique steady state, given by a…
We derive the variational limiting theory of thin films, parallel to the F\"oppl-von K\'arm\'an theory in the nonlinear elasticity, for films that have been prestrained and whose thickness is a general non-constant function. Using…
We investigate the dynamics of spreading of a small liquid droplet in gas in a one-component simple fluid, where the temperature is inhomogeneous around 0.9Tc and latent heat is released or generated at the interface upon evaporation or…
In this paper we consider a two-phase thin film consisting of two immiscible viscous fluids endowed with a layer of insoluble surfactant on the surface of the upper fluid. The governing equations for the two film heights and the surfactant…
We consider the Ginzburg-Landau heat flow without magnetic effect in a curved thin domain under the Naumann boundary condition. When the curved thin domain shrinks to a given closed hypersurface as the thickness of the thin domain tends to…
In this paper, we discuss a particular model arising from sinking of a rigid solid into a thin film of fluid, i.e. a fluid contained between two solid surfaces and part of the fluid surface is in contact with the air. The fluid is governed…
We consider here the effects of inertia on the instability of a flat liquid film under the effects of capillary and intermolecular forces (van der Waals interaction). Firstly, we perform the linear stability analysis within the long wave…
The inverse problem of backward diffusion is known to be ill-posed and highly unstable. Backward diffusion processes appear naturally in image enhancement and deblurring applications. It is therefore greatly desirable to establish a…
In any number of space variables, we study the Cauchy problem related to the thin-film equation in the simplest case of a linearly degenerate mobility. This equation, derived from a lubrication approximation, also models the surface tension…