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We analyze the Vlasov equation coupled with the compressible Navier--Stokesequations with degenerate viscosities and vacuum. These two equations arecoupled through the drag force which depends on the fluid density and therelative velocity between particle and fluid. We first establish the existenceand uniqueness of local-in-time regular solutions with arbitrarily largeinitial data and a vacuum. We then present sufficient conditions on the initialdata leading to the finite-time blowup of regular solutions. In particular, ourstudy makes the result on the finite-time singularity formation forVlasov/Navier-Stokes equations discussed by Choi [J. Math. Pures Appl., 108,(2017), 991-1021] completely rigorous.

On regular solutions and singularity formation for Vlasov/Navier-Stokes equations with degenerate viscosities and vacuum