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

TL;DR

This paper proves that weak solutions to the 3D barotropic compressible Navier-Stokes equations with large external forces decay exponentially to equilibrium, using a Lyapunov functional and density integrability properties.

Contribution

It introduces a novel approach to establish exponential decay for weak solutions with non-constant equilibrium states under large external forces.

Findings

Weak solutions decay exponentially to equilibrium.

Constructed a Lyapunov functional for stability analysis.

Extended stability results to non-constant density equilibria.

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.