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In this paper, we investigate a generic compressible two-fluid model withcommon pressure ($P^+=P^-$) in $\mathbb{R}^3$. Under some smallnessassumptions, Evje-Wang-Wen [Arch Rational Mech Anal 221:1285--1316, 2016]obtained the global solution and its optimal decay rate for the 3D compressibletwo-fluid model with unequal pressures $P^+\neq P^-$. More precisely, thecapillary pressure $f(\alpha^-\rho^-)=P^+-P^-\neq 0$ is taken into account, andis assumed to be a strictly decreasing function near the equilibrium. Asindicated by Evje-Wang-Wen, this assumption played an key role in theiranalysis and appeared to have an essential stabilization effect on the model.However, global well-posedness of the 3D compressible two-fluid model withcommon pressure has been a challenging open problem due to the fact that thesystem is partially dissipative and its nonlinear structure is very terrible.In the present work, by exploiting the dissipation structure of the model andmaking full use of several key observations, we establish global existence andlarge time behavior of classical solutions to the 3D compressible two-fluidmodel with common pressure. One of key observations here is that to closure thehigher-order energy estimates of non-dissipative variables (i.e, fractiondensities $\alpha_{\pm}\rho_\pm$), we will introduce the linear combination oftwo velocities ($u^\pm$): $v=(2\mu^++\lambda^+)u^+-(2\mu^-+\lambda^-)u^-$ andexplore its good regularity, which is particularly better than ones of twovelocities themselves.

Global well--posedness and large time behavior of classical solutions to a generic compressible two-fluid model