Convergence to equilibrium for a thin film equation on a cylindrical surface
arXiv:1011.2108 · doi:10.1080/03605302.2011.648704
Abstract
The degenerate parabolic equation u_t + [u^3(u_xxx + u_x - sin x)]_x=0 models the evolution of a thin liquid film on a stationary horizontal cylinder. It is shown here that for each given mass there is a unique steady state, given by a droplet hanging from the bottom of the cylinder that meets the dry region at the top with zero contact angle. The droplet minimizes the energy and attracts all strong solutions that satisfy certain energy and entropy inequalities. The distance of any solution from the steady state decays no faster than a power law.
26 pages, 6 figures. This is a greatly expanded version of our previous submission [arXiv:1009.4092v1]. The main result on convergence to equilbrium is new, and the bounds on the rate of convergence have been simplified. For the second version, we have changed the title, corrected Lemma 2.3, rewrittenSection 3, and added several references