Does Leray’s structure theorem withstand buoyancy-driven chemotaxis-fluid interaction?

Abstract

Going beyond this, the present work focuses on the possible extent of unboundedness phenomena also on short timescales, and hence investigates in more detail the set of times in $(0,\infty)$ at which solutions may develop singularities. The main results in this direction reveal the existence of a global weak energy solution which coincides with a smooth function throughout $\overline{\Omega}\times E$, where $E$ denotes a countable union of open intervals which is such that $|(0,\infty)\setminus E|=0$. In particular, this indicates that a similar feature of the unperturbed Navier--Stokes equations, known as Leray’s structure theorem, persists even in the presence of the coupling to the attractive and hence potentially destabilizing cross-diffusive mechanism in the full system ($\star$).