Well-Posedness and Singularity Formation for Inviscid Keller–Segel–Fluid System of Consumption Type

We consider the Keller–Segel system of consumption type coupled with an incompressible fluid equation. The describes dynamics oxygen and bacteria densities evolving within a fluid. establish local well-posedness in Sobolev spaces for partially inviscid fully cases. In latter, additional assumptions on initial data are required when either or density touches zero. Even though satisfies maximum principle due to consumption, we prove finite time blow-up its $$C^{2}$$ -norm certain data.