Seminar on Mathematics of Thin Liquid Films

Originating in the works of Laplace, Young, Neumann and Reynolds the mathematical theory of liquid films have newly attracted mathematicians in the last two decades during which the theoretical justification for the lubrication approximation was suggested and developed. Describing the evolution of liquid micro and nanoscopic films raises many mathematical challenges. These include modeling issues such as finding appropriate simplied descriptions of the systems that are physically correct, and analytically and numerically tractable. Furthermore partial differential equations describing thin liquid films are (due to their nonlinearity, higher order, and the degenerate nature of the diffusion terms) notoriously difficult to analytically understand and simulate numerically. Mathematics ranging from functional analysis, variational methods, asymptotic analysis to modeling and numerics is essential to gain insights about thin liquid films and will be considered in this seminar.

There will be weekly presentations on selected papers. The seminar is addressed to graduate students planning a deep in the area of applied analysis or numerics, Ph.D. students and everybody who is just interested in. Feel very welcome to participate.

Selected topics:

• Lubrication approximation for liquid films. The role of geometry and physical effects (e.g. Marangoni, gravity, evaporation and slippage).
• Special classes of solutions: bifurcations of stationary solutions; traveling waves; self-similar solutions.
• Existence of entropy solutions and their regularity. Example of non-uniqueness.
• Analysis of the moving boundary value problems.
• Numerical schemes for lubrication equations and their convergence properties.
• Analysis of finite time rupture.
• Analysis of coarsening dynamics of droplets: reduced ODE models and coarsening rates.