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.

翻译：Keller-Segel-Navier-Stokes系统描述了液体环境中的趋化性。该系统需对生物体密度、化学吸引剂密度以及流体速度和压力进行求解。已知若初始细胞总质量低于$2\pi$，则存在全局定义的一般化解，但对于超过该阈值是否存在爆破解及其最优性仍了解不足。基于此问题，本文研究了数值爆破场景。采用基于激波捕捉技术的稳定化有限元方法计算的近似解满足先验界以及细胞与化学吸引剂密度的下界与$L^1(\Omega)$界。特别地，后两个性质对于检测数值爆破结构至关重要，因为不满足这两个要求可能触发数值振荡，导致不现实的有限时间坍塌为持久狄拉克型测度。研究发现，细胞密度质量所遇到的临界阈值$2\pi$可能并非最优，因此推测$4\pi$这一临界阈值可能继承自无流体的Keller-Segel方程。此外还观察到，若增强流体流动，则可忽略奇点形成。