Pressure-relaxation limit for a one-velocity Baer-Nunziato model to a Kapila model

In this paper, we study a singular limit problem for a compressible one-velocity bifluid system. More precisely, we show that solutions of the Kapila system generated by initial data close to equilibrium are obtained in the pressure-relaxation limit from solutions of the Baer-Nunziato (BN) system. The convergence rate of this process is a consequence of our stability result. Besides the fact that the quasilinear part of the (BN) system cannot be written in conservative form, its natural associated entropy is only positive semi-definite such that it is not clear if the entropic variables can be used in the present case. Using an ad-hoc change of variables we obtain a reformulation of the (BN) system which couples, via low-order terms, an undamped mode and a non-symmetric partially dissipative hyperbolic system satisfying the Shizuta-Kawashima stability condition.