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

TL;DR

The paper addresses well-posedness, partial regularity, and asymptotic stability for the fourth-order degenerate thin-film equation $h_t+(h^n h_{zzz})_z=0$ in the complete-wetting regime with mobility exponent $n\in(1,3)\setminus\{\tfrac{3}{2}\}$. It develops a fixed-domain reformulation via the von Mises transform, establishing maximal $L^{p}$-regularity for the linearized operator and then a nonlinear fixed-point argument to obtain global well-posedness for small data and asymptotic stability of traveling waves or stationary states. Compared with prior $L^2$-based results, the $L^{p}$-in-time framework extends the allowable mobility range and yields partial regularity near the free boundary, together with explicit boundary expansions. The approach provides a concise, functional-analytic treatment of nonlinear mobilities in degenerate fourth-order parabolic equations and offers rigorous guidance for numerical and further theoretical investigations in complete wetting.

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, Knüpfer, 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.

Figures (5)

• Figure 1: A viscous thin film as described by \ref{['TFE']}.
• Figure 2: Surface tensions acting at the triple junction.
• Figure 3: For the two different cases of $n$ the zeros $\gamma_1,\dots,\gamma_4=\beta$ of $\mathfrak p(D)$ (solid lines) and the upper and lower bound in \ref{['coerc_range']} (dashed lines) are shown. The coercivity range for $\alpha$ contains the shaded area.
• Figure 4: For the two different cases of $n$ the zeros $\gamma_1,\dots,\gamma_4$ of $\mathfrak p(D)$ (solid line) and the upper and lower bound in \ref{['eq_coer_condition2_n_1_32']} and \ref{['eq_coer_condition2_n_32_3']} (dashed line) are shown. The coercivity range for $\alpha$ contains the shaded area according to \ref{['coerc_range']} of Lemma \ref{['lem_coercive_range']}. This area shifted up by 1 is shown in green (shaded darker, green color online).
• Figure 5: full coercivity range (color online). The areas and boundaries in the $n$-$\alpha$ plane are shown for which \ref{['it:cond_i']} or \ref{['it:cond_ii']} are valid and thus $\mathfrak p(D) = \mathfrak p_n(D)$ (given by \ref{['eq_def_p(D)']} and \ref{['eq_zeros_12']}) is coercive.

Theorems & Definitions (34)

• Lemma 2.1
• Lemma 2.2: Coercivity
• Theorem 2.3
• Lemma 3.1
• proof
• Lemma 3.2: cf. GGKO
• proof
• proof : Proof of Lemma \ref{['lem_coercive_range']}
• Lemma 3.3
• proof