Global bounded solutions for a class of generalized Hillen-Painter models near Couette flow in ${\mathbb{R}}^2$

We investigate the global well-posedness of a class of generalized Hillen–Painter systems – specifically, supercritical volume-filling chemotaxis models – in ${\mathbb{R}}^2$ under the influence of Couette flow. It is well established that, in the absence of fluid flow, solutions to this system may develop finite-time singularities (blow-up) for arbitrary initial cell mass. It is proved that the introduction of a Couette flow with sufficiently large amplitude guarantees the global existence of solutions for all initial masses. By employing a novel frequency decomposition technique, we successfully remove the mass threshold limitation presented in previous studies on the domain $\mathbb{T}\times \mathbb{R}$ (Wang et al., Commun. Contemp. Math.), thereby establishing global regularity in the whole space without any smallness assumptions.