Global boundedness in a 2D chemotaxis-Navier–Stokes system with flux limitation and nonlinear production

We consider the chemotaxis-Navier–Stokes system with gradient-dependent flux limitation and nonlinear production: Formula: see text, Formula: see text, Formula: see text and Formula: see text in a bounded domain Formula: see text, where the flux limitation function Formula: see text and the signal production function Formula: see text generalize the prototypes Formula: see text and Formula: see text with Formula: see text, Formula: see text and Formula: see text. For the linear production case of Formula: see text, the global boundedness of solutions has been verified in the related literature for Formula: see text. In this paper, we expand to prove that the corresponding initial-boundary value problem possesses a unique globally bounded solution if Formula: see text for Formula: see text, or if Formula: see text for Formula: see text, which shows that when Formula: see text, that is, the self-enhancement ability of chemoattractant is weak, the solutions still remain globally bounded even though the flux limitation is relaxed to permit proper Formula: see text; however, if Formula: see text, it is necessary to impose the stronger flux limitation than that in the case Formula: see text to inhibit the possible finite-time blow-up. This seems to be the first result on the global solvability in the chemotaxis-Navier–Stokes model with nonlinear production.