Fourth-order Degenerate Parabolic Equation Research Articles

Thin film flow dynamics on fiber nets

We analyze existence and qualitative behavior of non-negative weak solutions for fourth order degenerate parabolic equations on graph domains with Kirchhoff's boundary conditions at the inner nodes and Neumann boundary conditions at the boundary nodes. The problem is originated from industrial constructions of spray coated meshes which are used in water collection and in oil-water separation processes. For a certain range of parameter values we prove convergence toward a constant steady state that corresponds to the uniform distribution of coating on a fiber net.

Existence of Solutions to a Viscous Thin Film Equation

A fourth-order degenerate parabolic equation with a viscous term: is studied with the initial-boundary conditions ux=wx=0 on {-1,1}×(0,T), u(x,0)=u0(x) in (-1,1). It can be taken as a thin film equation or a Cahn-Hilliard equation with a degenerate mobility. The entropy functional method is introduced to overcome the difficulties that arise from the degenerate mobility m(u) and the viscosity term. The existence of nonnegative weak solution is obtained.

Weak Solutions for Nonlinear Parabolic Equations with Variable Exponents

Abstract In this work, we study the existence and uniqueness of weak solu- tions of fourth-order degenerate parabolic equation with variable exponent using the di erence and variation methods.

Existence of solutions for a class of fourth order cross-diffusion systems of gradient flow type

This article is concerned with the existence and the long time behavior of weak solutions to certain coupled systems of fourth-order degenerate parabolic equations of gradient flow type. The underlying metric is a Wasserstein-like transportation distance for vector-valued functions, with nonlinear mobilities in each component. Under the hypothesis of (flat) convexity of the driving free energy functional, weak solutions are constructed by means of the variational minimizing movement scheme for metric gradient flows. The essential regularity estimates are derived by variational methods.

EXISTENCE AND LARGE TIME BEHAVIOR OF SOLUTIONS TO A FOURTH-ORDER DEGENERATE PARABOLIC EQUATION

Abstract. The paper is devoted to studying a fourth-order degener-ate parabolic equation, which arises in fluid, phase transformation andbiology. Based on the existence and uniqueness of one semi-discrete prob-lem, two types of approximate solutions are introduced. By establishingsome necessary uniform estimates for those approximate solutions, theexistence and uniqueness of the corresponding parabolic problem are ob-tained. Moreover, the long time asymptotic behavior is established bythe entropy functional method. 1. IntroductionIn the paper, we mainly study the following fourth-order degenerate para-bolic problem:(1) u t +∆(|∆u| p−2 ∆u) −∇· (f(x)|∇u| q−2 ∇u) = g(x) in Q T ,(2) u= ∆u= 0 on Γ,(3) u(x,0) = u 0 (x) on Ω,where Ω ⊂ R N is a bounded domain, ∂Ω is smooth enough, Q T = Ω ×(0,T),Γ = ∂Ω×(0,T) and p,q>1. The equation (1) can be taken as the generalizedCahn-Hilliard equation or thin film equation. Particularly, if p= 2, it becomesthe Cahn-Hilliard model, which appears in the description of phase transfor-mation theory (see [3]). By replacing |∆u|

A thin film approximation of the Muskat problem with gravity and capillary forces

Existence of nonnegative weak solutions is shown for a thin film approximation of the Muskat problem with gravity and capillary forces taken into account. The model describes the space-time evolution of the heights of the two fluid layers and is a fully coupled system of two fourth order degenerate parabolic equations. The existence proof relies on the fact that this system can be viewed as a gradient flow for the 2-Wasserstein distance in the space of probability measures with finite second moment.

Existence of solutions and separation from singularities for a class of fourth order degenerate parabolic equations

A nonlinear parabolic equation of the fourth order is analyzed. The equation is characterized by a mobility coefficient that degenerates at 0. Existence of at least one weak solution is proved by using a regularization procedure and deducing suitable a-priori estimates. If a viscosity term is added and additional conditions on the nonlinear terms are assumed, then it is proved that any weak solution becomes instantaneously strictly positive. This in particular implies uniqueness for strictly positive times and further time-regularization properties. The long-time behavior of the problem is also investigated and the existence of trajectory attractors and, under more restrictive conditions, of strong global attractors is shown.

Erratum: “On a fourth order degenerate parabolic equation in higher space dimension” [J. Math. Phys. 50, 123524 (2009)

In this erratum we fix a flaw in the proof of Proposition 3.1 of Li [“On a fourth order degerate parabolic equation in higher space dimensions,” J. Math. Phys. 50, 123524 (2009)]10.1063/1.3272788.

On a fourth order degenerate parabolic equation in higher space dimensions

We are concerned with the existence of non-negative solutions to the following initial boundary value problem of a fourth order degenerate parabolic equation in higher space dimensions: ut+div(|u|n∇Δu)=0 in Ω×(0,T], ∂u/∂ν=(∂/∂ν)Δu=0 on ∂Ω×(0,T], and u(x,0)=u0(x) in Ω.

