The Cauchy problem for a coupling system of Vlasov-Fokker-Planck/compressible Navier-Stokes equations

This paper is concerned with the Cauchy problem for the compressible fluid-particle interaction system in $\mathbb{R}^3$, which is based on a Vlasov-Fokker-Planck equation to describe the microscopic motion of the particles coupled to the Navier-Stokes equations for a compressible fluid. Global well-posedness of the Cauchy problem is proved in $\mathcal{H}^N(\mathbb{R}^3)$-framework ($N\geq 2$), and optimal decay rates of all order spatial derivatives of the solution toward equilibrium are established where we only assume that the ${\mathcal H}^2$-norm of initial data is small. The proof is accomplished by virtue of refined energy estimates and a new observation for decay properties of low-frequency and high-frequency quantities.

Keywords

Vlasov-Fokker-Planck, Navier-Stokes equations, global existence, optimal decay rates, the low-frequency and high-frequency decomposition

2010 Mathematics Subject Classification

35Q30, 35Q83, 35Q84, 76N10

Full Text (PDF format)

Received 7 December 2022

Received revised 20 April 2023

Accepted 3 October 2023

Published 12 July 2024