### Abstract

Consider the thin-film equation h _{t} +(hh _{yyy} ) _{y} =0 with a zero contact angle at the free boundary, that is, at the triple junction where liquid, gas, and solid meet. Previous results on stability and well-posedness of this equation have focused on perturbations of equilibrium-stationary or self-similar profiles, the latter eventually wetting the whole surface. These solutions have their counterparts for the second-order porous-medium equation h _{t} −(h ^{m} ) _{yy} =0, where m>1 is a free parameter. Both porous-medium and thin-film equation degenerate as h↘0, but the porous-medium equation additionally fulfills a comparison principle while the thin-film equation does not. In this note, we consider traveling waves h=[Formula presented]x ^{3} +νx ^{2} for x≥0, where x=y−Vt and V,ν≥0 are free parameters. These traveling waves are receding and therefore describe de-wetting, a phenomenon genuinely linked to the fourth-order nature of the thin-film equation and not encountered in the porous-medium case as it violates the comparison principle. The linear stability analysis leads to a linear fourth-order degenerate-parabolic operator for which we prove maximal-regularity estimates to arbitrary orders of the expansion in x in a right-neighborhood of the contact line x=0. This leads to a well-posedness and stability result for the corresponding nonlinear equation. As the linearized evolution has different scaling as x↘0 and x→∞ the analysis is more intricate than in related previous works. We anticipate that our approach is a natural step towards investigating other situations in which the comparison principle is violated, such as droplet rupture.

Original language | English (US) |
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Pages (from-to) | 1173-1243 |

Number of pages | 71 |

Journal | Advances in Mathematics |

Volume | 347 |

DOIs | |

State | Published - Apr 30 2019 |

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### Keywords

- Degenerate-parabolic equations
- Fourth-order equation
- Free boundary problems
- Stability
- Thin film equation
- Traveling waves

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Advances in Mathematics*,

*347*, 1173-1243. https://doi.org/10.1016/j.aim.2019.01.028