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

Let $\Omega\subset\mathbb{R}^{N}$, N ≥2 be a bounded smooth domain and α > 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, a complete description of all continuous radially symmet...

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Bibliographic Details
Published in:European journal of applied mathematics 2007-04, Vol.18 (2), p.153-180
Main Authors: JIANG, HUIQIANG, NI, WEI-MING
Format: Article
Language:English
Subjects:
Applied mathematics
Fluid dynamics
Mathematical problems
Thin films
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Summary:Let $\Omega\subset\mathbb{R}^{N}$, N ≥2 be a bounded smooth domain and α > 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, a complete description of all continuous radially symmetric solutions is given. In particular, we construct 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 α = 3, the equation is used to model steady states of van der Waals force driven thin films of viscous fluids. We also consider the physical problem when total volume of the fluid is prescribed.
ISSN:0956-7925
1469-4425
DOI:10.1017/S0956792507006936