Asymptotic analysis for a homogeneous bubbling regime Vlasov–Fokker–Planck/Navier–Stokes system

The evolution of a cloud of particles in a compressible fluid can be modeled with a Vlasov–Fokker–Planck equation for the distribution function of the particles coupled with Navier–Stokes or Euler equations for the density and velocity of the fluid. Formal calculations have established the convergence of solution to the mesoscopic model to solutions to the macroscopic Navier–Stokes or Euler model coupled with a Smoluchowski equation as the ratio of the settling time for the microscopic velocity fluctuation of the particles to the characteristic macroscopic time scale goes to zero. This paper provides a rigorous asymptotic analysis for a homogeneous mesoscopic fluid–particle interaction model for particles dispersed in a compressible fluid is provided for the bubbling regime. A relative entropy inequality for a mixed hyperbolic/parabolic system of equations is employed.