A fourth order parabolic equation with nonlinear principal part

In this paper, we study a fourth order parabolic equation with nonlinear principal part modeling epitaxial thin film growth in two space dimensions. On the basis of the Schauder type estimates and Campanato spaces, we prove the global existence of classical solutions.     

A fourth-order parabolic equation modeling epitaxial thin film growth

We study the continuum model for epitaxial thin film growth from Phys. D 132 (1999) 520-542, which is known to simulate experimentally observed dynamics very well. We show existence, uniqueness and regularity of solutions in an appropriate function space, and we characterize the existence of nontrivial equilibria in terms of the size of the underlying domain. In an investigation of asymptotical behavior, we give a weak assumption under which the ω-limit set of the dynamical system consists only of steady states. In the one-dimensional setting we can characterize the set of steady states and determine its unique asymptotically stable element. The article closes with some illustrative numerical examples.     

一个广义薄膜方程弱解的存在性

讨论一个广义薄膜方程,在一些初值的假定下,用时间离散化方法证明其弱解的存在性.     

A GENERALIZED THIN FILM EQUATION

The authors study a generalized thin film equation. Under some assumptions on theinitial value, the existence of weak solutions is established by the time-discrete method.The uniqueness and asymptotic behavior of solutions are also discussed.     

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).     

Existence of Weak Solutions to a Class of Fourth Order Partial Differential Equations with Wasserstein Gradient Structure

We prove the global-in-time existence of nonnegative weak solutions to a class of fourth order partial differential equations on a convex bounded domain in arbitrary spatial dimensions. Our proof relies on the formal gradient flow structure of the equation with respect to the L²-Wasserstein distance on the space of probability measures. We construct a weak solution by approximation via the time-discrete minimizing movement scheme; necessary compactness estimates are derived by entropy-dissipation methods. Our theory essentially comprises the thin film and Derrida-Lebowitz-Speer-Spohn equations.     

Stability and regularity of weak solutions for a generalized thin film equation

In this paper we establish a stability result and an error estimate of weak solutions for the initial-boundary value problem of a generalized thin film equation and also obtain some higher regularity results for weak solutions.     

Cahn–Hilliard and thin film equations with nonlinear mobility as gradient flows in weighted-Wasserstein metrics

In this paper, we establish a novel approach to proving existence of non-negative weak solutions for degenerate parabolic equations of fourth order, like the Cahn–Hilliard and certain thin film equations. The considered evolution equations are in the form of a gradient flow for a perturbed Dirichlet energy with respect to a Wasserstein-like transport metric, and weak solutions are obtained as curves of maximal slope. Our main assumption is that the mobility of the particles is a concave function of their spatial density. A qualitative difference of our approach to previous ones is that essential properties of the solution – non-negativity, conservation of the total mass and dissipation of the energy – are automatically guaranteed by the construction from minimizing movements in the energy landscape.     

Existence and stability of weak solutions for a degenerate parabolic system modelling two-phase flows in porous media

We prove global existence of nonnegative weak solutions to a degenerate parabolic system which models the interaction of two thin fluid films in a porous medium. Furthermore, we show that these weak solutions converge at an exponential rate towards flat equilibria.     

A fourth‐order parabolic equation in two space dimensions

In this paper, we consider an initial‐boundary problem for a fourth‐order nonlinear parabolic equations. The problem as a model arises in epitaxial growth of nanoscale thin films. Based on the L^p type estimates and Schauder type estimates, we prove the global existence of classical solutions for the problem in two space dimensions. Copyright © 2007 John Wiley & Sons, Ltd.     

Finite Speed of Propagation of Perturbations for the Cahn-Hilliard Equation with Degenerate Mobility

This paper is devoted to the Cabn-Hilliard equation with degenerate mobility in two spatial variables with a typical case modelling thin viscous film spreading over a solid surface. We establish tbe existence of radial symmetric solutions with the property of finite speed of perturbations.     

Kinetic Equation for Soliton Gas: Integrable Reductions

We study the dynamic behaviour of two viscous fluid films confined between two concentric cylinders rotating at a small relative velocity. It is assumed that the fluids are immiscible and that the volume of the outer fluid film is large compared to the volume of the inner one. Moreover, while the outer fluid is considered to have constant viscosity, the rheological behaviour of the inner thin film is determined by a strain-dependent power-law. Starting from a Navier–Stokes system, we formally derive evolution equations for the interface separating the two fluids. Two competing effects drive the dynamics of the interface, namely the surface tension and the shear stresses induced by the rotation of the cylinders. When the two effects are comparable, the solutions behave, for large times, as in the Newtonian regime. We also study the regime in which the surface tension effects dominate the stresses induced by the rotation of the cylinders. In this case, we prove local existence of positive weak solutions both for shear-thinning and shear-thickening fluids. In the latter case, we show that interfaces which are initially close to a circle converge to a circle in finite time and keep that shape for later times.

