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

Abstract

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.