Establish rigorous zero-diffusivity asymptotics for active suspensions with general swimmer-wall interactions

Establish rigorous asymptotics for the Doi-Saintillan-Shelley Smoluchowski model in the zero-translational-diffusivity limit under the full range of swimmer-wall interactions, including incoming, outgoing, and grazing trajectories.

Background

The paper identifies the Doi-Saintillan-Shelley model as a higher-dimensional active-suspension analogue of the one-dimensional run-and-tumble systems studied. In this model, rod-like swimmers have continuous orientations, undergo translational and orientational diffusion, and interact hydrodynamically through an active stress coupled to a Stokes flow.

For related linear Fokker–Planck models, bulk–wall decompositions involving a wall-supported measure have been conjectured or established under restricted assumptions, particularly when swimmer motion is only toward the wall. The unresolved problem is to justify the corresponding asymptotic decomposition and effective limiting dynamics when swimmers may move into, away from, or tangentially along the wall.