Large time behavior of a gas-liquid two-phase flow with unequal phase velocities and degenerate viscosity

Abstract

The main concern of this paper is the long time behavior of weak solutions to the one-dimensional compressible gas-liquid drift-flux model with a slip law in Lagrangian coordinates. Motivated by the applications of the model in the wellbore flow system, we mainly focus on a scenario that the gas-liquid two-phase flow is separated by a gas-dominated region that holds a specific pressure $p^{⁎}>0$. Under appropriate smallness assumptions on the initial energy, we show that the velocity u tends to 0 as time goes to infinity, and that the pressure function P converges to $p^{⁎}$, whereas the mass-related function Q converges to a non-constant state. Besides, the pointwise interface behaviors, along with the exponential decay rates, of the solution are also studied. Our results reveal the prominent role of the pressure function in determining the asymptotic behavior of the two-phase flow that seems quite different from the one of the classical single-phase flow. The proof is based on some delicate energy estimates established by choosing some appropriate weight functions and adopting the Hardy inequality.