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

TL;DR

This paper investigates the long-term behavior of solutions to a gas-liquid model with external forces and vacuum, establishing convergence rates to stationary solutions under certain conditions on initial data and parameters.

Contribution

It extends previous analysis by providing convergence rates and stability results for solutions to a complex gas-liquid model with external forces and vacuum.

Findings

Solutions converge to stationary states asymptotically.

Established explicit stabilization rates in the $L^ abla$ norm.

Identified parameter conditions for convergence and stability.

Abstract

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.