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Well-posedness for the Navier slip thin-film equation in the case of partial wetting. (English) Zbl 1227.35146
The author addresses well-posedness and regularity for the Navier slip thin-film equation in the case of partial wetting. As many people know that if one considers a viscous liquid droplet spreading on a surface, the classical boundary condition for the Navier-Stokes equations at a liquid-solid interface is the no-slip condition. However, if a no-slip condition is imposed, then any movement of the contact line leads to infinite dissipation of energy at the moving contact line. For this reason, other slip conditions such as the Navier slip condition have been proposed. The author proves well-posedness for a reduced 1-D fluid model related to Navier slip. It turns out that the profile of the droplet cannot be described by a smooth function. By using some methods such as a fixed point argument, the theories on weighted Sobolev spaces, the Laplace transformation etc., the author also addresses the existence, uniqueness and regularity of the problem.
MSC:
35K25Higher order parabolic equations, general
35B65Smoothness and regularity of solutions of PDE
35A01Existence problems for PDE: global existence, local existence, non-existence
76A20Thin fluid films (fluid mechanics)
35Q35PDEs in connection with fluid mechanics