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

Zhendong Fang, Kunlun Qi, Huanyao Wen

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.

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

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.
Paper Structure (17 sections, 12 theorems, 210 equations)

This paper contains 17 sections, 12 theorems, 210 equations.

Table of Contents

  1. Introduction
  2. The model
  3. Previous results and our contributions
  4. Notations
  5. Main results
  6. Formal analysis
  7. Energy estimate
  8. The remainder system
  9. Energy and dissipation functionals
  10. Energy estimate of the remainder system
  11. Estimates of kinetic part of the remainder system
  12. Estimates of fluid part of the remainder system
  13. Proof of total energy estimate
  14. Proof of main results
  15. Proof of Theorem \ref{['Main-Limits']}
  16. ...and 2 more sections

Key Result

Proposition 1.1

Assume that the initial data $(m_0^{in},u_0^{in},h_0^{in})$ satisfy (i)$1+\inf_{x\in\mathbb{R}^3} h_0^{in}(x)>0$, (ii)$(m_0^{in},u_0^{in},h_0^{in})\in H^6_x \times H^6_x \times H^6_x$. Then, there exists a small constant $\delta_0 > 0$ such that if $\|(m_0^{in},u_0^{in},h_0^{in})\|_{H^6_x}^2 \leq \d Furthermore, for all $t>0$, there exists a constant $C_0 > 0$ such that and where macroscopic ene

Theorems & Definitions (22)

  • Proposition 1.1
  • Remark 1.1
  • Theorem 1.1
  • Remark 1.2
  • Corollary 1.1
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • Corollary 3.1
  • Lemma 3.1
  • ...and 12 more