The thin film equation with backwards second order diffusion

TL;DR

This paper analyzes a generalized thin film equation with backwards diffusion, establishing conditions for existence, regularity, and finite speed of propagation of solutions, relevant for viscous film dynamics with gravity and thermo-capillary effects.

Contribution

It provides new existence and regularity results for the thin film equation with backwards diffusion, including conditions for finite speed of propagation and entropy solutions.

Findings

Global existence of weak solutions under specified conditions.

Finite speed of propagation for certain parameter ranges.

Derivation of local energy and entropy estimates.

Abstract

In this paper, we focus on the thin film equation with lower order "backwards" diffusion which can describe, for example, the evolution of thin viscous films in the presence of gravity and thermo-capillary effects, or the thin film equation with a "porous media cutoff" of van der Waals forces. We treat in detail the equation $$u_t + \{u^n(u_{xxx} + \nu u^{m-n}u_x -A u^{M-n} u_x)\}_x=0,$$ where $\nu=\pm 1,$ $n>0,$ $M>m,$ and $A \ge 0.$ Global existence of weak nonnegative solutions is proven when $ m-n> -2$ and $A>0$ or $\nu=-1,$ and when $-2< m-n<2,$ $A=0,$ $\nu=1.$ From the weak solutions, we get strong entropy solutions under the additional constraint that $m-n> -{3}/{2}$ if $\nu=1.$ A local energy estimate is obtained when $2 \le n<3 $ under some additional restrictions. Finite speed of propagation is proven when $m>n/2,$ for the case of "strong slippage," $0<n<2,$ when $\nu=1$ based on local entropy estimates, and for the case of "weak slippage," $2 \le n<3,$ when $\nu=\pm 1$ based on local entropy and energy estimates.