One leitmotiv in the work of Günther Grün’s group are mathematical problems related to fourth-order (degenerate) parabolic equations. With respect to MODELING, this includes the derivation of thermodynamically consistent diffuse and sharp interface models in materials sciences — with special emphasis on species transport across fluidic interfaces or on electrokinetic phenomena with or without wall effects (electrowetting). Moreover, multiscale models for surfactant transport and effects of thermal noise in wetting (“stochastic thin-film equation”) are considered. Concerning MATHEMATICAL ANALYSIS, the group works on free boundary problems for (stochastic) degenerate parabolic equations of second and higher order. Results include existence of solutions, optimal estimates for propagation rates of free boundaries or for the size of waiting times. For phase-field models in fluidics, existence and higher regularity of solutions could be obtained (electrowetting). The group also studies the NUMERICAL APPROXIMATION of the above mentioned problems. The inhouse finite-element/finite-volume package EconDrop3D provides space-time adaptive schemes for coupled momentum-phase-field systems with species transport. The group successfully participated at benchmark computations for two-phase channel flow. Monte-Carlo simulations to validate models based on the stochastic thin-film equation have been performed as well. The group pioneered convergence analysis for numerical schemes related to degenerate fourth-order parabolic equations like the thin-film equation or equivalently like degenerate Cahn-Hilliard equations. Just recently, the convergence of self-tailored finite-element schemes for diffuse interface models for two-phase flow of incompressible, immiscible fluids with different mass densities has been established, too.
- J. Fischer, G. Grün: “Finite speed of propagation and waiting times for the stochastic porous medium equation — a unifying approach”, SIAM J. Num. Anal., 47(1), 825–854, 2015
- G. Grün: “On convergent schemes for diffuse interface models for two-phase flow of incompressible fluids with general mass densities”, SIAM J. Num. Anal. 51(6), 3036-3061, 2013
- M. Fontelos, G. Grün, S. Jörres: “On a phase-field model for electrowetting and other electrokinetic phenomena”, SIAM J.Math.Anal. 43(1):527–563, 2011
- G. Grün, K. Mecke, M. Rauscher: “Thin-film flow influenced by thermal noise”, J.Stat.Phys., 122(6):1261-1291, 2006
- G. Grün: “Droplet spreading under weak slippage: existence for the Cauchy problem”, Comm.Part.Diff.Equations, 29:1697-1744, 2004
- R. Dal Passo, L. Giacomelli, G. Grün: “A waiting time phenomenon for thin film equations”, Ann.Sc.Norm.Sup.Pisa, XXX:437-463, 2001