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

Abstract

Consider the thin-film equation $h_t+{\left(h{h}_{yyy}\right)}_y=0$ 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 $h\searrow 0$, but the porous-medium equation additionally fulfills a comparison principle while the thin-film equation does not. In this note, we consider traveling waves $h=\frac{V}{6}{x}^3+\nu x^2$ for $x\ge 0$, where $x=y-Vt$ and $V,\nu \ge 0$ 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 $x\searrow 0$ and $x\to \infty$, 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.

54 pages, revised version, minor changes and corrections, changed citation style to numbers

References in corpus (1)

• The Navier-slip thin-film equation for 3D fluid films: existence and uniqueness

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• On singularity formation in a Hele-Shaw model
• Relaxation to equilibrium in the one-dimensional thin-film equation with partial wetting
• Droplet motion with contact-line friction: long-time asymptotics in complete wetting