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The mathematics of moving contact lines in thin liquid films. (English) Zbl 0917.35100
The author studies the title problem by using the fourth-order degenerate diffusion equation $h_t+ \nabla(f(h)\nabla \Delta h)= 0$, where $h$ is the thickness of the liquid film, and $f(h)$ is a prescribed function related to the solution-dependent diffusion coefficient, $f(h)\to 0$ as $h\to 0$. First, a special discussion is devoted to the finite-time singularities and similarity solutions of a particular one-dimensional equation $h_t+ (h^n h_{xxx})_x= 0$. Then the author examines weak solutions of the general equation which correspond to various constitutive laws for moving contact lines (Young’s law, Greenspan-McKay law, van der Walls and superdiffusion models). Numerical results obtained by finite difference methods conclude the paper.
Reviewer: O.Titow (Berlin)

35Q35PDEs in connection with fluid mechanics
76D45Capillarity (surface tension)
76D08Lubrication theory
82B24Interface problems; diffusion-limited aggregation (equilibrium statistical mechanics)