Long-time behavior to the 3D isentropic compressible Navier-Stokes equations

We are concerned with the long-time behavior of classical solutions to the isentropic compressible Navier-Stokes equations in $\mathbb R^3$. Our main results and innovations can be stated as follows: Under the assumption that the density $\rho({\bf{x}}, t)$ verifies $\rho({\bf{x}},0)\geq c>0$ and $\sup_{t\geq 0}\|\rho(\cdot,t)\|_{L^\infty}\leq M$, we establish the optimal decay rates of the solutions. This greatly improves the previous result (Arch. Ration. Mech. Anal. 234 (2019), 1167--1222), where the authors require an extra hypothesis $\sup_{t\geq 0}\|\rho(\cdot,t)\|_{C^\alpha}\leq M$ with $\alpha$ arbitrarily small. We prove that the vacuum state will persist for any time provided that the initial density contains vacuum and the far-field density is away from vacuum, which extends the torus case obtained in (SIAM J. Math. Anal. 55 (2023), 882--899) to the whole space. We derive the decay properties of the solutions with vacuum as far-field density. This in particular gives the first result concerning the $L^\infty$-decay with a rate $(1+t)^{-1}$ for the pressure to the 3D compressible Navier-Stokes equations in the presence of vacuum. The main ingredient of the proof relies on the techniques involving blow-up criterion, a key time-independent positive upper and lower bounds of the density, and a regularity interpolation trick.