Exploring numerical blow-up phenomena for the Keller-Segel-Navier-Stokes equations

The Keller-Segel-Navier-Stokes system governs chemotaxis in liquid environments. This system is to be solved for the organism and chemoattractant densities and for the fluid velocity and pressure. It is known that if the total initial cell density mass is below $2\pi$ there exist globally defined generalised solutions, but what is less understood is whether there are blow-up solutions beyond such a threshold and its optimality. Motivated by this issue, a numerical blow-up scenario is investigated. Approximate solutions computed via a stabilised finite element method founded on a shock capturing technique are such that they satisfy \emph{a priori} bounds as well as lower and $L^1(\Omega)$ bounds for the cell and chemoattractant densities. In particular, this latter properties are essential in detecting numerical blow-up configurations, since the non-satisfaction of these two requirements might trigger numerical oscillations leading to non-realistic finite-time collapses into persistent Dirac-type measures. Our findings show that the existence threshold value $2\pi$ encountered for the cell density mass may not be optimal and hence it is conjectured that the critical threshold value $4\pi$ may be inherited from the fluid-free Keller-Segel equations. Additionally it is observed that the formation of singular points can be neglected if the fluid flow is intensified.