On a decay rate for 1D-viscous compressible barotropic fluid equations

Abstract

The Navier-Stokes equations of a compressible barotropic fluid in 1D with zero velocity boundary conditions are considered. We study the case of large initial data in H 1 as well as the mass force such that the stationary density is positive. The uniform lower bound for the density is proved. By constructing suitable Lyapunov functionals, decay rate estimates in L 2-norm and H 1-norm are given. The decay rate is exponential if so the decay rate of the nonstationary part of the mass force is. The results are proved in the Eulerian coordinates for a wide class of increasing state functions including $ p(\rho)=p_1\rho^\gamma $ with any γ > 0 as well as functions of arbitrarily fast growth. We also extend the results for equations of a multicomponent compressible barotropic mixture (in the absence of chemical reactions).