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# Communications in Mathematical Sciences

## Volume 16 (2018)

### Number 7

### A regularity criterion of strong solutions to the 2D Cauchy problem of the kinetic-fluid model for flocking

Pages: 1827 – 1847

DOI: http://dx.doi.org/10.4310/CMS.2018.v16.n7.a4

#### Authors

#### Abstract

In this paper, we consider the blow-up criterion for the two dimensional kinetic-fluid model in the whole space. For particle and fluid dynamics, we employ the Cucker–Smale–Fokker–Planck model for the flocking particle part, and the isentropic compressible Navier–Stokes equations for the fluid part, and the separate systems are coupled through the drag force. We show that the strong solution exists globally if the $L^{\infty} (0, T; L^{\infty})$ norm of the fluid density $\rho (t,x)$ is bounded.

#### Keywords

compressible Navier–Stokes equations, Cucker–Smale–Fokker–Planck equation, vacuum, blow-up criterion

#### 2010 Mathematics Subject Classification

35A20, 35B45, 76N10, 76T99

This work is supported by “the Fundamental Research Funds for the Central Universities” and by the grant from NNSFC under the contract 11671309.

Received 17 February 2018

Accepted 11 June 2018