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Quantitative light-particle limit for the Vlasov-Fokker-Planck-Navier-Stokes system

Young-Pil Choi, Jinwook Jung

Abstract

We investigate the hydrodynamic limit of the Vlasov--Fokker--Planck--Navier--Stokes system in the light particle regime, where the particle relaxation takes place on a singularly fast time scale. Using a relative entropy method adapted to this scaling, we develop the first quantitative convergence theory for the light particle limit. Our analysis yields explicit rates for the convergence of both the kinetic distribution and the fluid velocity, extending the qualitative compactness-based result of Goudon, Jabin, and Vasseur [Indiana Univ. Math. J., 53, (2004), 1495--1515]. Moreover, we show that these quantitative estimates propagate in weak topologies and, in particular, lead to optimal convergence rates in the bounded Lipschitz distance. The results apply on both the torus and the whole space, providing a unified quantitative description of the light particle hydrodynamic limit.

Quantitative light-particle limit for the Vlasov-Fokker-Planck-Navier-Stokes system

Abstract

We investigate the hydrodynamic limit of the Vlasov--Fokker--Planck--Navier--Stokes system in the light particle regime, where the particle relaxation takes place on a singularly fast time scale. Using a relative entropy method adapted to this scaling, we develop the first quantitative convergence theory for the light particle limit. Our analysis yields explicit rates for the convergence of both the kinetic distribution and the fluid velocity, extending the qualitative compactness-based result of Goudon, Jabin, and Vasseur [Indiana Univ. Math. J., 53, (2004), 1495--1515]. Moreover, we show that these quantitative estimates propagate in weak topologies and, in particular, lead to optimal convergence rates in the bounded Lipschitz distance. The results apply on both the torus and the whole space, providing a unified quantitative description of the light particle hydrodynamic limit.
Paper Structure (19 sections, 11 theorems, 218 equations)

This paper contains 19 sections, 11 theorems, 218 equations.

Table of Contents

  1. Introduction
  2. Formal asymptotic limit $\varepsilon \to 0$
  3. Main result
  4. Organization of paper
  5. Preliminaries
  6. Weighted Trudinger--Moser inequality
  7. Uniform bound estimates
  8. Weak compactness
  9. Stability estimates and modulated entropy structure
  10. Bounded Lipschitz distance stability
  11. Relative entropy inequality for the light-particle limit
  12. Proof of Theorem \ref{['thm_main']}
  13. Closing the relative entropy inequality
  14. Well-prepared case
  15. Scaled well-prepared case
  16. ...and 4 more sections

Key Result

Theorem 1.1

Let $T>0$, and let $(\rho,v)$ be the unique classical solution to the system lim_light satisfying Let $\{(f^\varepsilon, v^\varepsilon)\}_{\varepsilon > 0}$ be a family of weak entropy solutions to the equation main1 on the time interval $[0,T]$, with initial data $\{(f^\varepsilon_0, v^\varepsilon_0)\}_{\varepsilon > 0}$ satisfying the uniform bounds If the initial data is well-prepared in the

Theorems & Definitions (26)

  • Definition 1.1
  • Theorem 1.1
  • Remark 1.1
  • Remark 1.2
  • Remark 1.3
  • Remark 1.4
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • ...and 16 more