On the Well-Posedness and Decay Rates of Solutions to the Poisson-Nernst-Planck-Navier-Stokes System

We consider the initial value problem associated to the Poisson-Nernst-Planck-Navier-Stokes system which is first derived by Wang et al. (J Differ Equ 262:68-115, 2017) through an Energetic Variational Approach (EVA). Exploiting harmonic analysis tools (especially Littlewood-Paley theory), we first study the local and global well-posedness of the system in critical Besov spaces. Then, under a suitable condition involving only low-frequency of initial data, we use the Lyapunov-type inequality of the energy functionals to establish optimal time decay rates for the constructed global solutions. The proof crucially depends on a careful analysis for treating the extra effect of the distribution for the negative (positive) charge and non-standard product estimates, interpolation inequalities.