The Small Deborah Number Limit for the Fluid-Particle Flows: Incompressible Case

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.