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Phase transition and asymptotic behaviour of flocking Cucker-Smale model

Xingyu Li

Abstract

In this paper, we study a continuous ocking Cucker-Smale model with noise, which has isotropic and polarized stationary solutions depending on the intensity of the noise. The first result establishes the threshold value of the noise parameter which drives the phase transition. This threshold value is used to classify all stationary solutions and their linear stability properties. Using an entropy, these stability properties are extended to the non-linear regime. The second result is concerned with the asymptotic behaviour of the solutions of the evolution problem. In several cases, we prove that stable solutions attract the other solutions with an optimal exponential rate of convergence determined by the spectral gap of the linearized problem around the stable solutions. The spectral gap has to be computed in a norm adapted to the non-local term.

Phase transition and asymptotic behaviour of flocking Cucker-Smale model

Abstract

In this paper, we study a continuous ocking Cucker-Smale model with noise, which has isotropic and polarized stationary solutions depending on the intensity of the noise. The first result establishes the threshold value of the noise parameter which drives the phase transition. This threshold value is used to classify all stationary solutions and their linear stability properties. Using an entropy, these stability properties are extended to the non-linear regime. The second result is concerned with the asymptotic behaviour of the solutions of the evolution problem. In several cases, we prove that stable solutions attract the other solutions with an optimal exponential rate of convergence determined by the spectral gap of the linearized problem around the stable solutions. The spectral gap has to be computed in a norm adapted to the non-local term.

Paper Structure

This paper contains 30 sections, 26 theorems, 116 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Main results.
  3. Motivations and ideas of the proofs.
  4. Structure of the paper.
  5. Stationary solutions and phase transition
  6. A technical observation
  7. The one-dimensional case
  8. The case of a dimension $d\ge2$
  9. Classification of the stationary solutions and phase transition
  10. An important estimate
  11. The additional result of $u(D)$
  12. The linearized problem: local properties of the stationary solutions
  13. Stability of the isotropic stationary solution
  14. A coercivity result
  15. Properties of the free energy and consequences
  16. ...and 15 more sections

Key Result

Theorem 1.1

\newlabelThm:Main1 Let $d\ge 1$ and $\alpha>0$. There exists a critical intensity of the noise $D_*>0$ such that

Figures (2)

  • Figure A.1: \newlabelFig1 Plot of $u\mapsto\mathcal{H}(u)$ when $d=2$, $\alpha=2$, and $D=0.2$, $0.25$, … $0.45$. In this particular case, $D_*\approx 0.354$ solves $$8 Γ$\frac{3}{2},\frac{1}{8\,D}$-8 √π$D -\operatorname\Gamma$12,18 D$+2\,\sqrt\pi=0$.
  • Figure A.2: \newlabelfig2 Plot of $h_d(D)$ against $D$ when $d=1$ with $\alpha=0.5$, $1$, … $3.0$.

Theorems & Definitions (44)

  • Theorem 1.1
  • Proposition 1.1
  • Theorem 1.2
  • Proposition 2.1
  • proof
  • Lemma 2.1
  • proof
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • ...and 34 more