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The lubrication approximation for thin viscous films: The moving contact line wih a “porous media” cut-off of van der Waals interactions. (English) Zbl 0811.35045

Summary: We consider the effect of a second-order ‘porous media’ term on the evolution of weak solutions of the fourth-order degenerate diffusion equation

${h}_{t}=-\nabla ·\left({h}^{n}\nabla {\Delta }h-\nabla {h}^{m}\right)$

in one space dimension. The equation without the second-order term is derived from a ‘lubrication approximation’ and models surface tension dominated motion of thin viscous films and spreading droplets. Here $h\left(x,t\right)$ is the thickness of the film, and the physical problem corresponds to $n=3$. For simplicity, we consider periodic boundary conditions which has the physical interpretation of modelling a periodic array of droplets.

We discuss a physical justification for the ‘porous media’ term when $n=3$ and $1. We propose such behaviour as a cut off of the singular ‘disjoining pressure’ modelling long range van der Waals interactions. For all $n>0$ and $1, we discuss possible behaviour at the edge of the support of the solution via leading order asymptotic analysis of travelling wave solutions. This analysis predicts a certain ‘competition’ between the second- and fourth-order terms. We present rigorous weak existence theory for the above equation for all $n>0$ and $1. In particular, the presence of a second-order ‘porous media’ term in the above equation yields non-negative weak solutions that converge to their mean as $t\to \infty$ and that have additional regularity. Moreover, we show that there exists a time ${T}^{*}$ after which the weak solution is a positive strong solution. For $n>3/2$, we show that the regularity of the weak solutions is in exact agreement with that predicted by the asymptotics.

Finally, we present several numerical computations of solutions. The simulations use a weighted implicit-explicit scheme on a dynamically adaptive mesh. The numerics suggest that the weak solution described by our existence theory has compact support with a finite speed of propagation. The data confirms the local ‘power law’ behaviour at the edge of the support predicted by asymptotics.

##### MSC:
 35K55 Nonlinear parabolic equations 35K60 Nonlinear initial value problems for linear parabolic equations 35K65 Parabolic equations of degenerate type