Global existence for the Vlasov–Euler–Fokker–Planck system in low-regularity space

Abstract

This paper investigates the global well-posedness of the Cauchy problem for the Vlasov–Fokker–Planck equation coupled with the incompressible Euler system around a normalized global Maxwellian in a periodic spatial domain. The system describes the interaction between a fluid governed by Euler equations and a particle distribution evolving under the VFP dynamics, with coupling through a drag force. We establish the existence and uniqueness of global mild solutions for small initial data in a low regularity function space $L_k^1{L}_T^{\infty }{L}_v^2$ by employing Fourier analysis.

Compare to the Navier–Stokes–Vlasov-Fokker–Planck system (Tan and Fan, 2023) where velocity dissipation estimates can be directly derived from the viscous term, the Vlasov–Euler–Fokker–Planck system lacks such direct accessibility to velocity dissipation due to its inherent structural differences. To overcome this obstacle, we need to exploit the macroscopic dissipation $b$ inherent in the macroscopic equation. Then the dissipation of velocity is indirectly captured by combining the macroscopic dissipation of $b$ and the linear dissipation of $u-b$ within the equation. Finally the uniform energy functionals of the solution can be obtained by utilizing the refined energy estimate.