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Well-Posedness for a Class of Thin-Film Equations with General Mobility in the Regime of Partial Wetting

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Abstract

We establish well-posedness for the family of thin-film equations

$$\left \{\begin{array}{ll}h_t + (h^n h_{xxx})_x \ = \ 0 \quad \ {\rm in } \ \{ h > 0 \},\\ h \ = \ 0, \ |h_x| \ = \ 1 \quad\quad {\rm on } \ \partial \{ h > 0 \}\end{array}\right. $$
(1)

with \({n \in (0,\frac {14}{5}) \backslash \{ 1, 2 \}}\). The model (1) with \({n \in (0,3]}\) has been used to describe the evolution of a capillary driven thin liquid droplet on a solid substrate in terms of its height profile \({h \geqq 0}\). The family of thin-film equations (1) provides a model problem to investigate contact line propagation in fluid dynamics under relaxed slip conditions. The parabolicity of the fourth order parabolic problem degenerates at the free boundary, which leads to a loss of regularity at the moving contact point. Our solutions are regular in terms of the two variables d(x) and d(x)3−n, where d(x) is the distance to the free boundary. The main technical difficulty in the analysis of (1) is related to the loss of regularity at the contact points.

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  1. University of Heidelberg, Im Neuenheimer Feld 294, Heidelberg, Germany

    Hans Knüpfer

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Communicated by V. Šverák

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Knüpfer, H. Well-Posedness for a Class of Thin-Film Equations with General Mobility in the Regime of Partial Wetting. Arch Rational Mech Anal 218, 1083–1130 (2015). https://doi.org/10.1007/s00205-015-0882-x

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