Blowup criterion for the compressible fluid-particle interaction model in 3d with vacuum

In this article, we consider the blowup criterion for the local strong solution to the compressible fluid-particle interaction model in dimension three with vacuum. We establish a BKM type criterion for possible breakdown of such solutions at critical time in terms of both the L∞(0,T;L6)L∞(0,T;L6)-norm of the density of particles and the L1(0,T;L∞)L1(0,T;L∞)-norm of the deformation tensor of velocity gradient.