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

Crin-Barat T, Shou LY, Tan J (2024)

Publication Type: Journal article

Publication year: 2024

Journal

Nonlinearity Institute of Physics: Hybrid Open Access

Book Volume: 37

Article Number: 075002

Journal Issue: 7

DOI: 10.1088/1361-6544/ad3f66

Abstract

We derive a novel two-phase flow system in porous media as a relaxation limit of compressible multi-fluid systems. Considering a one-velocity Baer-Nunziato system with friction forces, we first justify its pressure-relaxation limit toward a Kapila model in a uniform manner with respect to the time-relaxation parameter associated with the friction forces. Then, we show that the diffusely rescaled solutions of the damped Kapila system converge to the solutions of the new two-phase porous media system as the time-relaxation parameter tends to zero. In addition, we also prove the convergence of the Baer-Nunziato system to the same two-phase porous media system as both relaxation parameters tend to zero. For each relaxation limit, we exhibit sharp rates of convergence in a critical regularity setting. Our proof is based on an elaborate low-frequency and high-frequency analysis via the Littlewood-Paley decomposition and includes three main ingredients: a refined spectral analysis of the linearized problem to determine the frequency threshold 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 renormalized energy estimates in the high-frequency region to cancel higher-order nonlinear terms. To justify the convergence rates, we discover several auxiliary unknowns allowing us to recover crucial O ( ϵ ) bounds.

Authors with CRIS profile

Timothée Crin-Barat Lehrstuhl für Dynamics, Control, Machine Learning and Numerics (Alexander von Humboldt-Professur)

Involved external institutions

Nanjing University of Aeronautics and Astronautics (NUAA) / 南京航空航天大学 CN China (CN) CY Cergy Paris University / CY Cergy Paris Université FR France (FR)

How to cite

APA:

Crin-Barat, T., Shou, L.Y., & Tan, J. (2024). Quantitative derivation of a two-phase porous media system from the one-velocity Baer-Nunziato and Kapila systems. Nonlinearity, 37(7). https://doi.org/10.1088/1361-6544/ad3f66

MLA:

Crin-Barat, Timothée, Ling Yun Shou, and Jin Tan. "Quantitative derivation of a two-phase porous media system from the one-velocity Baer-Nunziato and Kapila systems." Nonlinearity 37.7 (2024).

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