Boundedness and asymptotic stabilization in a two-dimensional Keller–Segel–Navier–Stokes system with sub-logistic source

Abstract

This paper mainly deals with a Keller–Segel–Navier–Stokes model with sub-logistic source in a two-dimensional bounded and smooth domain. For a large class of cell kinetics including sub-logistic sources, it is shown that under an explicit condition involving the chemotactic strength, asymptotic “damping” rate and initial mass of cells, the associated no-flux/no-flux/Dirichlet problem possesses a global and bounded classical solution. Moreover, a systematical treatment has been conducted on convergence of bounded solutions toward constant equilibrium in [Formula: see text] for sub- and standard logistic sources. In such chemotaxis-fluid setting, our boundedness improves known blow-up prevention by logistic source to blow-up prevention by sub-logistic source, indicating standard logistic source is not the weakest damping source to prevent blow-up, and our stability improves known algebraic convergence under quadratic degradation to exponential convergence under log-correction of quadratic degradation, implying log-correction of quadratic degradation quickens the decay of bounded solutions. These findings significantly improve and extend previously known ones.