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

• Published in: Nonlinearity Vol. 37; no. 7; pp. 75002 - 75056
• Main Authors: Crin-Barat, Timothée, Shou, Ling-Yun, Tan, Jin
• Format: Journal Article
• Language: English
• Published: IOP Publishing 01.07.2024
• Subjects: 35B40; 35Q35; 76N10; 76T17; Analysis of PDEs; Baer–Nunziato system; critical regularity; Kapila system; Mathematics; multi-fluid system; overdamping phenomenon; pressure-relaxation limit; two-phase flow in porous media; overdamping phenomenon; Baer-Nunziato system 2020 Mathematics Subject Classification-35Q35 35B40 76N10 76T17; pressure-relaxation limit; two-phase flow in porous media; Baer-Nunziato system 2020 Mathematics Subject Classification-35Q35; 35B40; 76T17; critical regularity; Multi-fluid system; 76N10; Kapila system
• ISSN: 0951-7715; 1361-6544
• DOI: 10.1088/1361-6544/ad3f66

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.