Asymptotic stability of the compressible gas-liquid model with well-formation interaction and gravity

In this paper we are concerned with the initial-boundary value problem of the compressible gas-liquid model with well-formation interaction and gravity. The asymptotic behavior of solutions to steady states is established. Also the time-decay rates of perturbed solutions in the sense of L∞ norm are obtained under some suitable assumptions on the initial date, if γ>1 (associated with pressure law of gas) and β∈(0,γ2]∩(0,γ−αγ)∩(0,γ+αγ3] where β characterizes the viscosity coefficient and α describes the mass decay rate at the boundary. A main purpose of this work is to clarify the role played by the well-reservoir interaction term. The analysis demonstrates that it is essential to take into account information about sign as well as size of the interaction term in order to obtain time-independent estimates when it operates in combination with gravity.