Low Mach number limit of the compressible Euler--Vlasov--Fokker--Planck system in the whole space

@misc{li2026mach,
  title = {{Low Mach number limit of the compressible Euler--Vlasov--Fokker--Planck system in the whole space}},
  author = {Fucai Li and Jinkai Ni and Zhipeng Zhang},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.12648},
  eprint = {2609.12648},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.12648},
}

TY  - UNPB
TI  - Low Mach number limit of the compressible Euler--Vlasov--Fokker--Planck system in the whole space
AU  - Fucai Li
AU  - Jinkai Ni
AU  - Zhipeng Zhang
PY  - 2026
DA  - 2026/09/11
PB  - arXiv
DO  - 10.48550/arxiv.2609.12648
UR  - https://arxiv.org/abs/2609.12648
AB  - Although there are many important contributions on compressible and incompressible fluid-particle interaction models respectively, how to connect the two-type fluid-particle models via the low Mach number limit remains a challenging open problem. In this paper, we resolve it for the compressible isentropic fluid-particle model (Euler--Vlasov--Fokker--Planck (Euler--VFP) system) in the whole space ℝ^3. First, we establish the global-in-time {\it a priori estimates} of strong solutions that are uniform with respect to the Mach number ε near the global Maxwellian. The proof relies on a refined energy method that combines the relaxation structure b^ε−u^ε induced by the fluid-particle interaction and the symmetrized acoustic structure of the compressible Euler part in the model. Under the assumption of well-prepared initial data, we derive a {\it global-in-time} uniform error estimate in the H^2 framework between the solution of the compressible Euler--VFP system and that of the limiting incompressible Euler--VFP system. A key point is to introduce the corrected acoustic variable $ q^\varepsilon-\varepsilon [P'(1)]^{-1}π$, which captures the pressure corrector in the low Mach number limit. This also allows us to exploit the exact cancellation of the singular acoustic terms and to close the {\it global-in-time} error estimate. The damping term b^ε−u^ε, which is absent in the pure Euler equations, plays an essential role in recovering the relative velocity dissipation and in controlling the coupled fluid-particle dynamics. As a consequence, we prove the low Mach number limit of the compressible Euler--VFP system with the convergence rate 𝒪(ε) in the time-continuous H^2 topology.
ER  -