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.