The small Deborah number limit for the compressible fluid-particle flows

In this paper, we consider the hydrodynamic limit for the fluid-particle flows governed by the Vlasov-Fokker-Planck equation coupled with the compressible Navier-Stokes equation as the Deborah number tends to zero.

The proof is based on a formal derivation via the Hilbert expansion around the limiting system, the rigorous justification of which is completed by the refined energy estimates involving the macro-micro decomposition.

Compared with the existing results obtained by the relative entropy argument ([A.

Mellet and A.

F.

Vasseur, Comm.

Math.

Phys., 281 (2008), pp.

573-596]), the present work extends to a pointwise convergence of the hydrodynamic limits with an explicit rate for the fluid-particle coupled model.