Stability of receding traveling waves for a fourth order degenerate parabolic free boundary problem

Manuel V. Gnann, Slim Ibrahim, Nader Masmoudi

Research output: Contribution to journalArticle

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 languageEnglish (US)
Pages (from-to)1173-1243
Number of pages71
JournalAdvances in Mathematics
Volume347
DOIs
StatePublished - Apr 30 2019

    Fingerprint

Keywords

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

ASJC Scopus subject areas

  • Mathematics(all)

Cite this