Exponential Decay for Lions-Feireisl's Weak Solutions to the Barotropic Compressible Navier-Stokes Equations in 3D Bounded Domains

TL;DR

This paper proves that weak solutions to the 3D barotropic compressible Navier-Stokes equations in bounded domains decay exponentially to equilibrium, extending previous work by leveraging density integrability and Lyapunov functionals.

Contribution

It establishes exponential decay for Lions-Feireisl weak solutions in 3D bounded domains, a significant advancement in understanding their long-term behavior.

Findings

Weak solutions decay exponentially to equilibrium

Utilizes density integrability and Lyapunov functionals

Extends decay results to 3D bounded domains

Abstract

For barotropic compressible Navier-Stokes equations in three-dimensional (3D) bounded domains, we prove that any finite energy weak solution obtained by Lions [Mathematical topics in fluid mechanics, Vol. 2. Compressible models(1998)] and Feireisl-Novotn\'{y}-Petzeltov\'{a} [J. Math. Fluid Mech. 3(2001), 358-392] decays exponentially to the equilibrium state. This result is established by both using the extra integrability of the density due to Lions and constructing a suitable Lyapunov functional just under the framework of Lions-Feireisl's weak solutions.