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The compressible Navier-Stokes-Smoluchowski equations under investigation concern the behavior of the mixture of fluid and particles at a macroscopic scale. We devote to the existence of the global classical solution near the stationary solution based on the energy method under weaker conditions imposed on the external potential compared with Chen et al. (Global existence and time–decay estimates of solutions to the compressible Navier-Stokes-Smoluchowski equations, Discrete Contin. Dyn. Syst. 36 (2016), no. 10, 5287–5307). Under further assumptions that the stationary solution(ρs(x),0,0)T{\left({\rho }_{s}\left(x),0,0)}^{T}is in a small neighborhood of the constant state(ρ¯,0,0)T{\left(\bar{\rho },0,0)}^{T}at infinity, we also obtain the time decay rates of the solution by the combination of the energy method and the linearLp{L}^{p}-Lq{L}^{q}decay estimates.

Global existence and decay estimates of the classical solution to the compressible Navier-Stokes-Smoluchowski equations in ℝ 3