Project part 9 -
Large-time and macroscopic asymptotics in kinetic transport (Christian Schmeiser)

[P9, Schmeiser] is concerned with kinetic transport equations, describing the dynamics of particle ensembles in terms of their distributions in phase (position-velocity) space. This Project Part will contribute to the effort of proving decay to equilibrium with explicit rates by hypocoercivity methods, typically relying on mathematical entropy vs. entropy dissipation approaches, based on modifications of physical entropies. A second research direction is the modeling of self-organisation phenomena in ensembles of biological cells, including model derivation, analysis, simulation, long time and macroscopic limits.

Various approaches for the analysis of hypocoercivity will be synthesized together with [P1, Arnold], J. Dolbeault (Paris), F. Herau (Nantes), C. Mouhot (Cambridge), and L. Neumann (Innsbruck), guided by the goals of minimal assumptions on initial data, optimal convergence rates, and inclusion of nonlinear problems. Entropy dissipation aspects will be guiding numerical approaches to Fokker-Planck equations (with [P2, Jüngel]) and kinetic models for gases and for myxobacteria (with [P10, Schöberl]).

A joint effort with P. Degond (London), Hittmeir, Markowich, and A. Manhart (New York) will be concerned with kinetic models for bacterial colonies, where signaling interactions are responsible for self-organisation phenomena such as aggregation, traveling waves, and branching patterns. A new class of models for local binary interactions (alignment and/or reversal) of myxobacteria (a group of social bacteria with gliding motility) will be formulated, analyzed, and solved numerically. The cell biology expertise of C. Schmeiser will also be used in cooperative work on cell membranes (with [P11, Stefanelli]) and on transportation networks (with [P4, Markowich]).

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