Intermediate Asymptotics of the Capillary-Driven Thin Film Equation

TL;DR

This paper investigates the intermediate asymptotic behavior of solutions to the capillary-driven thin film equation, deriving a Green's function and demonstrating convergence to a universal self-similar attractor, supported by numerical analysis.

Contribution

It provides an analytical derivation of the Green's function and shows convergence to a universal attractor for the linearized equation, with conjectures extending to the nonlinear case.

Findings

Rescaled solutions converge to a universal self-similar attractor.

Green's function derived for the linearized equation.

Numerical evidence suggests extension to nonlinear equation.

Abstract

We present an analytical and numerical study of the two-dimensional capillary-driven thin film equation. In particular, we focus on the intermediate asymptotics of its solutions. Linearising the equation enables us to derive the associated Green's function and therefore obtain a complete set of solutions. Moreover, we show that the rescaled solution for any summable initial profile uniformly converges in time towards a universal self-similar attractor that is precisely the rescaled Green's function. Finally, a numerical study on compact-support initial profiles enables us to conjecture the extension of our results to the nonlinear equation.