Zero-viscosity limit of the chemotaxis-Navier-Stokes equations with the Navier-slip boundary condition

The interplay between chemotaxis and the diffusion of nutrients or signaling chemicals in bacterial suspensions can produce localized structures with high cell concentrations, including phyllotactic patterns, filaments, and aggregations in fabricated microstructures, as described by the chemotaxis-Navier-Stokes model for bacterial flows proposed by Tuval et al. in 2005. Dombrowski et al. also observed that bacterial flow in a sessile drop, related to the Boycott effect of sedimentation, can carry bioconvective plumes; viewed from below through the bottom of a Petri dish, the horizontal turbulent-like line near the top corresponds to the air-water-plastic contact line. These physical behaviors motivate the mathematical study of boundary-layer phenomena in coupled chemotaxis-fluid systems. In this paper, we investigate the simultaneous vanishing-diffusivity and vanishing-viscosity limit for a two-dimensional chemotaxis-Navier-Stokes system in the half-space under Neumann boundary conditions for the chemotactic variables and a Navier-slip boundary condition for the fluid velocity. The primary contributions of this work are threefold: first, through matched asymptotic expansions, we identify new structural equations for the boundary-layer profiles to resolve the degeneracies in the chemotactic density layers; second, we construct new anisotropic conormal Sobolev spaces incorporating time derivatives to effectively close the coupled energy estimates; and third, we establish a rigorous 𝑂 ⁡ ( 𝜀 3 / 2 ) convergence rate of the exact viscous solution to the approximate solution in the 𝐿 ∞ norm.