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 organism density mass is below 2π 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 a priori bounds as well as lower and L 1(Ω) bounds for the organism and chemoattractant densities. In particular, these 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 2π encountered for the organism density mass may not be optimal and hence it is conjectured that the critical threshold value 4π 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.

中文翻译：

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探索 Keller-Segel-Navier-Stokes 方程的数值爆炸现象

Keller-Segel-Navier-Stokes 系统控制液体环境中的趋化性。该系统要求解生物体和化学引诱剂密度以及流体速度和压力。已知如果初始生物体总密度质量低于2π存在全局定义的广义解决方案，但较少了解的是是否存在超出此阈值及其最优性的爆炸解决方案。受此问题的启发，研究了数值爆炸场景。通过基于冲击捕获技术的稳定有限元方法计算的近似解满足先验界限以及下限和L 1(Ω) 生物体和趋化剂密度的界限。特别是，后面这些属性对于检测数值爆炸配置至关重要，因为不满足这两个要求可能会触发数值振荡，导致不现实的有限时间崩溃为持久的狄拉克型测量。我们的研究结果表明存在阈值 2π生物体密度质量遇到的可能不是最佳的，因此推测临界阈值 4π可以继承自无流体的 Keller-Segel 方程。此外，据观察，如果流体流动增强，奇异点的形成可以忽略不计。