2024/07/24 by Wu, Guochun, Zhong, Xin
#Analysis of PDEs (math.AP) #FOS: Mathematics
paper · doi:10.48550/arxiv.2407.17439
We are concerned with the long-time behavior of classical solutions to the isentropic compressible Navier-Stokes equations in \mathbb R3. Our main results and innovations can be stated as follows: Under the assumption that the density ρ(\bfx, t) verifies ρ(\bfx,0)≥ c>0 and supt≥ 0‖ρ(⋅,t)‖L^∞≤ 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 supt≥ 0‖ρ(⋅,t)‖Cα≤ M with α 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^∞-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.