On steady states of van der Waals force driven thin film equations

Details On steady states of van der Waals force driven thin film equations On steady states of van der Waals force driven thin film equations

2006-10-25

arxiv.org/abs/math/0610762v1

Mathematics

Analysis of PDEs

24 pages

Scientific paper

Let $\Omega\subset\mathbb{R}^{N}$, $N\geq2$ be a bounded smooth domain and $\alpha>1$. We are interested in the singular elliptic equation% \[ \triangle h=\frac{1}{\alpha}h^{-\alpha}-p\quad\text{in}\Omega \] with Neumann boundary conditions. In this paper, we gave a complete description of all continuous radially symmetric solutions. In particular, we constructed nontrivial smooth solutions as well as rupture solutions. Here a continuous solution is said to be a rupture solution if its zero set is nonempty. When N=2 and $\alpha=3$, the equation has been used to model steady states of van der Waals force driven thin films of viscous fluids. We also considered the physical problem when total volume of the fluid is prescribed.

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