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Asymptotic flocking dynamics of Relativistic-Cucker-Smale particles immersed in incompressible Navier-Stokes equations

Shenglun Yan, Weiyuan Zou

TL;DR

This work develops and analyzes a coupled relativistic kinetic-fluid model, the Relativistic Cucker-Smale-Navier-Stokes (RCS-NS) system, describing particle-fluid interactions with drag on the torus. By constructing a Lyapunov functional $\\mathcal{L}(t)$ that blends kinetic, fluid, and relative-energy contributions, the authors prove exponential alignment of particle and fluid velocities under precise hypotheses, and they derive weak flocking in probability. They establish global existence of weak solutions via a regularization and iterative scheme, providing a rigorous foundation for the well-posedness of the RCS-NS coupling even for large initial data. The results advance the mathematical understanding of relativistic flocking in viscous incompressible media and lay groundwork for numerical schemes and further extensions to other fluid regimes.

Abstract

In this paper, we propose a coupled system describing the interaction between the Relativistic Cucker-Smale model and the incompressible Navier-Stokes equations via a drag force, and establish a global existence theory as well as the time-asymptotic behavior of the proposed model in $\mathbb{T}^3$. It is shown that the coupled system exhibits an exponential alignment under some specific assumptions, and that weak solutions exist globally for general initial data.

Asymptotic flocking dynamics of Relativistic-Cucker-Smale particles immersed in incompressible Navier-Stokes equations

TL;DR

This work develops and analyzes a coupled relativistic kinetic-fluid model, the Relativistic Cucker-Smale-Navier-Stokes (RCS-NS) system, describing particle-fluid interactions with drag on the torus. By constructing a Lyapunov functional that blends kinetic, fluid, and relative-energy contributions, the authors prove exponential alignment of particle and fluid velocities under precise hypotheses, and they derive weak flocking in probability. They establish global existence of weak solutions via a regularization and iterative scheme, providing a rigorous foundation for the well-posedness of the RCS-NS coupling even for large initial data. The results advance the mathematical understanding of relativistic flocking in viscous incompressible media and lay groundwork for numerical schemes and further extensions to other fluid regimes.

Abstract

In this paper, we propose a coupled system describing the interaction between the Relativistic Cucker-Smale model and the incompressible Navier-Stokes equations via a drag force, and establish a global existence theory as well as the time-asymptotic behavior of the proposed model in . It is shown that the coupled system exhibits an exponential alignment under some specific assumptions, and that weak solutions exist globally for general initial data.

Paper Structure

This paper contains 13 sections, 21 theorems, 179 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Construction of RCS-NS model
  4. Known Results
  5. Some estimates of Lorentz transformation
  6. Conservation laws and energy dissipation
  7. Asymptomatic flocking estimate
  8. A global existence of weak solutions
  9. A brief introduction to a regularized system
  10. The solvability of the regularized system
  11. Convergence of iteration
  12. Proof of Theorem 1.2
  13. Conclusion

Key Result

Theorem 1.1

Let $[f (t, x, w),u(t,x)]$ be a smooth solution to eq1 satisfying (1) ${\int_{\mathbb{T}^3 \times \mathbb{R}^3}f^{\text{in}}(x,w)\mathrm{d}x\mathrm{d}w=1,\quad \bigcup\limits_{t\geq 0} \Omega_w (f,t)\subset B_{W_0}(0)\quad \text{for some} \quad W_0>0}$, (2) ${||\rho_f||_{L^\infty\left(\mathbb{R}_+ where $C:=C(\rho_f,\mu)$ is a positive constant independent of $t$.

Theorems & Definitions (45)

  • Theorem 1.1: Asymptotic stability of RCS-NS system
  • Remark 1.1
  • Remark 1.2
  • Remark 1.3
  • Theorem 1.2
  • Remark 1.4
  • Proposition 2.1
  • Proposition 2.2
  • Proposition 2.3
  • Proposition 2.4
  • ...and 35 more