An interpolation inequality involving $L\log L$ spaces and application to the characterization of blow‐up behavior in a two‐dimensional Keller–Segel–Navier–Stokes system

<jats:title>Abstract</jats:title><jats:p>In a smoothly bounded two‐dimensional domain and for a given nondecreasing positive unbounded , for each and the inequality <jats:disp-formula />is shown to hold for any positive fulfilling <jats:disp-formula />This is thereafter applied to nonglobal solutions of the Keller–Segel system coupled to the incompressible Navier–Stokes equations through transport and buoyancy, and it is seen that in any such blow‐up event the corresponding population density cannot remain uniformly integrable over near its explosion time.</jats:p>