Convergence Rates to Stationary Solutions of a Gas-liquid Model with External Forces and Vacuum

Details Convergence Rates to Stationary Solutions of a Gas-liquid Model with External Forces and Vacuum Convergence Rates to Stationary Solutions of a Gas-liquid Model with External Forces and Vacuum

2011-10-28

arxiv.org/abs/1110.6248v1

Mathematics

Analysis of PDEs

30 Pages

Scientific paper

In this paper, we study the asymptotic behavior of solutions to a Gas-liquid model with external forces and general pressure law. Under some suitable assumptions on the initial date and $\gamma>1$, if $\theta\in(0,\frac{\gamma}{2}]\cap(0,\gamma-1]\cap(0,1-\alpha\gamma]$, we prove the weak solution $(cQ(x,t),u(x,t))$ behavior asymptotically to the stationary one by adapting and modifying the technique of weighted estimates. In addition, if $\theta\in(0,\frac{\gamma}{2}]\cap(0,\gamma-1)\cap(0,1-\alpha\gamma]$, following the same idea in \cite{Fang-Zhang4}, we estimate the stabilization rate of the solution as time tends to infinity in the sense of $L^\infty$ norm.

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