# American Institute of Mathematical Sciences

November  2011, 30(4): 1263-1283. doi: 10.3934/dcds.2011.30.1263

## Asymptotic behavior of solutions to 1D compressible Navier-Stokes equations with gravity and vacuum

 1 Laboratory of Nonlinear Analysis, School of Mathematics and Statistics, Central China Normal University, Wuhan 430079, China, China

Received  June 2010 Revised  August 2010 Published  May 2011

In this paper, we study the asymptotic behavior of solutions to one-dimensional compressible Navier-Stokes equations with gravity and vacuum for isentropic flows with density-dependent viscosity $\mu(\rho)=c\rho^{\theta}$. Under some suitable assumptions on the initial date and $\gamma>1$, if $\theta\in(0,\frac{\gamma}{2}]$, we prove the weak solution $(\rho(x,t),u(x,t))$ behavior asymptotically to the stationary one by adapting and modifying the technique of weighted estimates. This result improves the one in [5] where Duan showed that the weak solution converges to the stationary one in the sense of integral for shallow water model. In addition, if $\theta\in(0,\frac{\gamma}{2}]\cap(0,\gamma-1]$, following the same idea in [9], we estimate the stabilization rate of the solution as time tends to infinity in the sense of $L^\infty$ norm, weighted $L^2$ norm and weighted $H^1$ norm.
Citation: Changjiang Zhu, Ruizhao Zi. Asymptotic behavior of solutions to 1D compressible Navier-Stokes equations with gravity and vacuum. Discrete & Continuous Dynamical Systems - A, 2011, 30 (4) : 1263-1283. doi: 10.3934/dcds.2011.30.1263
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