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In this paper we investigate two types of relaxation processes quantitativelyin the context of small data global-in-time solutions for compressibleone-velocity multi-fluid models. First, we justify the pressure-relaxationlimit from a one-velocity Baer-Nunziato system to a Kapila model as thepressure-relaxation parameter tends to zero, in a uniform manner with respectto the time-relaxation parameter associated to the friction forces modeled inthe equation of the velocity. This uniformity allows us to further consider thetime-relaxation limit for the Kapila model. More precisely, we show that thediffusely time-rescaled solution of the Kapila system converges to the solutionof a two-phase porous media type system as the time-relaxation parameter tendsto zero. For both relaxation limits, we exhibit explicit convergence rates. Ourproof of existence results are based on an elaborate low-frequency andhigh-frequency analysis via the Littlewood-Paley decomposition and it includesthree main ingredients: a refined spectral analysis for the linearized problemto determine the threshold of frequencies explicitly in terms of thetime-relaxation parameter, the introduction of an effective flux in thelow-frequency region to overcome the loss of parameters due to the overdampingphenomenon, and the renormalized energy estimates in the high-frequency regionto cancel higher-order nonlinear terms. To show the convergence rates, wediscover several auxiliary unknowns that reveal better structures. Inconclusion, our approach may be applied to a class of non-symmetric partiallydissipative hyperbolic system with rough coefficients which do not have anytime-integrability, in the context of overdamping phenomenon. It extends thelatest results of Danchin and the first author [15, 16].

Quantitative derivation of a two-phase porous media system from the one-velocity Baer-Nunziato and Kapila systems