Nonlinearly Exponential Stability for Lions-Feireisl's Weak Solutions to the Barotropic Compressible Navier-Stokes Equations with Large Potential External Forces

Abstract

The large time behavior for Lions-Feireisl's finite energy weak solutions to the barotropic compressible Navier-Stokes equations with large potential external forces in three-dimensional (3D) bounded domains is considered. Although the equilibrium state of density is not a constant anymore due to the non-constant external forces, by constructing a suitable Lyapunov functional and using the extra integrability of the density, after expanding the difference of the density and its steady state in a Taylor series with respect to the difference of some power function of density and that of the steady density, it is proved that any Lions-Feireisl's finite energy weak solution would decay exponentially to the equilibrium state as time tends to infinity.