Preprint no. 137
Mathematical Institute of the Academy of Sciences of the Czech Republic

On a decay rate for 1D-viscous compressible barotropic fluid equations: The case of positive stationary density

Ivan Straskraba, Alexander Zlotnik

Ivan Straskraba, Mathematical Institute, Acad. Sci.of the Czech Republic, Zitna 25, 115 67 Praha 1, Czech Republic, e-mail: strask@math.cas.cz;
Alexander Zlotnik, Mathematical Institute, Acad. Sci.of the Czech Republic, Zitna 25, 115 67 Praha 1, Czech Republic, e-mail: zlotnik@math.cas.cz

Summary: 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 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 $\gamma >0.$ We also extend the results for equations of multicomponent compressible barotropic mixture (in the absence of chemical reactions).
 
MSC (2000) Subject Classification: 35Q30, 35B40, 76N15
 
Keywords: Compressible fluid, Navier-Stokes equations, asymptotic behavior
 

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