Decay Estimates for Isentropic Compressible Navier-Stokes Equations in Bounded Domain

TL;DR

This paper establishes exponential decay rates for solutions to the isentropic compressible Navier-Stokes equations in bounded domains, allowing for vacuum states and relaxing regularity assumptions compared to classical results.

Contribution

It introduces a Lyapunov functional approach under the assumption of bounded density, extending previous decay results to include vacuum and lower regularity solutions.

Findings

Weak solutions decay exponentially in L^2 norm.

Vacuum states are permitted in the analysis.

The upper bound of density is crucial for the decay estimate.

Abstract

In this paper, under the hypothesis that $\rho$ is upper bounded, we construct a Lyapunov functional for the multidimensional isentropic compressible Navier-Stokes equations and show that the weak solutions decay exponentially to the equilibrium state in $L^2$ norm. This can be regarded as a generalization of Matsumura and Nishida's results in 1982, since our analysis is done in the framework of Lions 1998 and Feireisl et al. 2001, the higher regularity of $(\rho, u)$ and the uniformly positive lower bound of $\rho$ are not necessary in our analysis and vacuum may be admitted. Indeed, the upper bound of the density $\rho$ plays the essential role in our proof.