Higher-dimensional extensions

Extend the global well-posedness and suppression-of-blow-up results for Keller–Segel systems advected by sufficiently strong stochastic Couette flow from two dimensions to higher spatial dimensions.

Background

The paper is restricted to the two-dimensional cylinder T×R\mathbb T\times\mathbb R, where enhanced dissipation controls the nonzero horizontal modes and one-dimensional inequalities control the zero mode. The conclusion explicitly lists higher-dimensional extensions as an unresolved direction.

The problem asks whether analogous global existence and blow-up suppression can be obtained in higher dimensions; no such result is established in the paper.

References

Several open problems remain for future investigation. A natural question is whether our results for stochastic Couette flow extend to more general stochastic shear flows, in analogy with the deterministic setting studied by Bedrossian and He . Other directions include extensions to higher dimensions and the study of coupled chemotaxis–fluid systems in which the fluid velocity is governed by the Navier–Stokes equations.

— Suppression of Blow-up in Two-dimensional Keller--Segel Systems by Stochastic Couette Flows  (2609.25551 - Kong et al., 22 Sep 2026) in Conclusion