Global Existence and Large Time Behaviors of Strong Solutions to the Kinetic Cucker--Smale Model Coupled with the Three Dimensional Incompressible Navier--Stokes Equations

TL;DR

This paper proves the global existence and analyzes the large-time decay of strong solutions to a coupled kinetic Cucker--Smale and Navier--Stokes system in unbounded space, extending previous results to the whole space setting.

Contribution

It establishes global strong solutions for the coupled system in unbounded space and determines the optimal decay rates, overcoming challenges posed by unbounded domains.

Findings

Global existence of strong solutions under small initial data.

Optimal decay rates matching Navier--Stokes behavior.

Use of Fourier splitting method for unbounded domain analysis.

Abstract

In this paper, we study global existence and large time behaviors of strong solutions to the kinetic Cucker--Smale model coupled with the three dimensional incompressible Navier--Stokes equations in the whole space. Using the maximal regularity estimate on the Stokes equations, global-in-time strong solutions to the Cauchy problem of the coupled system are obtained under small initial data regime. The optimal decay rate of the system, in the sense that the energy of the system decays at a rate coinciding with the one corresponding to the underling incompressible Navier--Stokes equations without the coupling term, is also analyzed. Compared with previous results set in spatial-periodic or bounded domains, we circumvent the difficulty caused by unboundedness of the domain using the Fourier splitting method.