The relaxation limit of a homogeneous two-phase flow model: isothermal case

This paper investigates the asymptotic behavior of a hyperbolic relaxation system designed for homogeneous two-phase flows in the limit of vanishing relaxation time. The governing equations comprise conservation laws for mixture mass and momentum, supplemented by a transport equation for the gas phase mass that includes a stiff relaxation source term. This source term drives the system toward local thermodynamic equilibrium. Under the assumptions of constant liquid density and an ideal isothermal gas phase, we demonstrate that, as the relaxation parameter $\epsilon \rightarrow 0$, a subsequence of solutions $(p^{\epsilon},u^{\epsilon})$ converges strongly in $L_{\mathrm{loc}}^{1}$ to an entropy solution of the equilibrium Euler system. The proof integrates several analytical techniques: the construction of a suitable entropy pair and associated energy estimates, a transport equation approach for representing the error, commutator estimates, and the theory of compensated compactness. This work provides a rigorous justification of the relaxation limit for the homogeneous two-phase flows model.