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Derivation of the Monokinetic Vlasov-Stokes equations

Richard M. Höfer, A. Mecherbet, R. Schubert

TL;DR

The paper rigorously derives the monokinetic Vlasov-Stokes equations as the mean-field limit of a microscopic system of inertial spheres in a Stokes flow, in the regime $R\to0$, $6\pi N R\to1$ with inertia fixed and initial data where the velocity field is Lipschitz. It introduces a monokinetic ansatz $f=f(t,dx,dv)=\rho(t,dx)\otimes\delta_{\mathrm{w}(t,x)}(dv)$ and proves a quantitative convergence result for the mesoscopic and macroscopic descriptions, using a nonperturbative buckling argument that controls the 2-Wasserstein distance $\mathcal{W}_2(f^N,f)$ and the minimal interparticle distance. The proof hinges on an approximate Stokes law $F_i=6\pi R(V_i-(u)_i)$, intermediate velocity fields $w_N^d$, and a sequence of estimates (Wasserstein, density separation, and drag-velocity bounds) to derive a Gronwall-type bound on the transport cost $\eta(t)$. The main result provides a concrete convergence bound $\mathcal{W}_2(f^N(t),f(t)) \le\big((1+\|\mathrm{w}_0\|_{W^{1,\infty}})\mathcal{W}_2(\rho^N_0,\rho_0)+C N^{-1/2}\big) e^{Ct}$ and a uniform lower bound on particle distances for times up to $T^*$, thus extending previous perturbative results to a fully nonlinear, monokinetic setting with finite inertia and reinforcing the connection between microscopic dynamics and the monokinetic Vlasov-Stokes system.

Abstract

We consider a microscopic model of spherical particles with inertia in a Stokes flow. As the particle number grows to infinity and their size goes to zero we derive the monokinetic Vlasov-Stokes equations as mean-field limit. We do this under the assumption that the particles have initial velocities given by a Lipschitz velocity profile and prove the mean-field limit for times of the order of the inverse Lipschitz constant. Notably this is not a perturbative result. In particular, we do not require the inertia of the particles to vanish in the limit. Thereby the result improves upon the perturbative derivation in [HS23] in the case of a monokinetic limit density.

Derivation of the Monokinetic Vlasov-Stokes equations

TL;DR

The paper rigorously derives the monokinetic Vlasov-Stokes equations as the mean-field limit of a microscopic system of inertial spheres in a Stokes flow, in the regime , with inertia fixed and initial data where the velocity field is Lipschitz. It introduces a monokinetic ansatz and proves a quantitative convergence result for the mesoscopic and macroscopic descriptions, using a nonperturbative buckling argument that controls the 2-Wasserstein distance and the minimal interparticle distance. The proof hinges on an approximate Stokes law , intermediate velocity fields , and a sequence of estimates (Wasserstein, density separation, and drag-velocity bounds) to derive a Gronwall-type bound on the transport cost . The main result provides a concrete convergence bound and a uniform lower bound on particle distances for times up to , thus extending previous perturbative results to a fully nonlinear, monokinetic setting with finite inertia and reinforcing the connection between microscopic dynamics and the monokinetic Vlasov-Stokes system.

Abstract

We consider a microscopic model of spherical particles with inertia in a Stokes flow. As the particle number grows to infinity and their size goes to zero we derive the monokinetic Vlasov-Stokes equations as mean-field limit. We do this under the assumption that the particles have initial velocities given by a Lipschitz velocity profile and prove the mean-field limit for times of the order of the inverse Lipschitz constant. Notably this is not a perturbative result. In particular, we do not require the inertia of the particles to vanish in the limit. Thereby the result improves upon the perturbative derivation in [HS23] in the case of a monokinetic limit density.

Paper Structure

This paper contains 14 sections, 5 theorems, 117 equations.

Table of Contents

  1. Introduction
  2. Notation
  3. Main result
  4. Strategy
  5. Preliminaries
  6. Reminder on the 2-Wasserstein distance
  7. Definition of the intermediate transport plans and velocity fields
  8. Proof of the main Theorem
  9. Step 1: Setting of the buckling argument.
  10. Step 2: Control of the Wasserstein distance.
  11. Step 3: Control of the minimal distance.
  12. Step 4: Control of the $|\cdot|_\infty$ norm of the drag forces and velocities.
  13. Step 5: Conclusion of the buckling argument.
  14. Proof of Lemmas \ref{['lem1']} and \ref{['lem2']}

Key Result

Theorem 1.1

Assume ass:gamma and hyp_S_2(0), $\mathrm{w}_0 \in W^{1,\infty}(\mathbb{R}^3,\mathbb{R}^3)$, and $\rho_0\in \mathcal{P}(\mathbb{R}^3) \cap L^\infty(\mathbb{R}^3)$. Let $\bar{T}$ be the time of existence of a solution $(\rho,\mathrm{w})\in L^\infty(0,\bar{T},\mathcal{P}(\mathbb{R}^3) \cap L^\infty(\m

Theorems & Definitions (6)

  • Theorem 1.1
  • Remark 1.2
  • Proposition 2.1
  • Corollary 2.2
  • Lemma 2.3
  • Lemma 2.4