Smooth zero-contact-angle solutions to a thin-film equation around the steady state

Abstract

In the simplest case of a linearly degenerate mobility, we view the thin-film equation as a classical free boundary problem. Our focus is on the regularity of solutions and of their free boundary in the "complete wetting" regime, which prescribes zero slope at the free boundary. In order to rule out of the analysis possible changes in the topology of the positivity set, we zoom into the free boundary by looking at perturbations of the stationary solution. Our strategy is based on a priori energy-type estimates which provide "minimal" conditions on the initial datum under which a unique global solution exists. In fact, this solution turns out to be smooth for positive times and to converge to the stationary solution for large times. As a consequence, we obtain smoothness and large-time behavior of the free boundary.

Well-posedness for the Navier-slip thin-film equation in the case of complete wetting

“…[3,4]. We remark that the case of complete wetting for mobility exponent n = 1 was investigated by three, respectively two, of the authors in earlier works within weighted L 2 -and Hölder-spaces, respectively [5,6]. Moreover, this problem in arbitrary space dimensions, i.e.…”

“…Note that the leading-order asymptotics as x 0 of the traveling-wave solution (1.5) and the stationary solution (1.3) differ, unlike in the case of the thin-film equation with n = 1 [5].…”

“…Indeed, these solutions play an important role in the analysis of the thin-film equation for n ∈ 0, 3 2 . In particular the mentioned regularity results [5][6][7] linearize around the profile of the quadratic stationary solution (1.3). For the range of mobility exponents n ∈ 3 2 , 3 , however, the expected generic solutions are traveling waves:…”

“…We refer to [5,11] for further discussions on analog results for the porous medium equation and the choice of boundary conditions, such as [34][35][36].…”

“…Proposition 5.5). A standard way to obtain smooth solutions to the resolvent problem for smooth right hand side was carried out in [5] by starting with the Lax-Milgram solution and proving regularity afterwards. Instead, in this work we opted for ODE-based arguments.…”