Uniform Exponential Stability for Finite Energy Weak Solutions to the Barotropic Compressible Navier--Stokes Equations with Large Potential External Forces

We study the large-time behavior of Lions--Feireisl finite energy weak solutions to the barotropic compressible Navier--Stokes equations in three-dimensional bounded domains with Dirichlet boundary conditions and large potential external forces. For $γ>3/2$, we prove that these weak solutions converge exponentially to the steady state as time tends to infinity, provided that the steady density is bounded away from zero. The main novelty of our analysis is a compactness argument based on Young measures. Furthermore, we prove that the Young measures generated by the density are Dirac measures. Together with the extra integrability of the density, this yields strong convergence to the nonconstant steady density and uniform smallness of the energy $w$ for sufficiently large time. We use Taylor expansion and the Bogovskii operator to construct a Lyapunov functional, and use this smallness of $w$ to obtain exponential decay with a time $T_0$, which is independent of the particular weak solution, thereby answering the question raised by Kang, Peng and Xiong in \cite[Remark~1.1]{KPX}. Corresponding decay estimates for the density, momentum, kinetic energy and the viscous dissipation integrated from large times to infinity also hold, allowing vacuum.