Global solvability with arbitrary mass for a chemotaxis--fluid model with sensitivity growth between linear and quadratic

We study a Patlak–Keller–Segel–Navier–Stokes system on $\mathbb{T}\times \mathbb{R}$ with density-dependent diffusion and chemotactic sensitivity growing like $n^{1+\alpha }$, where $0<\alpha <1$. The coefficients satisfy the constitutive relation of Painter and Hillen (Can. Appl. Math. Q. 2002; 10(4): 501–543). Near a sufficiently strong Couette flow, we prove global classical solvability and uniform density bounds for smooth initial data with arbitrary finite cell mass. A key ingredient is a joint energy estimate for the full density and the non-zero-mode vorticity, which preserves the positive nonlinear diffusion and avoids the mixed diffusion terms arising from separate estimates of the density modes. Bounded Fourier corrections yield enhanced dissipation within this energy, allowing the non-zero-mode density–fluid interactions to be controlled. For the density zero mode, which is not mixed by Couette transport, we observe that $\alpha <1$ allows one-dimensional interpolation to absorb its chemotactic production into diffusion for every finite mass. Together, these estimates control the nonlinear diffusion and fluid coupling without imposing a smallness condition on the initial cell mass.