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This paper is devoted to the asymptotic analysis of a thin film equation that describes the evolution of a thin liquid droplet on a solid support driven by capillary forces. We propose an analytic framework to rigorously investigate the connection between this model and Tanner's law, which claims: the edge velocity of a spreading thin film on a prewetted solid is approximately proportional to the cube of the slope at the inflection. More precisely, we investigate the asymptotic limit of the thin film equation when the slippage coefficient is small and at an appropriate time scale. We show that the evolution of the droplet can be approximated by a moving free boundary model (the so‐called quasi‐static approximation), and we present some results pointing to the validity of Tanner's law in that regime. Several papers have investigated a similar connection between the thin film equation and Tanner's law. Our main contribution is finding the effective self‐contained equation for the evolution of the apparent support of the droplet in the limit when the slip coefficient vanishes. © 2020 Wiley Periodicals LLC. [ABSTRACT FROM AUTHOR]

We analyze free energy functionals for macroscopic models of multi-agent systems interacting via pairwise attractive forces and localized repulsion. The repulsion at the level of the continuous description is modeled by pressure-related terms in the functional making it energetically favorable to spread, while the attraction is modeled through nonlocal forces. We give conditions on general entropies and interaction potentials for which neither ground states nor local minimizers exist. We show that these results are sharp for homogeneous functionals with entropies leading to degenerate diffusions while they are not sharp for fast diffusions. The particular relevant case of linear diffusion is totally clarified giving a sharp condition on the interaction potential under which the corresponding free energy functional has ground states or not. [ABSTRACT FROM AUTHOR]