A Fourth-order Degenerate Parabolic Equation with Variable Exponent

A Fourth-order Degenerate Parabolic Equation with Variable Exponent

Example of a displacement convex functional of first order

We present a family of first-order functionals which are displacement convex, that is convex along the geodesics induced by the quadratic transportation distance on the circle. The displacement convexity implies the existence and uniqueness of gradient flows of the given functionals. More precisely, we show the existence and uniqueness of gradient-flow solutions of a class of fourth-order degenerate parabolic equations with periodic boundary data. Moreover, positivity of the absolutely continuous part of the solutions is preserved along the flow.

A fourth order nonlinear degenerate parabolic equation

In this paper, we study a fourth order degenerate parabolic equation, which arises in epitaxial growth of nanoscale thin films. We establish the existence of weak solutions, based on the uniform estimates for the approximate solutions. The nonnegativity and the finite speed of propagation of perturbations of solutions are also discussed.

The Gradient Flow of ∫ M |Rm|2

We study the gradient flow of the Riemannian functional ℱ(g):=∫M|Rm|2. This flow corresponds to a fourth-order degenerate parabolic equation for a Riemannian metric. We prove that the degeneracies may be accounted for entirely by diffeomorphism flow, and hence we show short-time existence using the DeTurck method. We prove L2 derivative estimates of Bernstein-Bando-Shi type and use these to give a basic obstruction to long time existence and prove a compactness theorem.

Weak Solutions for a Fourth Order Degenerate Parabolic Equation

We consider an initial-boundary value problem for a fourth order degenerate parabolic equation. Under some assumptions on the initial value, we establish the existence of weak solutions by the discrete-time method. The asymptotic behavior and the fini

Surface-tension-driven dewetting of Newtonian and power-law fluids

The dewetting over a planar substrate of a thin layer of highly viscous fluid under the action of surface tension is considered, with a doubly-nonlinear fourth-order degenerate parabolic equation governing the flow of a power-law fluid. Asymptotic methods are applied to analyse the motion in the shear-thinning, shear-thickening and Newtonian cases, the last of these corresponding mathematically to a critical value of the relevant exponent. In particular, the role played by the local behaviour in the neighbourhood of the contact line is analysed and the dependence of the one-dimensional large-time dewetting behaviour on the fluid’s constitutive properties characterised. Stability issues are also touched upon.

A kinetic approximation of Hele–Shaw flow

In this Note we consider a fourth order degenerate parabolic equation modeling the evolution of the interface of a spreading droplet. The equation is approximated trough a collisional kinetic equation. This permits to derive numerical approximations that preserves positivity of the solution and the main relevant physical properties. A Monte Carlo application is also shown. To cite this article: L. Pareschi et al., C. R. Acad. Sci. Paris, Ser. I 338 (2004).

Shear-thinning liquid films: macroscopic and asymptotic behaviour by quasi-self-similar solutions

We consider the spreading of a thin droplet of viscous liquid on a plane surface driven by capillarity in the complete wetting regime. In the case of constant viscosity, the no-slip condition leads to a force singularity at advancing contact lines. It is well known nowadays that the introduction of appropriate slip conditions removes this paradox and alters only logarithmically the macroscopic behaviour of solutions at intermediate timescales. Here, we investigate a different approach, which consists in keeping the no-slip condition and assuming instead a shear-thinning rheology. This relaxation leads, in lubrication approximation, to fourth order degenerate parabolic equations of quasilinear type. By analysing a class of quasi-self-similar solutions to these equations in the limit of Newtonian rheology, we obtain a scaling law in time for macroscopic quantities (such as macroscopic profile, effective contact-angle) which is only logarithmically affected by the shear-thinning parameters. As opposed to positive slippage models, the scaling law is uniform for large times as far as the macroscopic support is well defined, and thus could also describe the asymptotic behaviour of a large class of solutions for fixed shear-thinning rheology.

Long-Time Asymptotics for Strong Solutions¶of the Thin Film Equation

In this paper we investigate the large-time behavior of strong solutions to the one-dimensional fourth order degenerate parabolic equation u t =−(u u xxx ) x , modeling the evolution of the interface of a spreading droplet. For nonnegative initial values u 0(x)∈H 1(ℝ), both compactly supported or of finite second moment, we prove explicit and universal algebraic decay in the L 1-norm of the strong solution u(x,t) towards the unique (among source type solutions) strong source type solution of the equation with the same mass. The method we use is based on the study of the time decay of the entropy introduced in [13] for the porous medium equation, and uses analogies between the thin film equation and the porous medium equation.

Nonnegativity preserving convergent schemes for the thin film equation

We present numerical schemes for fourth order degenerate parabolic equations that arise e.g. in lubrication theory for time evolution of thin films of viscous fluids. We prove convergence and nonnegativity results in arbitrary space dimensions. A proper choice of the discrete mobility enables us to establish discrete counterparts of the essential integral estimates known from the continuous setting. Hence, the numerical cost in each time step reduces to the solution of a linear system involving a sparse matrix. Furthermore, by introducing a time step control that makes use of an explicit formula for the normal velocity of the free boundary we keep the numerical cost for tracing the free boundary low.

A fourth-order degenerate parabolic equation describing thin viscous flows over an inclined plane

We prove the existence of nonnegative weak solutions for a fourth-order nonlinear degenerate parabolic equation which arises in the description of thin viscous flows over an inclined plane. Regularity, positivity properties, and large-time behaviour of these solutions are also discussed.