Asymptotic behavior of solutions to 1D compressible Navier-Stokes equations withgravity and vacuum

Changjiang Zhu; Ruizhao Zi

doi:10.3934/dcds.2011.30.1263

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2011, Volume 30, Issue 4: 1263-1283. Doi: 10.3934/dcds.2011.30.1263

This issue Previous Article Multiple solutions for superlinear elliptic systems of Hamiltonian type Next Article Investigating the consequences of global bifurcations for two-dimensional invariant manifolds of vector fields

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

Changjiang Zhu^1, and
Ruizhao Zi^1,

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

Received: May 31, 2010

Revised: July 31, 2010

Published: September 2011

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Abstract

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.

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Mathematics Subject Classification: Primary: 76D05, 35B45, 35B40.

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