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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
\[
h_ t = - \nabla \cdot (h^ n \nabla \Delta h - \nabla h^ m)
\]
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 \(h(x,t)\) 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 \(1 < m < 2\). 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 \(1 < m < 2\), 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 \(1 < m < 2\). 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.

We discuss a physical justification for the ‘porous media’ term when \(n = 3\) and \(1 < m < 2\). 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 \(1 < m < 2\), 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 \(1 < m < 2\). 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, boundary and initial-boundary value problems for linear parabolic equations |

35K65 | Degenerate parabolic equations |