Global existence and vanishing viscosity limit of the compressible Navier-Stokes-Vlasov-Fokker-Planck system in critical spaces

RESUMO

We study multidimensional compressible fluid-particle systems at critical regularity, in which a carrier fluid and a particle phase with Fokker-Planck diffusion are coupled through a drag force. We prove the existence and uniqueness of strong solutions for the Cauchy problems of the Navier-Stokes-Vlasov-Fokker-Planck and Euler-Vlasov-Fokker-Planck systems near equilibrium in their respective critical Besov spaces. Moreover, we establish regularity estimates for the Navier-Stokes-Vlasov-Fokker-Planck system uniform with respect to the common viscosity parameter $\mu=\lambda=\varepsilon$ and justify the global-in-time vanishing-viscosity limit with the convergence rate $\mathcal O(\varepsilon)$. Finally, under an additional lower-order Besov assumption on the initial data, we obtain optimal time-decay estimates for both systems and derive enhanced decay rates for the relative velocity and the microscopic part of the distribution function.