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.

arXiv commentWe have thoroughly revised and improved the entire manuscript.The modifications primarily involved:1.Rigorization of certain parts of the proof (e.g., Equations (3.45) and (3.46) in Section 3, Page 15), 2.Language refinement and polishing, 3. Standardization of mathematical notations, 4.Comprehensive correction of typographical errors in the previous version of the paper