Lorenzo Giacomelli

Regularity of source-type solutions to the thin-film equation with zero contact angle and mobility exponent between 3/2 and 3

In one space dimension, we consider source-type (self-similar) solutions to the thin-film equation with vanishing slope at the edge of its support (zero contact-angle condition) in the range of mobility exponents n ∈ ( 3 2 , 3 ) . This range contains the physically relevant case n = 2 (Navier slip). The existence and (up to a spatial scaling) uniqueness of these solutions has been established in [F. Bernis, L.A. Peletier & S.M. Williams, Nonlinear Anal. 18 (1992), 217-234]. There, it is also shown that the leading order expansion near the edge of the support coincides with that of a travelling-wave solution. In this paper we substantially sharpen this result, proving that the higher order correction is analytic with respect to two variables: the first one is just the spatial variable, whereas the second one is a (generically irrational, in particular for n = 2) power of it, which naturally emerges from a linearisation of the operator around the travelling-wave solution. This result shows that — as opposed to the case of n = 1 (Darcy) or to the case of the porous medium equation (the second-order analogue of the thin-film equation) — in this range of mobility exponents, ‡

THE THIN FILM EQUATION WITH NONLINEAR DIFFUSION

We consider the fourth order degenerate parabolic equation ut þ div ður u uruÞ 1⁄4 0, u 0, ð1:1Þ where m 2 R and n 2 Rþ (precise assumptions on the parameters will be stated in the sequel). Equation (1.1) is introduced to describe the evolution of the height u of a thin liquid film spreading on a solid surface. The fourth-order term accounts for the effects of surface tension. The second-order term accounts, if m 1⁄4 n, for the effect of gravity; it has also been proposed, for different values of m, as a ‘‘porous-media cut-off ’’ of Van der Waals forces. The case n 1⁄4 m 1⁄4 1 describes the extent u of the region occupied by a liquid in the half-space Hele-Shaw cell in the lubrication regime. We refer to the review paper of Oron, Davis and Bankoff [1] for details. The case n 1⁄4 1,

A Free Boundary Problem of Fourth Order: Classical Solutions in Weighted Hölder Spaces

We prove short-time existence and uniqueness of classical solutions in weighted Hölder spaces for the thin-film equation with linear mobility, zero contact angle, and compactly supported initial data. We furthermore show regularity of the free boundary and optimal regularity of the solution in terms of the regularity of the initial data. Our approach relies on Schauder estimates for the operator linearized at the free boundary, obtained through a variant of Safonovs method that is solely based on energy estimates.

Lower bounds on waiting times for degenerate parabolic equations and systems

We extend the method in [19] to obtain quantitative estimates of waiting times for free boundary problems associated with degenerate parabolic equations and systems. Our approach is multidimensional, it applies to a large class of equations, including thin-film equations, (doubly) degenerate equations of second and of higher order and also systems of semiconductor equations. For these equations, we obtain lower bounds on waiting times which we expect to be optimal in terms of scaling. This assertion is true for the porous-medium equation which seems to be the only PDE for which two-sided quantitative estimates of the waiting time have been established so far.

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.

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

Abstract 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.

Waiting Time Phenomena for Degenerate Parabolic Equations — A Unifying Approach

The aim of this contribution is to present a new approach to establish the occurrence of waiting time phenomena for solutions to degenerate parabolic equations. Originally developed by the authors in [15] for the thin film equation

Torsion in Strain-Gradient Plasticity: Energetic Scale Effects

{\text{u}}_t + div(u^n \nabla \Delta u) = 0,