Localization, smoothness, and convergence to equilibrium for a thin film equation

Abstract

We investigate the long-time behavior of weak solutions to the thin-film type equationwhich arises in the Hele-Shaw problem. We estimate the rate of convergence of solutions to the Smyth-Hill equilibrium solution, which has the formWe obtain exponential convergence in the ||| · ||| m,1 norm for all m with 1 ≤ m < 2, thus obtaining rates of convergence in norms measuring both smoothness and localization. The localization is the main novelty, and in fact, we show that there is a close connection between the localization bounds and the smoothness bounds: Convergence of second moments implies convergence in the H 1 Sobolev norm. We then use methods of optimal mass transportation to obtain the convergence of the required moments. We also use such methods to construct an appropriate class of weak solutions for which all of the estimates on which our convergence analysis depends may be rigorously derived. Though our main results on convergence can be stated without reference to optimal mass transportation, essential use of this theory is made throughout our analysis.

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

“…A very interesting question concerns stability, i.e., the convergence of solutions H of (3.9) to the self-similar profile (4.3). This issue has already been raised in [10, §6] and [27, §8] in a similar context and could be faced, starting from n = 1, either by energy-entropy methods [8][9][10]41], by studying global-in-time classical solutions for perturbations of special solutions like a self-similar profile [21,27,50], a traveling wave [20,28], or an equilibrium-stationary solution [7,19,23,[34][35][36]39]. In the three latter cases, the difficulty lies in finding suitable estimates for the linearized evolution of perturbations.…”

Droplet motion with contact-line friction: long-time asymptotics in complete wetting

“…the convergence of solutions H of (1.18) to the self-similar profile (2.3). This issue has already been raised in ([71], §6) and ([36], §8) in a similar context and could be faced, starting from n=1, either by energy–entropy methods [71–74], by studying global-in-time classical solutions for perturbations of special solutions like a self-similar profile [35,36,39], a travelling wave [40,42] or an equilibrium-stationary solution [33,34,46,47,49–51]. In the three latter cases, the difficulty lies in finding suitable estimates for the linearized evolution of perturbations.…”

“…In the absence of contact-line friction the thin-film equation (f ≡ 0), (1.11) admits self-similar solutions [70], which are expected to describe the long-time dynamics of generic solutions (however, rigorous results are available for n = 1 only [36,39,[71][72][73][74]). If f ≡ 0, (1.11c) breaks the self-similar structure of (1.11a).…”