Classical solutions to the thin-film equation with general mobility in the perfect-wetting regime

Abstract: We prove well-posedness, partial regularity, and stability of the thin-film equation $h_t + (m(h) h_{zzz})_z = 0$ with general mobility $m(h) = h^n$ and mobility exponent $n\in (1,\tfrac{3}{2})\cup (\tfrac{3}{2},3)$ in the regime of perfect wetting (zero contact angle). After a suitable coordinate transformation to fix the free boundary (the contact line where liquid, air, and solid coalesce), the thin-film equation is rewritten as an abstract Cauchy problem and we obtain maximal $L^{p}_t$-regularity for the linearized evolution. Partial regularity close to the free boundary is obtained by studying the elliptic regularity of the spatial part of the linearization. This yields solutions that are non-smooth in the distance to the free boundary, in line with previous findings for source-type self-similar solutions. In a scaling-wise quasi-minimal norm for the initial data, we obtain a well-posedness and asymptotic stability result for perturbations of traveling waves. The novelty of this work lies in the usage of $L^{p}$-estimates in time, where $1 < p < \infty$, while the existing literature mostly deals with $p = 2$ at least for nonlinear mobilities. This turns out to be essential to obtain for the first time a well-posedness result in the perfect-wetting regime for all physical nonlinear slip conditions except for a strongly degenerate case at $n = \tfrac 3 2$ and the well-understood Greenspan-slip case $n = 1$. Furthermore, compared to [J. Differential Equations, 257(1):15-81, 2014] by Giacomelli, the first author of this paper, Kn\"upfer, and Otto, where a PDE approach yields $L^2_t$-estimates, well-posedness, and stability for $1.8384 \approx \tfrac{3}{17}(15-\sqrt{21}) < n < \tfrac{3}{11}(7+\sqrt{5}) \approx 2.5189$, our functional-analytic approach is significantly shorter while at the same time giving a more general result.