| Abstract: |
| This talk investigates the stability and singularity formation (blow-up) in the Keller-Segel-Navier-Stokes system, which models chemotaxis in fluid environments. While existing theory establishes a mass threshold of $2\pi$ for global existence, its optimality remains unresolved. We introduce a stabilized finite element method with shock-capturing to explore numerical blow-up scenarios, ensuring the preservation of key physical properties such as positivity and mass conservation. Our results indicate that the $2\pi$ threshold may not be sharp; we conjecture that the true critical value could be $4\pi$, as in the fluid-free Keller-Segel system. Additionally, numerical experiments reveal that stronger fluid flows can suppress singularity formation, effectively preventing chemotactic collapse. |
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