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Small inertia limit for coupled kinetic swarming models

Young-Pil Choi, Simone Fagioli, Valeria Iorio

Abstract

We investigate various versions of multi-dimensional systems involving many species, modeling aggregation phenomena through nonlocal interaction terms. We establish a rigorous connection between kinetic and macroscopic descriptions by considering the small-inertia limit at the kinetic level. The results are proven either under smoothness assumptions on all interaction kernels or under singular assumptions for \emph{self-interaction} potentials. Utilizing different techniques in the two cases, we demonstrate the existence of a solution to the kinetic system, provide uniform estimates with respect to the inertia parameter, and show convergence towards the corresponding macroscopic system as the inertia approaches zero.

Small inertia limit for coupled kinetic swarming models

Abstract

We investigate various versions of multi-dimensional systems involving many species, modeling aggregation phenomena through nonlocal interaction terms. We establish a rigorous connection between kinetic and macroscopic descriptions by considering the small-inertia limit at the kinetic level. The results are proven either under smoothness assumptions on all interaction kernels or under singular assumptions for \emph{self-interaction} potentials. Utilizing different techniques in the two cases, we demonstrate the existence of a solution to the kinetic system, provide uniform estimates with respect to the inertia parameter, and show convergence towards the corresponding macroscopic system as the inertia approaches zero.
Paper Structure (20 sections, 32 theorems, 245 equations)

This paper contains 20 sections, 32 theorems, 245 equations.

Table of Contents

  1. Introduction
  2. Preliminaries and main results
  3. Preliminaries and assumptions
  4. Main results
  5. Smooth potential case
  6. Singular potential case
  7. Smooth interaction potentials
  8. Well-posedness for the kinetic system for $\varepsilon>0$ fixed
  9. A priori estimates on the characteristics system
  10. Existence and uniqueness for smooth potentials
  11. Uniform estimates in $\varepsilon$
  12. Estimate for smooth solutions
  13. Estimate for measure solutions
  14. Small inertia limit
  15. Singular interaction potentials
  16. ...and 5 more sections

Key Result

Proposition 2.1

Let $(X,d)$ be a complete and separable metric space. Let $\mu_n$ be a sequence of probability measures in $\mathcal{P}_1(X)$, and let $\mu \in \mathcal{P}_1(X)$. Then, the following are equivalent:

Theorems & Definitions (56)

  • Definition 2.1
  • Remark 2.1: Lipschitz constant
  • Proposition 2.1
  • Definition 2.2
  • Theorem 2.1
  • Theorem 2.2
  • Remark 2.2
  • Definition 3.1: Measure solution to \ref{['eq:kinetic']}
  • Lemma 3.1
  • Lemma 3.2
  • ...and 46 more