多维分数阶薄膜方程解的存在性与有限传播速度

Existence and behavior of asymmetric traveling wave solutions to thin film equation

We proved the existence and uniqueness of a traveling wave solution to the thin film equation with a Navier slip condition at the liquid-solid interface. We obtain explicit lower and upper bounds for the solution and an absolute error estimate of approximation of a solution to the thin films equation by the traveling-wave solution.

Justification of a nonlinear sixth-order thin-film equation as the reduced model for a fluid -- structure interaction problem

Starting from a nonlinear 2D/1D fluid-structure interaction problem between a thin layer of a viscous fluid and a thin elastic structure, on the vanishing limit of the relative fluid thickness, we rigorously derive a sixth-order thin-film equation describing the dynamics of vertical displacements of the structure. The procedure is essentially based on quantitative energy estimates, quantified in terms of the relative fluid thickness, and a uniform no-contact result between the structure and the solid substrate. The sixth-order thin-film equation is justified in the sense of strong convergence …

二维情况下薄膜方程弱解的Hölder连续性

Hölder continuity of weak solutions to the thin-film equation in $d=2$

本文研究了二维情况下薄膜方程弱解的Hölder连续性，通过孔填充技术克服了四阶偏微分方程的退化抛物性挑战。

Some thin film models based on Shikhmurzaev's approach

We derive several lubrication approximation models, using Shikhmurzaev's approach to the contact line problem, obtained in \cite{GNV}. The first two lubrication models describe thin film flow of incompressible fluids on solid substrates, based on different orders of magnitude of the slip length parameter. The third lubrication model describes a meniscus formation where a wedge-shaped solid immerses in a thin film of fluid.

On the thin-film equation with nonlinear surface tension term

In this thesis, we consider the thin-film equation with nonlinear surface tension term in one space dimension. Relying on the corresponding energy and entropy estimates, we prove existence of weak solutions as well as nonnegativity results.

分数Sobolev空间在低积分性 regime 下的渐进行为

Asymptotic analysis of fractional Sobolev spaces on thin films in the low-integrability regime

研究薄片上分数Sobolev空间的渐进行为，证明其趋近于H^{s+1/2}(ω)空间，通过维度缩减收敛概念和适当缩放的Gagliardo半范数定义等 coercive 函数。

Thin Film Equations with Soluble Surfactant and Gravity: Modeling and Stability of Steady States

A thin film on a horizontal solid substrate and coated with a soluble surfactant is considered. The governing degenerate parabolic equations for the film height and the surfactant concentrations on the surface and in the bulk are derived using a lubrication approximation when gravity is taken into account. It is shown that the steady states are asymptotically stable.

A Mathematical Justification of a Thin Film Approximation for the Flow down an Inclined Plane

We consider a two-dimensional motion of a thin film flowing down an inclined plane under the influence of the gravity and the surface tension. In order to investigate the stability of such flow, we often use a thin film approximation, which is an approximation obtained by the perturbation expansion with respect to the aspect ratio of the film. The famous example of the approximate equations are the Burgers equation, Kuramoto--Sivashinsky equation, KdV--Burgers equation, KdV--Kuramoto--Sivashinsky equation, and so on. In this paper, we give a mathematically rigorous justification of a thin film…

Uniform Estimates for the Flow of a Viscous Incompressible Fluid down an Inclined Plane in the Thin Film Regime

We consider a two-dimensional motion of a thin film flowing down an inclined plane under the influence of the gravity and the surface tension. In order to investigate the stability of such flow, it is hard to treat the Navier--Stokes equations directly, so that a thin film approximation is often used. It is an approximation obtained by the perturbation expansion with respect to the aspect ratio $δ$ of the film under the thin film regime $δ\ll1$. Our purpose is to give a mathematically rigorous justification of the thin film approximation by establishing an error estimate between the solution o…

Local Bifurcation-Branching Analysis of Global and "Blow-up" Patterns for a Fourth-Order Thin Film Equation

The quasilinear fourth-order thin film equation in $R^N$ is shown to admit infinite countable families of global and blow-up similarity solutions, which are originated in the limit from eigenfunctions of the corresponding linear non-self-adjoint operators.

On self-similar collapse of discontinuous data for thin film equations with doubly degenerate mobility

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) settings are taken into account.

Branching analysis of a countable family of global similarity solutions of a fourth-order thin film equation

We show that various asymptotic properties of global solutions of a fourth-order quasilinear thin film equation can be described by branching from corresponding solutions of the linear bi-harmonic equation. This includes a countable family of self-similar solutions and other aspects.

Asymptotic analysis for a second order curved thin film

We consider a second order thin curved film whose behavior is governed by an energy made up of a first order nonlinear part depending on the gradient of the deformation augmented by a quadratic second order part depending on the tensor of second derivatives of the deformation. We carry out a 3D-2D analysis through an asymptotic expansion in powers of the thickness of the film as it tends to zero.

Singular Limits for Thin Film Superconductors in Strong Magnetic Fields - Maan Field Model for Thin Films

We consider singular limits of the three-dimensional Ginzburg-Landau functional for a superconductor with thin-film geometry, in a constant external magnetic field. The superconducting domain has characteristic thickness on the scale $\eps>0$, and we consider the simultaneous limit as the thickness $\eps\rightarrow 0$ and the Ginzburg-Landau parameter $κ\rightarrow\infty$. We assume that the applied field is strong (on the order of $\eps^{-1}$ in magnitude) in its components tangential to the film domain, and of order $\logκ$ in its dependence on $κ$. We prove that the Ginzburg-Landau energy $…

A note on the homogenization of incommensurate thin films

Dimension-reduction homogenization results for thin films have been obtained under hypotheses of periodicity or almost-periodicity of the energies in the directions of the mid-plane of the film. In this note we consider thin films, obtained as sections of a periodic medium with a mid-plane that may be incommensurate; that is, not containing periods other than $0$. A geometric almost-periodicity argument similar to the cut-and-project argument used for quasicrystals allows to prove a general homogenization result.