On the Ostwald ripening of thin liquid films (Q390236)

the author proposes a mean-field model for the Ostwald ripening of a 2D thin liquid film, with underying solid substrate, in order to determine the coarsening rate under suitable assumptions on the droplets and the distance among them. In the derivation of the mean field model, which develops in the same spirit of \textit{B. Niethammer} and \textit{F. Otto} [Commun. Pure Appl. Math. 54, No. 3, 361--384 (2001; Zbl 1029.82026)], it has been taken into account that the ripening mechanism dominates the dynamics and the droplets have paraboloid shapes and a fixed contact angle between their surface and the residual liquid film (see [\textit{K. B. Glasner}, SIAM J. Appl. Math. 69, No. 2, 473--493 (2008; Zbl 1159.76012)] and [\textit{K. Glasner} et al., Eur. J. Appl. Math. 20, No. 1, 1--67 (2009; Zbl 1155.76057)]). The proposed model is a transport equation for the distribution of the radii of droplets' bases. In the discrete case, it leads to a system of ODEs.NEWLINENEWLINENEWLINEThe technique to deduce an upper bound for the coarsening rate in terms of the average volume of droplets with respect to time is based on a method proposed in [\textit{R. V. Kohn} and \textit{F. Otto}, Commun. Math. Phys. 229, No. 3, 375--395 (2002; Zbl 1004.82011)] for Cahn-Hilliard equations. The method relies on three steps: an interpolation inequality between the energy and a suitable `dual quantity', deduced in the same spirit as in [the author and \textit{R. L. Pego}, SIAM J. Math. Anal. 37, No. 2, 347--371 (2005; Zbl 1107.82031)], a dissipation inequality to relate the rate of change of the free energy and the `dual quantity', and an ODE argument.NEWLINENEWLINENEWLINEThe achieved upper bound for the coarsening rate in this mean-field model provides a justification of the temporal power-logaritmic law, which describes the behaviour of the average volume of droplets, as proposed in [\textit{K. B. Glasner}, SIAM J. Appl. Math. 69, No. 2, 473--493 (2008; Zbl 1159.76012)] and [\textit{K. Glasner} et al., Eur. J. Appl. Math. 20, No. 1, 1--67 (2009; Zbl 1155.76057)]. In particular, the results improve those in [the author, Nonlinearity 23, No. 2, 325--340 (2010; Zbl 1191.35153)], and [\textit{F. Otto} et al., SIAM J. Math. Anal. 38, No. 2, 503--529 (2007; Zbl 1298.76027)]. NEWLINENEWLINENEWLINEMigration, collision of droplets, and PDE models are left to a next investigation.