The Cox–Voinov law for traveling waves in the partial wetting regime*

We consider the thin-film equation $\partial_t h + \partial_y \left(m(h) \partial_y^3 h\right) = 0$ in $\{h > 0\}$ with partial-wetting boundary conditions and inhomogeneous mobility of form $m(h) h^3+\lambda^{3-n}h^n$, where $h \ge is film height, $\lambda slip length, $y denotes lateral variable, $n \in (0,3)$ exponent parameterizing nonlinear condition. The regime implies condition $\partial_y \mathrm{const.} at triple junction $\partial\{h (nonzero microscopic contact angle). Existence uniqueness traveling-wave solutions to this problem under constraint $\partial_y^2 \to as \infty$ have been proved previous work by Chiricotto Giacomelli [Commun. Appl. Ind. Math., 2(2):e-388, 16, 2011]. are interested asymptotics \downarrow \infty$. By reformulating a dynamical system for difference between solution angle, values $n$ found which linear well resonances occur. These lead different asymptotic behavior $h\downarrow0$ depending on $n$. Together $h\to\infty$ characterizing Cox-Voinov law velocity-dependent macroscopic angle Giacomelli, first author work, Otto [Nonlinearity, 29(9):2497-2536, 2016], rigorous partial wetting can be characterized. Furthermore, our approach enables us analyze relation angle. It that depends continuously differentiably