Global existence and time-decay estimates of solutions to the compressible Navier-Stokes-Smoluchowski equations

This paper is concerned with the Cauchy problem of the compressible Navier-Stokes-Smoluchowski equations in $\mathbb{R}^3$. Under the smallness assumption on both the external potential and the initial perturbation of the stationary solution in some Sobolev spaces, the existence theory of global solutions in $H^3$ to the stationary profile is established. Moreover, when the initial perturbation is bounded in $L^p$-norm with $1\leq p< \frac{6}{5}$, we obtain the optimal convergence rates of the solution in $L^q$-norm with $2\leq q\leq 6$ and its first order derivative in $L^2$-norm.