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

TL;DR

This paper rigorously analyzes the hydrodynamic limit of compressible fluid-particle flows governed by coupled Vlasov-Fokker-Planck and Navier-Stokes equations as the Deborah number approaches zero, providing explicit convergence rates.

Contribution

It introduces a new method for proving the hydrodynamic limit with explicit convergence rates, improving upon previous relative entropy approaches.

Findings

Established global-in-time validity of the limit under small initial perturbations.

Derived explicit convergence rates for the hydrodynamic limit.

Provided stronger pointwise convergence results compared to prior work.

Abstract

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 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 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 provides a stronger pointwise convergence of the hydrodynamic limits with an explicit rate for the fluid-particle coupled model.