Extension to more general stochastic shear flows

Establish whether the global well-posedness and almost-sure suppression of chemotactic blow-up proved for the two-dimensional Keller–Segel system advected by stochastic Couette flow extend to more general stochastic shear flows, in analogy with the deterministic setting.

Background

The paper proves that sufficiently strong stochastic Couette mixing yields a unique global mild solution for the two-dimensional Keller–Segel system with arbitrary-mass initial data, by combining enhanced dissipation of nonzero Fourier modes with estimates for the remaining zero mode. The conclusion identifies extension beyond the specific Couette structure as an unresolved direction.

The proposed problem concerns stochastic shear flows more broadly and is explicitly framed as a natural question motivated by analogous deterministic results studied by Bedrossian and He. It is not resolved by the results established in the paper.