Eventual smoothness in a chemotaxis-Navier–Stokes system with indirect signal production involving Dirichlet signal boundary condition

This paper deals with a chemotaxis-Navier–Stokes model with indirect signal production involving Dirichlet signal boundary condition in a bounded domain with smooth boundary. A recent literature has asserted that for all reasonably regular initial data, the associated no-flux/saturation/no-flux/no-slip problem possesses at least one globally defined weak solution in the logistic-type degradation here is weaker than quadratic case. But the knowledge on regularity properties of solution has not yet exceeded some information on fairly basic integrability features. The present study reveals that each of these weak solutions becomes eventually classical and bounded under some suitably strong sub-quadratic degradation assumption and an explicit smallness condition. Furthermore, in comparison with the related contributions in the case of the direct signal production, our findings inter alia rigorously reveal that the indirect signal production mechanism genuinely contributes to the global solvability and eventual smoothness of the chemotaxis-Navier–Stokes system.