## 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:

 35K25 Higher-order parabolic equations 35B65 Smoothness and regularity of solutions to PDEs 35A01 Existence problems for PDEs: global existence, local existence, non-existence 76A20 Thin fluid films 35Q35 PDEs in connection with fluid mechanics

### Keywords:

one space dimension; fixed point argument
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### References:

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