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

TL;DR

This paper rigorously derives a two-phase porous media model from a compressible multi-fluid system through pressure and time relaxation limits, providing explicit convergence rates and advanced analytical techniques.

Contribution

It introduces a comprehensive analysis of relaxation limits in multi-fluid models, including explicit convergence rates and novel spectral and energy estimates.

Findings

Pressure-relaxation limit from Baer-Nunziato to Kapila system justified.

Convergence of Kapila system to porous media model established.

Explicit rates of convergence provided for both relaxation processes.

Abstract

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