     

Lubrication Approximation for Thin Viscous Films: Asymptotic Behavior of Nonnegative Solutions

We use standard regularized equations and adapted entropy functionals to prove exponential asymptotic decay in the H ¹ norm for nonnegative weak solutions of fourth-order nonlinear degenerate parabolic equations of lubrication approximation for thin viscous film type. The weak solutions considered arise as limits of solutions for the regularized problems. Relaxed problems, with second-order nonlinear terms of porous media type are also successfully treated by the same means. The problems investigated here are one-dimensional in space, with power-law nonlinearities. Our approach is direct and natural, as it is adapted to deal with the more complex nonlinear terms occurring in the regularized, approximating problems.     

Epitaxial growth and electrical transport properties of La0.5Sr0.5Co03 thin films prepared by pulsed laser deposition

Epitaxial growth of the La_0.5Sr_{0.5 3}(LO) thin films has been realized on Lin3, SrTiC₃ and MgO substrates by pulsed laser deposition. The epitaxial growth behavior and the electrical transport properties of these films were studied systematically. The temperature dependencies of the resistivity of the film have been determined. Studies indicate that close dependencies exist between the crystal structures and the electrical transport properties of the epitaxial LSCO films, and that the epitaxial thin films are of low resistivity and metallic conductive features. The epitaxial films deposited on the LaA10₃ substrates at about 700 °C possess the optimal properties compared with the others. Discussions of the dependencies and the mechanisms of the epitaxial structures on the electrical transport properties of the LSCO films have been made. Project supported by the National Natural Science Foundation of China (Grant No. 19574003 and No. 19674001).     

On a non-isothermal,non-Newtonian lubrication problem with Tresca law: Existence and the behavior of weak solutions

We consider a problem describing the motion of an incompressible, non-isothermal, and non-Newtonian fluid in a three-dimensional thin domain. We first establish an existence result for weak solutions of this problem. Then we study the asymptotic analysis when one dimension of the fluid domain tends to zero. A specific weak Reynolds equation, the limit of Tresca fluid–solid boundary conditions, and the limit boundary conditions for the temperature are obtained. The uniqueness result for the limit problem is also proved.     

Finite-amplitude long-wave instability of Bingham liquid films

The long-wave perturbation method is employed to investigate the weakly nonlinear hydrodynamic stability of a thin Bingham liquid film flowing down a vertical wall. The normal mode approach is first used to compute the linear stability solution for the film flow. The method of multiple scales is then used to obtain the weak nonlinear dynamics of the film flow for stability analysis. It is shown that the necessary condition for the existence of such a solution is governed by the Ginzburg–Landau equation. The modeling results indicate that both the subcritical instability and supercritical stability conditions can possibly occur in a Bingham liquid film flow system. For the film flow in stable states, the larger the value of the yield stress, the higher the stability of the liquid film. However, the flow becomes somewhat unstable in unstable states as the value of the yield stress increases.     

Fluid accumulation in thin-film flows driven by surface tension and gravity

In this note we derive a model describing the two-dimensional viscous flow driven by surface tension and gravity of a thin liquid film near a stagnation point. In the thin-film approximation of such a flow, accumulation takes place where the combined effects of gravity and surface tension stop the flow. These stagnation points are characterised to leading order by the geometry of the substrate. We first derive the thin-film approximation that describes the flow away from such accumulation regions. Then, assuming the existence of isolated stagnation points, we derive the boundary layer equation describing the inner structure of solutions describing accumulation. The existence of these solutions has been proved by the authors elsewhere. Finally, in order to justify the model we prove the existence of curves that give a substrate with an isolated stagnation point.     

On the membrane approximation in isothermal film casting

In this work, a one-dimensional model for isothermal film casting is studied. Film casting is an important engineering process to manufacture thin films and sheets from a highly viscous polymer melt. The model equations account for variations in film width and film thickness, and arise from thinness and kinematic assumptions for the free liquid film. The first aspect of our study is a rigorous discussion of the existence and uniqueness of stationary solutions. This objective is approached via the argument principle, exploiting the homotopy invariance of a family of analytic functions. As our second objective, we analyze the linearization of the governing equations about stationary solutions. It is shown that solutions for the associated boundary-initial value problem are given by a strongly continuous semigroup of bounded linear operators. To reach this result, we cast the relevant Cauchy problem in a more accessible form. These transformed equations allow us insight into the regularity of the semigroup, thus yielding the validity of the spectral mapping theorem for the semigroup and the spectrally determined growth property.     

Exploding solutions for three-dimensional rimming flows

We examine the linear stability of a thin film of viscous fluidon the inside of a cylinder with horizontal axis, rotating aboutthis axis. Unlike previous models, both axial and azimuthalcomponents of the hydrostatic pressure gradient are taken intoaccount, which yields solutions which collapse in both dimensions.Two types of such solutions are found: disturbances with zeroand non-zero net mass (the former have greater explosion rates,that is, their amplitudes grow faster than those of the latter).It is also shown that, despite the existence of exploding disturbances,all solutions with harmonic dependence on time (eigenmodes)are neutrally stable