On a thin film model with insoluble surfactant

Abstract

This paper studies the existence and asymptotic behavior of global weak solutions for a thin film equation with insoluble surfactant under the influence of gravitational, capillary and van der Waals forces. We prove the existence of global weak solutions for medium sized initial data in large function spaces. Moreover, exponential decay towards the flat equilibrium state is established, where an estimate on the decay rate can be computed explicitly.2010 Mathematics Subject Classification. 35D30, 35B40, 35K52, 35K65, 76A20 .

Well‐posedness and stability for a mixed order system arising in thin film equations with surfactant

“…It thereby improves the existing result in [4], which only states asymptotic stability for initial data in H 4 × H 2 . We would like to mention that in [6] an asymptotic stability result for systems including (1.1) is proved for initial data, which even permit high oscillations -the price to pay are more specific size restrictions and solutions in lower regularity spaces.…”

“…Concerning the fourth‐order counter part, that is system (1.1), when capillary effects instead of gravitation form the driving force, the existence of nonnegative global weak solutions is studied in [7, 11, 12]. Moreover, the existence and asymptotic behavior of global weak solutions of a thin film equation with insoluble surfactant under the influence of gravitational, capillary as well as van der Waals forces (the system of evolution equations derived by Jensen & Grotberg [13]) is subject of [6]. Eventually, a corresponding analysis concerning modeling, well‐posedness, asymptotic stability of equilibria, and weak solutions is carried out in [4, 5] for a two‐phase thin film equation with insoluble surfactant under consideration of capillary effects.…”