The small Deborah number limit for the fluid-particle flows: incompressible case
Abstract
This work is devoted to the study of the hydrodynamic limit for the fluid-particle flows governed by the Vlasov-Fokker-Planck (VFP) equation coupled with the incompressible Navier-Stokes (INS) equation as the Deborah number approaches to zero. The limit is valid globally in time provided that the initial perturbation is small in a neighborhood of a steady state. The proof is based on a formal derivation of the limiting system via the Hilbert approach, followed by a rigorous justification via introducing a novel decomposition involving some macroscopic quantities and a refined energy estimate motivated by macro-micro decomposition. In contrast to the existing results for the same scaled model, the present work provides the first one on the hydrodynamic limits in a strong sense with an explicit convergence rate.
Versions
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BibTeX
@misc{fang2026small,
title = {{The small Deborah number limit for the fluid-particle flows: incompressible case}},
author = {Zhendong Fang and Kunlun Qi and Huanyao Wen},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.39204},
eprint = {2610.39204},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.39204},
}
RIS
TY - UNPB
TI - The small Deborah number limit for the fluid-particle flows: incompressible case
AU - Zhendong Fang
AU - Kunlun Qi
AU - Huanyao Wen
PY - 2026
DA - 2026/10/29
PB - arXiv
DO - 10.48550/arxiv.2610.39204
UR - https://arxiv.org/abs/2610.39204
AB - This work is devoted to the study of the hydrodynamic limit for the fluid-particle flows governed by the Vlasov-Fokker-Planck (VFP) equation coupled with the incompressible Navier-Stokes (INS) equation as the Deborah number approaches to zero. The limit is valid globally in time provided that the initial perturbation is small in a neighborhood of a steady state. The proof is based on a formal derivation of the limiting system via the Hilbert approach, followed by a rigorous justification via introducing a novel decomposition involving some macroscopic quantities and a refined energy estimate motivated by macro-micro decomposition. In contrast to the existing results for the same scaled model, the present work provides the first one on the hydrodynamic limits in a strong sense with an explicit convergence rate.
ER -